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Mathematical Improving Stents Getting It Together
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The Mathematical Moments program promotes The Mathematical Moments program promotes appreciation and understanding of the role mathematics appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture. plays in science, nature, technology, and human culture. lated into other languages, and some feature MM/72.s www.ams.org /mathmoments MM/68.s www.ams.org/mathmoments podcast interviews with experts in the fi eld. Listen Up! — Podcast Interviews with Experts
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5VUSPULHY7HY[PHS 5L^1V\YUHSMVY 5V^(]HPSHISLPU:VM[JV]LY +PMMLYLU[PHS,X\H[PVUZ 1V\YUHSVM *HYKPUHS0U]HYPHU[ZVU (Z`TW[V[PJ)LOH]PVYVM:VS\[PVUZ 7ZL\KV+PMMLYLU[PHS )VVSLHU(SNLIYHZ HUK:LSM:PTPSHY:VS\[PVUZ 6WLYH[VYZHUK +VUHSK14VUR &$// ÷)$; ÷(0$,/ RUGHUV#ELUNKDXVHUFRP÷ZZZELUNKDXVHUFRP 7SLHZLTLU[PVUWYVTV[PVU ^OLUVYKLYPUN7YPJLZHYL]HSPKPU[OL(TLYPJHZVUS`HUKHYLZ\IQLJ[[VJOHUNL^P[OV\[UV[PJL -VYWYPJLHUKVYKLYPUNPUMVYTH[PVUV\[ZPKL[OL(TLYPJHZWSLHZLJVU[HJ[)PYROp\ZLY=LYSHN(.I`,THPS!VYKLYZ'IPYROH\ZLYJO [ Notices of the American Mathematical Society February 2010 Communications 244 236 WHAT IS…a Halting Probability? Cristian S. Calude and G. J. Chaitin 248 Simons Foundation Launches US$40-Million Program for Theoretical Research Allyn Jackson 208 208 250 2009 Annual Survey of the Mathematical Sciences (First Report) Ideas of differential geometry have permeated all parts Polly Phipps, James W. of mathematics. Partial differential equations, topology, Maxwell, and Colleen Rose analysis, and many other branches of our subject have taken on a geometric patina. The recent solution of the Poincaré Commentary conjecture used geometry in surprising ways. This issue of the Notices showcases some recent developments in the 205 Opinion: Collaborating geometry of Riemann. on Research with —Steven G. Krantz Mathematicians from Less Editor Developed Countries Melvin Henriksen Features 207 Letters to the Editor 239 What Then? Plato’s 208 An Invitation to Cauchy-Riemann and Ghost: The Modernist Sub-Riemannian Geometries Transformation of John P. D’Angelo and Jeremy T. Tyson Mathematics—A Book Review 220 Mathematics of the Gateway Arch Reviewed by Yuri Manin Robert Osserman 244 Solving Mathematical Problems: A Personal 230 Model Theory and Complex Geometry Perspective—A Book Review Rahim Moosa Reviewed by Loren Larson Notices Departments of the American Mathematical Society About the Cover ...... 266 EDITOR: Steven G. Krantz Mathematics People ...... 260 ASSOCIATE EDITORS: MacPherson Awarded Hopf Prize, Lindenstrauss and Villani Awarded Krishnaswami Alladi, David Bailey, Jonathan Borwein, Fermat Prize, Pujals Awarded TWAS Prize in Mathematics, Dereich Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert J. Daverman, Susan Friedlander, Robion Kirby, Receives 2009 Information-Based Complexity Young Researcher Award, Rafe Mazzeo, Harold Parks, Lisette de Pillis, Peter NSF CAREER Awards Made, Memories of Eddie Nussbaum. Sarnak, Mark Saul, Edward Spitznagel, John Swallow ...... SENIOR WRITER and DEPUTY EDITOR: Mathematics Opportunities 264 Allyn Jackson Summer Program for Women Undergraduates, NSF Support for MANAGING EDITOR: Sandra Frost Undergraduate Training in Biological and Mathematical Sciences, CONTRIBUTING WRITER: Elaine Kehoe Monroe H. Martin Prize, Clay Mathematics Institute 2010 Summer CONTRIBUTING EDITOR: Randi D. Ruden School, News from CRM, Everett Pitcher Lectures. EDITORIAL ASSISTANT: David M. Collins Inside the AMS ...... 268 PRODUCTION ASSISTANT: Muriel Toupin PRODUCTION: Kyle Antonevich, Teresa Levy, From the AMS Public Awareness Office, AMS Email Support for Stephen Moye, Erin Murphy, Lori Nero, Karen Frequently Asked Questions, Deaths of AMS Members. 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NOTICES ON THE AMS WEBSITE: Supported by the 2010 AMS Election—Nominations by Petition ...... 301 AMS membership, most of this publication is freely available electronically through the AMS website, the Society’s resource for delivering electronic prod- ucts and services. Use the URL http://www.ams. org/notices/ to access the Notices on the website. [Notices of the American Mathematical Society (ISSN 0002- 9920) is published monthly except bimonthly in June/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248 USA.] Publication here of the Society’s street address and the other information in brackets I thank Randi D. Ruden for her splendid editorial work, and above is a technical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: [email protected]. for helping to assemble this issue. She is essential to everything © Copyright 2010 by the American Mathematical Society. that I do. All rights reserved. Printed in the United States of America. The paper used —Steven G. Krantz in this journal is acid-free and falls within the guidelines Editor established to ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Opinion like encourage their citizens to study and earn advanced Collaborating on degrees abroad. Should you visit your co-author, there are many books named “Culture Shock (name of country)” written to describe local customs with a view to warn you Research with of things that might insult your hosts. For example, in many countries your host will feel obligated to urge you Mathematicians to accept as a gift something that you admire—and resent it when you accept. from Less Developed One communication difficulty pertaining to mathemat- ics is that few of these foreign co-authors have access to Countries good library facilities or even an effective interlibrary loan system. References mailed to potential collaborators often For a decade, I have collaborated on research with math- result in a polite reply saying that some book or journal is ematicians from Jordan, Iran, Pakistan, India, and South unavailable. This is a serious problem everywhere because Africa. Hardly any of my collaborators have high math- the number of distinct mathematics journals and books ematical prestige or have positions at institutions that has grown substantially and gotten both more specialized have it. By and large, these mathematicians make up for and more expensive. deficiencies that include high teaching loads and very The best substitute for a poor library is a subscription poor libraries with enthusiasm, patience, goodwill, and to MathSciNet, which is an electronic version of Math hard work. Most, though not all, such efforts have led to Reviews. Its reviews and papers can be read and printed publication in respectable journals published in western mostly, but not always, sometimes for an additional countries. charge. Were it up to me, I would provide open access to Common sense told me to expect that collaborating MathSciNet to all Third World AMS members even if some with mathematicians from other cultures using different dues had to be increased. The main problem for these technologies would bring me problems, but these prob- mathematicians is cost. lems were far more difficult than I expected. I am most fearful at this prospect: The quantity of re- Since the end of the Second World War, most published search in the less developed countries is growing rapidly, mathematics is written in English, which is a second lan- while many journals’ articles are becoming available only guage for most people. Even if your foreign co-author’s in electronic form. In some countries ideology, econom- knowledge of English is very good, it is often not idiomatic ics, or technological limitations restrict a mathematician’s and needs to be corrected in a way that is effective without access to Web portals. Commercial publisher-maintained being pedantic or patronizing. Treat your co-author with sites are only available to subscribers at costs that some respect and make it easy for your co-author to be critical institutions cannot afford. Ignoring this trend may result of what you have written. A co-author’s English may be less in serious negative consequences for Third World authors, than perfect, a co-author from a poor country may know indeed for all of us. less mathematics than you do, but still may be extremely We want a worldwide community of research math- gifted. Once I overcame their reluctance to criticize my ematicians. Quality of published research in mathematics writing or mathematics because of their fear of being that is independent of geographic location or financial discourteous, my Third World co-authors could be very wealth should be our common goal. Perhaps there are good at detecting errors. less expensive methods for making mathematical research Avoid political discussions or comparison of govern- and thoughts readily available. What is most required is a ments. Ignoring this, especially in writing, can get you into desire to reach out to all mathematicians in all countries trouble. Stick to mathematics. as a way to make our combined research efforts easier to Keep in mind two differences in how mathematicians implement. are “rewarded for publishing papers” in some countries. Collaborating with international mathematicians is The author whose name appears first gets more of a re- both important and rewarding. It is difficult to exaggerate ward (sometimes financial) than the remaining authors. the enjoyment I feel after a successful collaboration with There is also an Impact Factor assigned to journals that is someone from a different culture. I hope that a committee ill understood. Publishing in a journal with a high impact of the AMS will study this important problem. factor gets a bigger reward. Communication, electronic or otherwise, is difficult —Melvin Henriksen for a number of reasons. Email works most of the time, but power failures are much more frequent than in North America or Europe. Regular mail tends to be slow, and fax Editor’s Note: Professor Henriksen died after a long machines are usually turned off at night. Acknowledge all illness on October 14, 2009. emails and encourage your co-authors to do the same. If possible, get a list of PDF files of your own publications to potential co-authors. Even governments you may not FEBRUARY 2010 NOTICES OF THE AMS 205 GOOD MATH. IT’S WEBASSIGN, ONLY BETTER. Only WebAssign is supported by every publisher and supports every course in the curriculum from developmental math to calculus and beyond. That’s good. Now we’ve added the latest homework tools to bring you the ultimate learning environment for math. That’s better. Welcome to a better WebAssign. There are new tools —including calcPad, that lets students enter higher level mathematical functions, then automatically assesses their calcPad lets students easily work. There are more books, over 170 titles from every major enter higher-level mathematical publisher. And there’s more server capacity with an expanded functions, then automatically assesses them. support team just waiting to serve you and over half a million loyal WebAssign users each term. It’s the culmination of a team-wide effort to build the best WebAssign ever. The ingenuity of WebAssign for math. It’s available now and it’s free to faculty at WebAssign.net. Now what could be better than that? Visit www.webassign.net to sign up for your free faculty account today. Letters to the Editor Andrew Gleason questions, scored +1 for a correct Gleason mentioned to me that he Many readers of the Notices are, I answer, -3 for a wrong answer, zero was studying Hilbert’s axioms of am sure, grateful for the fine ar- for no answer. One question was so geometry, in particular the axioms ticles about the late Andrew Glea- tricky that no one among the crew on order. son and his accomplishments, in of past, present, and future deep- Our teacher and friend is gone. But the November 2009 issue. I myself thinking and Putnam Prize-winning the example he set for us lives on for was particularly interested in Paul students dared answer! The professor us to follow. Chernoff’s article, for I once spent still knew best! much effort to completely master Later I wanted to enter a team in —Eric Altschuler, M.D., Ph.D. Gleason’s profound theorem charac- the COMAP Mathematical Contest New Jersey Medical School– terizing abstractly defined “states” in Modeling. Knowing that Gleason University of Medicine & Dentistry of in quantum mechanics, and I even was a three-year Putnam champion New Jersey hoped to be able to shorten the proof, and literally wrote the book on this (Received October 2009) something in which, however, I failed. pure math contest, I approached him Because Gleason is probably most about being our faculty advisor with widely known for his path-breaking some trepidation, but he immediately contribution to solving Hilbert’s Fifth agreed. Problem, I thought that a personal ex- We solved the discrete problem perience of mine may be of interest. for the specific given set of condi- Gleason once visited Indiana Uni- tions, and also showed that with versity in Bloomington, and I met him general conditions the problem was at a post-lecture party. I mentioned NP-complete. On Monday morning to him my admiration for “Gleason’s after Gleason mailed our sealed write- Theorem”. “Which one?” he asked. up, we proudly told him our results. Submitting Letters to the Upon my replying that I meant the He listened with a knowing look. He Editor too had found the same, but didn’t one about states in quantum theory, The Notices invites readers to look nearly as tired as we were! We his face darkened, he thought for a submit letters and opinion pieces few moments, and then said “You were awarded an Outstanding Paper. on topics related to mathemat- know, that was the most difficult Gleason’s legacy is that Harvard now ics. Electronic submissions are thing I ever did in my life.” regularly competes in this applied preferred (notices-letters@ math contest. ams.org); see the masthead for —Andrew Lenard I have done some work on J. J. postal mail addresses. Opinion Indiana University Thomson’s century-old problem of pieces are usually one printed [email protected] the minimum energy configuration of page in length (about 800 words). charges on the surface of a sphere. I Letters are normally less than (Received October 2009) discussed with Gleason one paper in one page long, and shorter letters which we were studying the differ- are preferred. As a former physics major, now phy- ence in energies of configurations for sician and neuroscientist, and one of certain numbers of charges that we Professor Gleason’s “last students” had found admitted more than one Identifications in a couple of different ways, I would nice lattice configuration. He asked Affiliations of authors of “Let- just like to add to the recent recount- why we hadn’t given the general ters to the Editor” are provided ings in the Notices. formula for such N—his legendary for identification purposes only. ability to generalize your results I was the only nonmathematician Opinions expressed in letters on the spot! My co-author Richard taking Gleason’s graduate real analy- are those of the authors and do Stong was then able to produce the sis class. Gleason passed out a set of not necessarily reflect those of notes from a prior time he had taught formula. I have never seen reviewers their employers or, in the case of the class and requested that we bring so impressed! I continued to have American Mathematical Society any errors to his attention—there the pleasure and good fortune of officers or committee members, were less than a handful. I mentioned discussing work on other problems policies of the Society. Commit- that he must be pleased that when in physics and neuroscience with tee reports to the Council of the he taught the class again, the notes Andrew Gleason. Society and official communica- would be error free! He responded From his lectures and comments, tions of officers of the Society, that he would never teach the class it seems to me that Andrew Gleason when published in the Notices, again, but he continued to collect was never fully comfortable with appear in the section of the No- errors with interest and concern. The the notion of completeness of the tices “From the AMS Secretary”. final for the class was half true/false real numbers. When last we talked FEBRUARY 2010 NOTICES OF THE AMS 207 An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries John P. D’Angelo and Jeremy T. Tyson certain amount of complex analysis, in both one and several variables, com- bines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe Athat foundation and then invite the reader to follow specific geometric paths based upon it. We discuss such topics as the failure of the Riemann mapping theorem in several variables, the geometry of real hypersurfaces in complex Euclidean space, the Heisenberg group, sub-Riemannian manifolds and their metric space structure, the Hopf fibration, sub-Riemannian geodesics on the three-sphere, CR mappings invariant under a finite group, and finite type conditions. All of these topics fall within the branch of mathematics described by the title, but of course they form only a small part of it. We hope that the chosen topics are representative and appealing. The connections among them provide a fertile ground for future research. We modestly hope that this article inspires others to develop these connections further. To this end, our refer- Figure 1. Georg Friedrich Bernhard Riemann. ence list includes a diverse collection of books and accessible articles. Given Riemann’s contributions to geometry, it There Is No Riemann Mapping Theorem in is not surprising that his name occurs twice in the Higher Dimensions title. It is more surprising that the uses of his name In an attempt to understand complex analysis here come from different parts of mathematics. We in several variables, especially questions revolv- find it delightful that contemporary mathematics ing around the failure of the Riemann mapping is forging connections that even Riemann could theorem, many mathematicians have focused at- not have anticipated. tention on geometric properties of the boundaries John P. D’Angelo is professor of mathematics at the Uni- of domains and how those geometric properties versity of Illinois. His email address is [email protected]. influence the complex analysis on the domain. edu. Such investigations have led to the fields of CR Jeremy T. Tyson is associate professor of mathemat- geometry and sub-Riemannian geometry. ics at the University of Illinois. His email address is We begin with the Riemann mapping theo- [email protected]. rem. Let Ω be an open, connected, and simply 208 Notices of the AMS Volume 57, Number 2 connected proper subset of C. Then there is a is no biholomorphic map from any polydisc to bijective holomorphic mapping f from Ω to the unit any ball. One classical proof of this statement disc D. The inverse mapping is also holomorphic. involves showing that the groups of biholomorphic We say that Ω and D are biholomorphically equiva- automorphisms of the ball and of the polydisc lent. The entire plane must of course be excluded, have different dimensions. because a bounded holomorphic function on C re- A second proof (see [D1, page 14]) proceeds in the duces to a constant. This theorem has applications following manner. First consider a biholomorphic throughout both pure and applied mathematics. In (or even proper) image Ω of the Cartesian product of many of these applications, such as uniform fluid bounded domains; one shows that bΩ must contain flow, the upper half plane substitutes for the unit complex analytic sets. Then an easy computation, disc. The explicit linear fractional transformation given in Lemma 6, shows that the definiteness of f (the Cayley transformation) defined by the Levi form precludes the existence of positive- i − w dimensional complex analytic subsets of b . This (1) z = f (w) = Ω i + w computation helps unify several of the ideas in this maps the upper half plane biholomorphically to article. For example, higher order commutators of the unit disc. complex vector fields can determine obstructions We mention in passing one of the most funda- to the existence of complex analytic sets in a real mental geometric generalizations of the Riemann hypersurface. Such obstructions lead to geometric mapping theorem, although we will not head in finite type conditions for subelliptic estimates. In that direction. The uniformization theorem states two complex dimensions a subelliptic estimate (see that the universal cover of a compact Riemann page 217) holds if and only if the bracket-generating surface must be one of three objects: the unit disc property (see page 212) holds. D, the complex plane C, or the Riemann sphere C. Consider a domain Ω with smooth boundary As a reference for analysis in one complex variable, bΩ. In order that geometric results about bΩ be we need only mention the classical text by Ahlfors meaningful for complex analysis on Ω, one requires [Ahl]. results concerning the boundary smoothness of Let us now consider complex Euclidean space Cn biholomorphic maps. Two early references in this of arbitrary dimension. We equip Cn with the usual area are [Bel] and [Fef]; we do not attempt to list any P Hermitian inner product given by hz, wi = zj w j of the many newer references. In general, smoothly and the corresponding norm given by |z| = hz, zi1/2. bounded domains, topologically equivalent to The unitary group U(n) consists of linear mappings the unit ball Bn, are seldom biholomorphically preserving the inner product. The unit ball Bn is equivalent to it. On the other hand, Fridman [Fri] the set of z for which |z| < 1. has established a wonderful approximate Riemann In dimension at least two the unit ball is not mapping theorem: given any topological ball and a biholomorphically equivalent to a half space. There positive , there is a biholomorphic mapping to a are many possible proofs. One approach takes into domain whose boundary is always within distance account the geometry of the boundaries. When the of the unit sphere. boundaries are smooth manifolds (as they are in this case), the Levi form plays a crucial role. The The Unit Sphere and the Heisenberg Group Levi form for a real hypersurface in Cn (for n ≥ 2) The quintessential principle connecting CR geom- is the complex variable analogue of the second etry and sub-Riemannian geometry arises from n fundamental form for a real hypersurface in R . the correspondence between the unit sphere bBn We define and discuss the Levi form on page 211. and the Heisenberg group, which bounds the For now we simply note that the Levi form is unbounded Siegel upper half space Hn. We will n−1 a Hermitian form defined on part of the tangent obtain a useful identification of bHn with C × R, spaces of the boundary. There is no Levi form in on which we will impose a nonabelian group law one dimension, because the tangent space is one arising from the biholomorphic automorphisms of real dimension and the part on which the Levi form Hn. The equivalence between Hn and Bn generalizes acts does not exist. For a half plane in dimension the one-dimensional Cayley transformation. at least 2 the Levi form is identically zero. For The Siegel upper half space Hn is the set of w the unit sphere it is positive definite. As a result in Cn such that the boundary geometries are so different that no n−1 biholomorphic mapping can exist. By contrast, X 2 (2) Im(wn) > |wj | . in one dimension, different boundary curves are j=1 indistinguishable from this point of view. A polydisc is a Cartesian product of discs. For The following elementary but useful identity is the 2 key to discovering the biholomorphic map: example, the subset of C defined by |z1| < 1 and 2 2 |z2| < 1 is a polydisc. It is also not biholomor- i + ζ i − ζ (3) Im(ζ) = − . phically equivalent to B2. In fact, for n ≥ 2 there 2 2 February 2010 Notices of the AMS 209 With ζ = wn we plug (3) into (2) and rewrite the CR Structures result to obtain We next discuss the geometry of a general real n 2 n−1 2 hypersurface M in C . The beautiful interplay i + wn X i − wn (4) > |w |2 + . between real and complex geometry dominates the 2 j 2 j=1 discussion. See [BER], [D1], [DT], [Jac], [Tre] and their references for more about CR geometry. After dividing by | i+wn |2 and changing notation, 2 We wish to consider complex vector fields and inequality (4) becomes therefore start by considering the complexified n X tangent bundle 1 > |z |2. j n = n ⊗ j=1 (6) CT(C ) T(C ) C. 2wj i−wn As usual in complex analysis we define the complex Here zj = for j < n and zn = . The reader i+wn i+wn partial derivative operators should check that this transformation f : w , z is ! ∂ 1 ∂ ∂ biholomorphic from Hn to Bn and also compare it = − i with the transformation in (1). ∂zj 2 ∂xj ∂yj and ! The automorphism group of Hn ∂ 1 ∂ ∂ = + i . We alluded earlier to the biholomorphic automor- ∂zj 2 ∂xj ∂yj phism groups of the ball and the polydisc. For any Using these operators we decompose the exterior complex manifold the automorphism group is a derivative d as fundamental algebraic invariant. For example, the n n X ∂f X ∂f automorphism group of the upper half plane in (7) df = ∂f + ∂f = dz + dz . ∂z j ∂z k C is PSL(2, R). Its elements act on the (one-point j=1 j k=1 k compactification of the) boundary of the upper half 2 Since d2 = 0, it follows that ∂2 = ∂ = ∂∂ + ∂∂ = 0. plane as linear fractional transformations x , ax+b , cx+d A section of the bundle (6) is a complex vector − = for ad bc 1. Any such transformation can be field. Each such vector field is a combination of represented as the composition of dilations x , rx, both z and z derivatives with smooth coefficients: translations x , x + x0, and possibly the inversion n n 1 X ∂ X ∂ x , x . Automorphisms fixing the point at infinity L = aj (z) + bj (z) . do not require the inversion. j=1 ∂zj j=1 ∂zj We now describe the analogous situation on We obtain two naturally defined integrable sub- bHn, defined by equality in (2). The last variable bundles of CT(Cn). Let T 10(Cn) denote the bundle n in (2) plays a different role, and hence for w ∈ C whose sections are vector fields L of the form 0 we write w = (w , wn). Two natural families of n X ∂ biholomorphic self-maps of Hn are the dilations (8) L = aj (z) , 0 ∂zj δr : Hn → Hn, for r > 0, given by δr (w , wn) = j=1 0 2 (rw , r wn), and the rotations RA : Hn → Hn, for where the a are smooth complex-valued functions. 0 0 j A ∈ U(n − 1), given by RA(w , wn) = (A(w ), wn). We note that this bundle is integrable in the sense To introduce an analogue of translation we consider of Frobenius: if K,L are sections of T 10(Cn), then the biholomorphisms τp : Hn → Hn, for p = (ζ, t) ∈ so is their commutator, or Lie bracket, given by n−1 C × R, given by [K, L] = KL − LK. The complex conjugate bundle, 0 0 0 2 denoted T 01(Cn), is also integrable. Since the zero (5) τp(w , wn) = (w +ζ, wn+t+2ihw , ζi+i|ζ| ). section is the only section of both bundles, we All of the preceding maps extend to self-maps of have T 10(Cn) ∩ T 01(Cn) = 0. We obtain a splitting 0 0 2 bHn = {(w , wn) : wn = |w | }. The action of the n 10 n 01 n n−1 CT(C ) = T (C ) ⊕ T (C ). family {τp : p = (ζ, t) ∈ C × R} on Hn ∪ bHn is faithful and the action on bHn is simply transitive. This splitting of the tangent bundle plays a crucial By this method we equip Cn−1 × R with a group role in all aspects of complex geometry. CR law (p, q) , p · q, characterized by the identity geometry studies the extent to which this splitting holds on real manifolds. τp ◦τq = τp·q. The resulting space is the Heisenberg n group. Often in the literature Cn−1 is replaced by Let M be a smooth real hypersurface of C , or R2n−2 and the group law is expressed using real more generally, of a complex manifold. The tangent variables. Finally we observe that the group of spaces of M then inherit some of the ambient complex structure. We write CTM for T (M) ⊗ C. biholomorphic automorphisms of Hn which fix 10 10 n the point at infinity is generated by dilations, We define T (M) = T (C ) ∩ CTM, and denote 01 rotations, and translations. See Stein’s textbook in its complex conjugate bundle by T (M). Again we harmonic analysis [Ste, Chapters XII and XIII] for have the complete story. (9) T 10(M) ∩ T 01(M) = 0. 210 Notices of the AMS Volume 57, Number 2 Let M be an abstract real manifold M with complex- other hand, there is no natural way to resolve the ified tangent bundle CTM. We say that a subbundle ambiguity of sign for an abstract CR manifold. T 10(M) of CTM defines a CR structure on M if it is Next we express the Levi form on a hypersurface integrable and its intersection with its conjugate in terms of partial derivatives. In a neighborhood bundle satisfies (9). We then call M a CR manifold. of a given point we suppose that M is the zero set Its horizontal bundle is the direct sum of r, where r is smooth and dr 6= 0 where r = 0. A (10) H (M) = T 10(M) ⊕ T 01(M). complex vector field L is tangent to M if and only if L(r) = hdr, Li = 0 on M. For the one-form η we We say that M is of hypersurface type if the fibers 1 may use (∂ − ∂)(r). In this case dη = −∂∂r. of H (M) have codimension one in CTM. 2 The Cartan formula for the exterior derivative In general, the CR codimension of M is the of a 1-form η states that codimension of H (M) in CTM. Real submanifolds of Cn of arbitrary dimension define CR manifolds of (11) hdη, A ∧ Bi = Ahη, Bi − Bhη, Ai − hη, [A, B]i. various CR codimensions. Consider a k-dimensional n The commutator term ensures that dη is linear over real subspace of C . As a real submanifold it has the module of smooth functions. Let L and K be codimension 2n − k. On the other hand, its CR local sections of T 10(M). Then hη, Li = hη, Ki = 0. codimension can be any integer ` between 0 and k By the Cartan formula, such that k − ` is even. The subspace has maximal (12) CR codimension (equal to k) if it is totally real, λ(L, K) = hη, [L, K]i = h−dη, L∧Ki = h∂∂r, L∧Ki. and it has minimal CR codimension (equal to zero) if it is complex. Any complex manifold can be We have interpreted the Levi form as the restriction regarded as a CR manifold of CR codimension zero. of the complex Hessian of r to sections of T 10(M). Such manifolds are precisely the integrable almost It is therefore analogous to the real Hessian, which complex manifolds. governs Euclidean convexity. Euclidean convexity We focus on CR manifolds of hypersurface implies pseudoconvexity, but the converse fails. See type. On such manifolds there is a nonvanishing [Hör] and [Kra] for lengthy discussion, especially differential one-form η, defined up to a multiple, for the meaning of pseudoconvexity on a domain annihilating H (M). We may assume that η is whose boundary is not smooth. purely imaginary. Each of the summands in (9) Example 3. The zero set of r(z) = Im(z ) is a half- is integrable, but something new happens; their n H sum is not integrable. The Levi form measures the space Σ. Its horizontal space z (Σ) decomposes failure of integrability of the sum and leads to both into holomorphic and conjugate holomorphic sub- CR geometry and sub-Riemannian geometry. spaces, spanned respectively by the vector fields Lj = ∂/∂zj , j = 1, . . . , n − 1, and their conjugates. Definition 1. Let M be a CR manifold of hypersur- In this case, the complex Hessian of the defining face type. The Levi form λ is the Hermitian form function is identically equal to zero, and hence the on T 10(M) defined by Levi form also vanishes identically. λ(L, K) = hη, [L, K]i. Pn 2 Example 4. The zero set of r(z) = j=1 |zj | − 1 is 2n−1 2n−1 Any vector field T for which hη, T i 6= 0 has a the sphere S . The horizontal space Hz (S ) 10 component in the missing direction. The Levi form decomposes into the holomorphic subspace Tz 2n−1 λ(L, Li gives the component of [L, L] in the missing (S ) = span{L1,...,Ln−1} and its conjugate 01 2n−1 direction. In the following definition we identify λ Tz (S ), where 10 with a Hermitian linear transformation of T (M). ∂ ∂ (13) Lj = zj − zn . Definition 2. A CR manifold of hypersurface type ∂zn ∂zj is pseudoconvex if all nonzero eigenvalues of λ have The annihilating one-form can be taken to be the same sign. It is called strongly pseudoconvex if n λ is definite, that is, all eigenvalues have the same 1 X (14) η = z dz − z dz . nonzero sign. j j j j 2 j=1 Observe that there is an ambiguity of sign in The Levi form, after dividing out a nonzero fac- the definition of the Levi form. Even for a real tor, satisfies λ(Lj , Lk) = δjk + zj zk. Hence the unit hypersurface in real Euclidean space the sign of the sphere is strongly pseudoconvex. The Heisenberg second fundamental form is defined only modulo group is also strongly pseudoconvex. Its Levi form, the choice of normal. For a compact hypersurface computed similarly, is λ(Lj , Lk) = δjk. in Rn consider a point p farthest from the origin. Since the sphere centered at the origin osculates Remark 5 (Hans Lewy equation). In 1957 Hans M to order two at p, the second fundamental Lewy produced his famous example of an unsolv- form at p agrees with that of the sphere, thus able first-order linear PDE. His equation is Lu = f , determining its sign. The story is the same for where L is the type (1, 0) vector field tangent to S3 real hypersurfaces in Cn and the Levi form. On the given in (13). See [Tre] for an elegant and readable February 2010 Notices of the AMS 211 account of developments in PDE and CR geometry a differential equation of high order to a first-order based on this operator. system. The references [Mon], [Str], [Gro], [Bel], [FS], [CCG], [CDPT], and [CC] provide many examples Positive definiteness of the second fundamental and discussion of other sub-Riemannian manifolds. form for a real hypersurface M in Rn is a curvature condition: it precludes the existence of straight Differential geometric aspects line segments in M. The next lemma provides a subtle analogue for the Levi form. Let M be a smooth real manifold and let H (M) be a distribution (subbundle) in the tangent bundle Lemma 6. Let M be a strongly pseudoconvex real T (M). The classical Frobenius theorem deals n hypersurface in C . Then M contains no complex with the case when H (M) is integrable, that analytic sets of positive dimension. is, closed under the Lie bracket. In this case M Proof. We prove the contrapositive statement: if M is foliated by H (M)-integral submanifolds. We contains such sets, then M cannot be strongly pseu- consider the opposite extreme, when H (M) is doconvex. Assume M contains a complex analytic completely nonintegrable: the Lie bracket span of set V of positive dimension. Let p be a nonsingular H (M), at any point p ∈ M, coincides with Tp(M). point of V. We can then find a one-dimensional In this case we say that the pair (M, H (M)) satisfies nonsingular holomorphic curve t → z(t) ∈ Cn with the bracket-generating property, also known as z(0) = p, with tangent vector z0(0) 6= 0, and with Hörmander’s condition. Put another way, for each z(t) ⊂ M for |t| < 1. For each local defining func- p ∈ M there exists an integer s = s(p) < ∞ so that tion r for M near p we have r(z(t)) = 0 for |t| the values at p of all s-fold iterated Lie brackets small. of vector fields valued in H (M) fill out the entire ∂ = tangent space T (M). We call s(p) the step of M Taking ∂t of the identity r(z(t)) 0 and evalu- p ating at t = 0 gives at p. We emphasize that s(p) may depend on the point p. In the case when s is uniformly bounded (15) h∂r(p), z0(0)i = 0. on M, we call supp∈M s(p) the step of M. ∂2 Taking the Laplacian ∂t∂t of the same identity and We elaborate with some examples. If M is evaluating at t = 0 gives a CR manifold of hypersurface type, we define 0 H (M) as in (10). The missing direction might or (16) h∂∂r, z (0) ∧ z0(0)i = 0. might not be obtained via iterated Lie brackets 0 10 Equation (15) says that z (0) ∈ Tp (M) and equa- of horizontal vector fields. Example 3 shows that 0 tion (16) says that z (0) is in the null space of Levi flat hypersurfaces do not satisfy the bracket- 0 the Levi form. Since z (0) 6= 0, M is not strongly generating property. The opposite extreme is the pseudoconvex at p. strongly pseudoconvex case. If L is a (1, 0) vector field and not zero at p, then the bracket [L, L] Sub-Riemannian Geometry has a component in the missing direction. In this It has been evident so far that the stratification of case the missing direction arises upon taking a the complexified tangent spaces of a CR manifold single Lie bracket of horizontal vector fields, and given by the horizontal bundle together with the the induced sub-Riemannian structure is of step missing direction is at the heart of CR geometry. two. To be step two we need only one such vector In this section we will consider real manifolds field; we recover the missing direction at p with equipped with a similar horizontal subbundle a single Lie bracket whenever the Levi form is of the (uncomplexified) tangent bundle. In both not zero at p. The simplest example of a higher cases there are horizontal directions which are step structure is the pseudoconvex hypersurface infinitesimally accessible and missing directions. 2m 2 defined by Re(z2) = |z1| in C near the origin. When the Levi form is nonzero we can recover One requires an iterated commutator the missing direction by commutators. The ability to recover missing directions via commutators of [. . . [L, L], . . . , L] horizontal vector fields is the defining property of with 2m total brackets to obtain the missing a sub-Riemannian manifold. This nonintegrability direction. In this case the origin is called a point condition translates to a connectivity condition of type 2m, and one says that the vector field L whereby such manifolds are equipped with a is of type 2m there. Notice in this case that the singular metric. See Theorem 9. step at most points is 2. See page 217 for related CR manifolds provide a natural class of examples, information. but the general theory covers a much wider class of spaces arising in PDE, control theory, geometric Example 7. The Heisenberg group bHn provides group theory, and many other settings. Example the canonical example of a sub-Riemannian mani- 8 puts a sub-Riemannian structure on spaces of fold. The group law was discussed in the paragraph kth order Taylor polynomials (jets). This example following (5). As usual, the left invariant vector formalizes the well-known procedure for reducing fields define the Heisenberg Lie algebra. As a basis 212 Notices of the AMS Volume 57, Number 2 for the horizontal distribution H (bHn) in T (bHn) [0, ∞) (the so-called Carnot-Carathéodory distance): we take the vector fields for p, q ∈ M, d(p, q) is the infimum of (17) Z b ∂ ∂ ∂ ∂ 0 0 1/2 = + = − = − (18) hγ (t), γ (t)i dt Xj 2yj ,Yj 2xj , j 1, . . . , n 1, a ∂xj ∂t ∂yj ∂t over all smooth horizontal curves γ : [a, b] → M = + = = ∂ where wj xj iyj and t Re(wn). Set T ∂t . with γ(a) = p and γ(b) = q. The fundamental Then [Xj ,Yk] = −4T δjk for j, k = 1, . . . , n − 1. By result is Theorem 9 below, the Chow–Rashevsky = 1 − = 1 + putting Lj 2 (Xj iYj ) and Lj 2 (Xj iYj ) and theorem. See the preface in [Mon] for the history regarding them as sections of CT (bHn), we return leading to this theorem. precisely to the CR setting. Theorem 9. Assume that Hörmander’s condition Example 8. Jet spaces provide a geometric inter- is satisfied for the pair (M, H (M)). Then all pairs pretation for Taylor polynomials, by viewing the of points in M are horizontally connected. Conse- equivalence classes of Cm functions modulo m-jets quently, the Carnot-Carathéodory distance function (mth order Taylor approximations) as points in an on a sub-Riemannian manifold M endows M with abstract space. We illustrate in the one-dimensional the structure of a metric space. case. Consider Cm functions f : R → R. The under- lying space M is Rm+2, with coordinates x (cor- An adapted Riemannian metric on M is an extension of the sub-Riemannian metric to the full responding to the base point x0 for the Taylor tangent bundle. If the step of M is finite, then the approximation of f ) and um, . . . , u0 (corresponding (m) topologies defined by the sub-Riemannian metric to f (x0), . . . , f (x0)). The sub-Riemannian struc- ture is defined by a family of smooth 1-forms and any adapted Riemannian metric coincide. However, the geometric and analytic properties ηj = duj − uj+1 dx, j = 0, . . . , m − 1, corresponding to the differential equalities df (j−1)(x) = f (j)(x) dx. of these two metrics differ substantially, unless the step is equal to one. For instance, if M has These forms, together with dx and du , frame the m step at least two, then these two metrics are cotangent bundle T ∗(M). Introduce the dual frame never bi-Lipschitz equivalent. (They are bi-Hölder X,Um,...,U0 for T (M). Let H (M) be spanned by equivalent with Hölder exponent 1 , where s is the X and Um. The Cartan formula (11) gives s step of M.) hηj−1, [Uk,X]i = −hdηj−1,X ∧ Uki Sub-Riemannian geodesics are smooth horizon- = hduj ∧ dx, Uk ∧ Xi = δjk, tal curves γ in M which locally minimize the length `(γ) defined in (18). The horizontality condition which yields the Lie bracket identities [Uj ,X] = is a family of nonlinear constraints which can be Uj−1, j = 1, . . . , m. We observe that H (M) is a understood in the language of control theory. The bracket-generating horizontal distribution of rank path planning problem for wheeled motion (includ- two inducing a sub-Riemannian structure on M = ing such concrete subproblems as parallel parking m+2 R of step m + 1, called the mth order jet space an automobile) can be reinterpreted as the problem (on the real line). The first-order jet space is iso- of finding geodesics joining prespecified points on morphic (as a Lie group) to the Heisenberg group certain sub-Riemannian manifolds modeled locally bH2. For additional details see [CDPT, Chapter 3.1] on the first Heisenberg group. Again, we reference or [Mon, Chapter 6]. [Mon] for more details. Sub-Riemannian geodesics behave rather dif- The fundamental theorem of sub-Riemannian ferently from their Riemannian counterparts. geometry Abnormal geodesics are locally length minimiz- Recall that a Riemannian metric on a smooth ing curves which fail to satisfy the geodesic manifold M is a smoothly varying family of equations. Such curves cannot exist in Riemannian inner products defined on the tangent spaces. geometry. We refer to [Mon] for a thorough discus- We say that a pair (M, H (M)) satisfying the sion of the geodesics in sub-Riemannian spaces Hörmander condition is a sub-Riemannian manifold and especially the issue of abnormal geodesics. if it is equipped with a smoothly varying family of We remark, however, that all geodesics in the inner products h·, ·ip (the sub-Riemannian metric) sub-Riemannian geometries arising from CR struc- defined on the horizontal tangent spaces Hp(M). tures on the boundaries of strictly pseudoconvex Such a metric permits us to define notions of length, domains are necessarily normal. Additional in- volume, angle and other geometric concepts for formation on the structure of sub-Riemannian objects taking values in the horizontal subbundle. geodesics can be found in [CCG] and the recent For instance, a smooth curve γ : (a, b) → M is publication [CC]. 0 termed horizontal if γ (t) ∈ Hγ(t)M for all t. A rich vein of current research activity in sub- As in the Riemannian case, the sub-Riemannian Riemannian geometry that we do not address is metric induces a distance function d : M × M → the study of the Carnot-Carathéodory geometry of February 2010 Notices of the AMS 213 submanifolds of sub-Riemannian manifolds. For more information see [CDPT, Chapter 4]. Example 10. We equip the Heisenberg group bHn with a Carnot-Carathéodory metric as follows. The left invariant vector fields from (17) define the horizontal distribution. We declare them to be an orthonormal basis. The geodesics for this metric have a beautiful geometric description. For sim- plicity, we restrict to the case n = 2. In view of the group structure it suffices to describe the geodesics from the identity element o = (0, 0, 0) to an ar- bitrary point p = (x1, y1, t1) with t1 ≤ 0. Each geodesic is the horizontal lift of a circular arc in the (x, y) plane with endpoints (0, 0) and (x1, y1) so that the area of the planar region bounded by the arc together with the line segment joining these two points is equal to −4t1. The sub-Riemannian distance from o to p is just the Euclidean length of the projected arc. When t1 = 0 the geodesic is just a ray in the t = 0 plane from o to p. When (x1, y1) = (0, 0) geodesics are not unique; the fam- ily of geodesics from o to (0, 0, t1) is rotationally symmetric about the t-axis. The Unit 3 Sphere Just like the circle and the two-sphere, the three-sphere is very round. But there are some beautiful, classical aspects to its roundness Figure 2. Sir William Rowan Hamilton and the that are not easy to guess from its lower- plaque on Dublin’s Brougham Bridge honoring dimensional sisters. Hamilton’s invention of the quaternions. The Mathematics Department of the National William Thurston [Thu] University of Ireland at Maynooth We illustrate the preceding circle of ideas commemorates the occasion with an annual by describing the sub-Riemannian geodesics in walk to the site of this plaque on October 16, 3 the three-sphere S equipped with the Carnot- the anniversary of Hamilton’s discovery. Carathéodory metric arising from its canonical CR structure. These geodesics have recently been computed by several authors using different tech- elementary identities z · j = j · z and z · k = k · z niques. See [HR], [BR], and [CC, Chapter 10]. We for z ∈ C. loosely follow the presentation by Hurtado and We view S3 as the set of unit quaternions Rosales [HR]. The geodesics admit a simple and p = z + z j, where |z |2 + |z |2 = 1. With the elegant description in terms of the Hopf fibration. 1 2 1 2 preceding conventions in place, S3 is equipped with the Lie group law S3 as a Lie group and the Hopf fibration · 0 = + · 0 + 0 Hamilton’s search for an “algebra of vectors in p p (z1 z2j) (z1 z2j) 3 0 0 0 0 R ”, parallel to the identification of plane vectors = (z1z1 − z2z2) + (z1z2 + z1z2)j. with complex numbers, is mathematical folklore. 3 His search led him to develop the theory of the It is well known that S is parallelizable: its tangent quaternions, a noncommutative algebraic structure bundle admits a smoothly varying orthonormal for vectors in R4. An arbitrary quaternion takes frame. The quaternionic presentation provides an the form a + bi + cj + dk, where a, b, c, d ∈ R elegant description for such a frame: just consider = · and i, j, and k are indeterminates satisfying i2 = the vector fields U, V , and W , where U(p) i p, = · = · j2 = k2 = ijk = −1. We will identify quaternions V (p) j p, and W (p) k p. These vector fields with pairs of complex numbers by the rule p = are linked by the bracket relations z + z j ↔ (z , z ) ∈ C2. Thus we identify C with (19) 1 2 1 2 [U, V ] = −2W , [W , U] = −2V , [V , W ] = −2U. the subspace of the quaternions H obtained by setting the coefficients of j and k equal to zero. In particular, setting H (S3) = span{V,W } defines Note that the quaternionic conjugate p is equal to a bracket-generating distribution on S3. If we z1 − z2 · j in this presentation. We point out the set p = z1 + z2 · j, then V (p) = j · (z1 + z2j) = 214 Notices of the AMS Volume 57, Number 2 −z2 + z1j, which coincides with the CR vector field The sub-Riemannian geodesics are critical points L from (13). We have already observed that S3 is for the first variation of (horizontal) length by strongly pseudoconvex, hence step two. Formula horizontal curves with fixed endpoints. Let γ : I → (19) provides the same information. We define a S3 be a C2 horizontal curve defined on a compact sub-Riemannian structure by introducing a family interval I. An admissible variation of γ is a C2 map 3 of inner products on the horizontal bundle for G : I × (−0, 0) → S satisfying which V and W are orthonormal. Our goal is to (1) G(s, 0) = γ(s) for all s ∈ I, describe the geodesics of this sub-Riemannian (2) G(s, ) = G(s, 0) for all and all s ∈ ∂I, structure. (3) for each , γ(s) := G(s, ) is horizontal. The Lie group S3 is isomorphic to the matrix Let X be the vector field along γ whose value at group SU(2) via the identification γ(s) is equal to (∂G/∂)(s, )| . Admissibility of ! =0 z1 z2 the variation can be characterized in terms of X: z1 + z2 · j ←→ . −z2 z1 G is admissible if and only if X vanishes at the endpoints of γ and We can realize SU(2) as the double cover of 0 0 the three-dimensional real rotation group SO(3). (20) γ (hX,Ui) = 2hπH X, J(γ )i, The most elegant description, which dates back to where π denotes projection into the horizontal Cayley, uses the quaternionic presentation: identify H space. R3 with the space of purely imaginary quaternions The classical formula for variation of length in via the basis {i, j, k}, and observe that the map (S3, g), specialized to admissible variations, reads q , p · q · p defines a rotation of R3. In the Z coordinates for the quaternions H described above, ∂ 0 `(γ)|=0 = − hDγ0 γ ,Ui. the map in question takes the form ∂ I ! Considering variations of the form X = f J(γ0) z1 z2 p = ∈ SU(2) , R 1 − p for C functions f : I → R with f |∂I = 0 and z2 z1 R f = 0, and taking into account (20), one finds 2 2 I |z1| − |z2| −2 Im(z1z2) 2 Re(z1z2) that the local minimizers for the sub-Riemannian 2 + 2 − 2 + 2 := 2 Im(z1z2) Re(z1 z2 ) Im(z1 z2 ) length functional (18) are characterized by the 2 2 2 2 −2 Re(z1z2) Im(z1 + z2 ) Re(z1 + z2 ) Euler–Lagrange equation ∈ SO(3). 0 0 (21) Dγ0 γ + 2κJ(γ ) = 0, This map is two-to-one, as it is invariant under the where κ is a real constant. This constant κ can antipodal map p , −p on S3. be interpreted as a curvature of the geodesic γ. Observe that the first row vector of R lies in p Geodesics of curvature zero are just great circles on S2 for every p. We obtain a map π : S3 → S2. The S3. Geodesics of nonzero curvature are horizontal circle group S1 acts on S3 via the diagonal action lifts of non-great circles on S2 through the Hopf eiθ · (z , z ) = (eiθz , eiθz ), leaving the first row 1 2 1 2 fibration. Indeed, consider a smooth curve γ of the corresponding rotation matrix invariant. taking values in S3 and parameterized by arc Thus we have exhibited S3 as an S1-bundle over S2, length. The tangent space to C2 at p = γ(t) called the Hopf bundle. The induced fibration of S3 splits as T (C2) = T (S3) ⊕ T (S3)⊥, which yields by (geometric) circles is the Hopf fibration. p p p the following decomposition for the acceleration vector: The sub-Riemannian geodesics on S3 00 0 We begin by observing that the annihilating one- (22) γ (t) = Dγ0(t)γ (t) − γ(t). form η from (14) and the corresponding horizontal Then (21) reads γ00 + 2κJ(γ0) + γ = 0. Viewing S3 3 1 distribution H (S ) are invariant under the S as a subset of C2, this equation takes the form action. This remark suggests that the structure of 00 0 the sub-Riemannian geodesics should be related to (23) γ + 2iκγ + γ = 0. the Hopf fibration. We now confirm this suggestion. Equation (23) is a system of constant coefficient Extend the inner product to a Riemannian second-order ODE’s whose explicit solution is easily metric g on S3 by declaring U, V and W to be obtained. Rather than presenting this solution, orthonormal. Let D be the Levi-Civitá connection however, we focus on its geometric content. for this metric g. Define a bundle isomorphism 2 J : H (S3) → H (S3) by setting J(V ) = W and Theorem 11 (Hurtado–Rosales). Let γ be a C hor- 3 J(W ) = −V . The data (S3, H (S3), η, J) defines a izontal curve on S parameterized by arc length. pseudo-Hermitian structure on S3, and the ensuing Then the following are equivalent: discussion can be framed in the language of pseudo- (1) γ is a geodesic of curvature κ in the sub- Hermitian geometry. See, for instance, [CHMY] and Riemannian metric on S3, [DT]. (2) hγ00, J(γ0)i = −2κ, February 2010 Notices of the AMS 215 (3) γ is the horizontal lift under the Hopf fibra- Group-invariant Mappings between tion S3 → S2 of a piece of a circle in S2 of Spheres in Complex Spaces constant geodesic curvature κ. The unit sphere S3 has dominated our discussion, We recall that the geodesic curvature of a space because of its symmetry and its role in both curve γ contained in S2 is its curvature computed CR and sub-Riemannian geometry. Earlier we 3 relative to the embedding in S2. For a curve γ observed an action of the circle on S defined by iθ iθ parameterized at arbitrary speed, this curvature is (z1, z2) → (e z1, e z2). In this section we study 3 given explicitly by the formula γ · (γ0 × γ00)/|γ0|3. symmetries of S via actions of finite subgroups of U(2). The simplest case is when the group is cyclic; For |h| < 1, the geodesic√ curvature of the circle γ = S2 ∩ {z = h} is h/ 1 − h2. even in the cyclic case there is a remarkable depth We sketch the proof of Theorem 11. To see and breadth of ideas. Given a positive integer p and that (1) and (2) are equivalent, note that the triple an integer q with 1 ≤ q < p, we choose a primitive {γ0, J(γ0), U(γ)} forms an orthonormal frame p-th root of unity α, and consider the group action 3 q for T S along γ. The fact that γ is horizontal (24) (z1, z2) → (αz1, α z2). and arc length parameterized immediately gives 0 The quotient of the sphere by this group is called a that the components of Dγ0 γ in the directions 0 0 lens space, written L(p, q). Lens spaces, introduced γ and U(γ) are equal to zero. Thus D 0 γ is γ by Tietze in 1908, are important in topology. The a multiple of J(γ0) and (21) can be rewritten 0 0 fundamental group of L(p, q) is cyclic of order hDγ0 γ , J(γ )i = −2κ. Equation (22) shows that 00 0 0 0 p for all q, and thus lens spaces with different hγ , J(γ )i = hDγ0 γ , J(γ )i, since γ(t) is normal to the sphere and J(γ0(t)) is tangent to the sphere. p are not homotopy equivalent. Famous work of To see the equivalence of (1) and (3), one calculates Alexander in 1919 showed that the lens spaces the binormal of the Hopf projection of γ, which L(5, 1) and L(5, 2) are not homeomorphic even turns out to be a constant. Thus this projection though they have isomorphic fundamental groups is a circle on S2. Another calculation shows that and the same homology. Necessary and sufficient its geodesic curvature agrees with the curvature κ conditions on p and q for being homeomorphic from (21). are known; similarly, necessary and sufficient con- The topological structure of these geodesics is ditions on p and q for being homotopy equivalent quite interesting. are also known. In Figure 4, the colors represent successive iterations of the group action (24). See Proposition 12 (Hurtado–Rosales). If h := √ κ 1+κ2 http://lukyanenko.net/math/heisenberg09/ is a rational number, then γ is a closed curve, iso- lens.html for animation and additional details. topic to a torus knot inside a group translate of the Clifford torus Th. If h is irrational, then γ is diffeo- morphic to R and is dense in some group translate of Th. Figure 4. A provocative view of the lens space L(5, 2). Figure 3. The Hopf fibration and sub-Riemannian geodesics. Lens spaces also arise in CR geometry. We naturally ask the following question. Given a finite subgroup Γ of U(n), is there an integer N and a Curves of the latter type are the well-known nonconstant rational mapping, invariant under Γ , skew lines on the torus. The Clifford tori in S3 from S2n−1 to S2N−1? The answer is, in general, no. In 3 2 are tori of the form Tρ := {(z1, z2) ∈ S : |z1| = fact, according to a theorem of Lichtblau [Lic], such 2 (1 + ρ)/2, |z2| = (1 − ρ)/2}, where |ρ| < 1. Figure a map can exist only when Γ is cyclic. Furthermore, 3 illustrates the Hopf fibration, sub-Riemannian (see [D1]), most representations of cyclic groups geodesics, and the Clifford tori. are also ruled out. We describe the results in the 216 Notices of the AMS Volume 57, Number 2 special case when n = 2. Roughly speaking, the Finite Type and Higher Brackets only possibilities for nonconstant invariant rational Iterated commutators of vector fields provide maps to spheres are the special cases when the a systematic method for dealing with higher quotient space is L(p, 1) or L(p, 2). One amazing order conditions, and their role in sub-Riemannian consequence is that there are homeomorphic lens geometry has been evident in this article. Such spaces, for example L(4, 1) and L(4, 3), only one of conditions are also significant in partial differential which (in this case L(4, 1)) admits a nonconstant equations. Given a finite set of vector fields d rational CR mapping to a sphere. X1,...,Xk in a neighborhood of a point in R , P 2 We do not give up. In order to get more groups consider the operator T = j Xj . Such sums of involved, we can proceed in two ways. First, we can squares of vector fields arise throughout partial weaken the assumptions on the mapping. Smooth differential equations and probability. When the CR mappings in this context must be rational. Sec- Xj form a basis for the tangent space, T is ond, we could allow continuous functions satisfying elliptic, and hence hypoelliptic. Hypoellipticity the tangential Cauchy-Riemann equations in the means that solutions u to the equation T u = f are sense of distributions. We then obtain spheres as smooth when f is smooth. Subellipticity is almost targets of group-invariant maps for any fixed-point as useful as ellipticity, because it also implies free subgroup of U(n). Such results require con- hypoellipticity. An elliptic operator of order two siderable machinery in function theory of several gains two derivatives in the Sobolev scale; a subelliptic operator gains fewer derivatives than its complex variables. If we weaken the assumption order, but the gain suffices to establish regularity on the target by allowing hyperquadrics, then more results. The Hörmander condition, applied to the elementary but fascinating ideas appear. Given a span of the X ’s, provides the necessary and finite subgroup of U(n), by [D1] we can always j Γ sufficient condition for subellipticity of T . find an N and a nonconstant polynomial mapping On the other hand, subellipticity for systems is p : Cn → CN such that p is -invariant, and p maps Γ much more difficult. In two dimensions, subelliptic- the unit sphere to a hyperquadric. The polynomial ity for the Cauchy-Riemann equations reduces to p arises from the following construction. First, one estimates for a scalar sum of squares operator. The defines the Hermitian symmetric polynomial as ΦΓ following theorem precisely relates the gain in a follows: subelliptic estimate with the step of the horizontal Y (z, w) = 1 − (1 − hγz, wi). distribution on the boundary. We omit the technical ΦΓ definition of a subelliptic estimate with gain . γ∈Γ Kohn established the first subelliptic estimate for By elementary linear algebra, there exist holo- weakly pseudoconvex domains in C2 under the morphic vector-valued Γ -invariant polynomial condition of finite step, and Greiner established mappings g, h such that the necessity of this condition. The sharp result N+ N− in Theorem 13 relies on a sophisticated theory X X (z, z) = |g (z)|2 − |h (z)|2. of estimates developed by Rothschild and Stein ΦΓ j j j=1 j=1 [RS] based on the sub-Riemannian geometry of Then the polynomial map p = (g, h) has the nilpotent Lie groups; their theory made it possible desired properties. Even for the cyclic subgroups to relate the geometry and the estimates in a precise fashion. Γ (p, q) used to define lens spaces, the computation of is interesting. For (p, 1) the result is ΦΓ (p,q) Γ Theorem 13. Let Ω be a smoothly bounded pseu- 2 2 p 2 (|z1| + |z2| ) . For Γ (p, 2) and Γ (p, p − 1) explicit doconvex domain in C , and suppose p ∈ bΩ. The formulas exist. For all q, the formula for Γ (p, q) is a following are equivalent: 2 2 polynomial fp,q in the variables |z1| and |z2| with • There is a subelliptic estimate at p with gain integer coefficients. In general these polynomials = 1 2m , but for no larger value of . have many interesting properties; we give here a • For a (1, 0) vector field L on bΩ with L(p) 6= few examples. By [D2], for each q, the congruence 0, we have type(L, p) = 2m. • H = 10 ⊕ 01 f (|z |2, |z |2) |z |2p + |z |2p (mod p) The bundle (M) T (M) T (M) has p,q 1 2 1 2 step 2m at p. holds if and only if p is prime. A combinatorial • The maximum order of contact at p of a interpretation of the integer coefficients appears complex analytic curve with bΩ equals 2m. in [LWW]. See [Mus] for an application to counting In higher dimensions things are more subtle; points on elliptic curves. See [D2] for many addi- algebraic geometry enters the story. Consider the tional properties of these polynomials. See [BR] for hypersurface M in C3 defined by a study of sub-Riemannian geometries on the lens 2 3 2 spaces L(p, q); these geometries are obtained by Re(z3) = |z1 − z2 | . projecting the horizontal distribution from S3 to The step at the origin is 4, and each (1, 0) vector L(p, q) via the natural quotient map. field has type either 4 or 6 there. Thus each such February 2010 Notices of the AMS 217 vector field is of finite type at the origin. On Acknowledgements the other hand, M contains the complex analytic We would like to thank Harold Parks for encour- 2 3 variety V defined by z1 − z2 = z3 = 0, which has a aging us to write an article for the Notices and for cusp at the origin. (See Figure 5.) At all nonsingular his useful comments on a preliminary draft. We points of V , the step is 2, but there is a (1, 0) vector also acknowledge valuable suggestions from two field (tangent to V ) of infinite type. In particular the referees. The Graduate Program at UIUC provided condition that each (1, 0) vector field be of finite us the opportunity to speak in the Math 499 type holds at the origin but fails at nearby points. series. Luca Capogna, Joe Kohn, Chris Leininger, This failure of openness has consequences in the and Tom Nevins have invariably shared useful study of subelliptic estimates. The condition for geometric insights with us over the years and thus subellipticity must be an open finite-type condition. indirectly aided the preparation of this article. The answer turns out to be that there is a bound on The pictures of lens spaces, the Hopf fibration, the order of contact of (possibly singular) complex and the sub-Riemannian geodesics on S3 were analytic one-dimensional varieties with M at p. See produced by Anton Lukyanenko using Mathemat- [Cat], [CD], [D1], [D3], [DK], and [Koh1]. ica; we gratefully recognize his contribution. We acknowledge the National Science Foundation for research funding under NSF Grants DMS-0500765 and DMS-0753978 (JPD) and Grants DMS-0555869 (JTT). References [Ahl] L. V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., 1978. [BER] M. S. Baouendi, P. Ebenfelt, and L. P. Roth- schild, Real submanifolds in complex space and their mappings, vol. 47 of Princeton Mathemati- Figure 5. A singular variety in the cal Series, Princeton University Press, Princeton, 3 2 3 2 hypersurface M = {z ∈ C : Re(z3) = |z1 − z2 | }. NJ, 1999. [Bel] S. R. Bell, Biholomorphic mappings and the ∂- problem, Annals of Math. 114 (1981), 103–113. [BR] U. Boscain and F. Rossi, Invariant Carnot- Carathéodory metrics on S3, SO(3), SL(2) and Concluding Remarks lens spaces, SIAM J. Control Optim. 47 (2008), Math 499, a required course for first-year graduate 1851–78. students at the University of Illinois, exposes [CC] O. Calin and D.-C. Chang, Sub-Riemannian students to a wide variety of mathematical points of Geometry: General Theory and Examples, view through engaging invitations to the research Cambridge University Press, Cambridge, UK, areas of the speakers. This note arose from 2009. [CCG] O. Calin, D.-C. Chang and P. Greiner, Geometric discussing the background common to talks we analysis on the Heisenberg group and its gener- have given in this course but includes material well alizations, vol. 40 of AMS/IP Studies in Advanced beyond what we actually presented. We hope that Mathematics, American Mathematical Society, we have provided sufficient insights and references Providence, RI; International Press, Somerville, to motivate further study. MA, 2007. Much of the material here is only tangentially [CDPT] L. Capogna, D. Danielli, S. D. Pauls, and J. T. related to differential geometry, especially if one Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, appeals to the hilarious definition “Differential vol. 259 of Progress in Mathematics, Birkhäuser, geometry is the study of those properties invariant Basel, 2007. under change of notation” sometimes attributed [Cat] D. W. Catlin, Subelliptic estimates for the ∂- to Calabi. 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February 2010 Notices of the AMS 219 Mathematics of the Gateway Arch Robert Osserman ne of the oldest problems to have been assume the form of a parabola.” The reason that solved using the calculus of variations virtually everyone refers to this passage rather than was to find the equation for the shape the one in which Galileo spells out his conclusions formed by a hanging chain or flexible in far greater detail and complete accuracy is not cord. Most often it is said that Galileo as clear. wasO the first to pose the problem and that he Some of Galileo’s earliest experiments with claimed that the resulting curve was a parabola. hanging chains apparently date from the summer (See, for example, Goldstine [5], p. 32, footnote of 1602, when he was working with Guidobaldo del 42, and Truesdell [18], pp. 43–44.) Both of these Monte. Had he thought to pose and try to answer statements are at best half true. Galileo’s exact the question of the exact equation for the shape of words (in a classic English translation of his book a hanging chain, he would have encountered two Two New Sciences [4], p. 290) are serious obstacles. First, the essential tool needed— …a cord stretched more or the calculus—had not yet been invented. Second, less tightly assumes a curve the equation for the curve involves logarithms, which closely approximates the which had also not yet been invented. parabola.…the coincidence is more After the problem was finally explicitly stated, exact in proportion as the parabola it was treated frequently by a number of authors is drawn with less curvature or, so between 1690 and 1720 and the correct answer to speak, more stretched; so that derived, although some of the initial reasoning using parabolas described with used was decidedly suspect. The curve became ◦ elevations less than 45 the chain known as a catenary. Its equation is y = cosh x, fits its parabola almost perfectly.… the hyperbolic cosine, up to change of coordinates. Nowhere, to my knowledge, does Galileo address One method used to arrive at this result was the the question of finding the exact shape of the curve calculus of variations, finding the shape of a curve formed by the hanging chain. And he is exactly whose center of gravity is the lowest among all right that for a parabola with elevation less than curves having a prescribed length and prescribed 45◦ the curve formed by the chain is an extremely endpoints. close approximation to the parabola. (See Figure 1.) This passage follows immediately after an The Gateway Arch extended discussion during which Galileo presents It was Robert Hooke who in 1675 made the one of his most famous discoveries: that the path connection between the ideal shape of an arch of a projectile in a vacuum under the influence and that of a hanging chain in an aphorism that of gravity is a parabola ([4], pp. 244–290). In the says, in abbreviated form, “As hangs the chain, so course of the discussion, he points out that for a stands the arch.” In other words, the geometry of a fixed initial velocity, the range will be maximized standing arch should mirror that of a hanging chain. by an initial elevation of 45◦. Finally he addresses The horizontal and vertical forces in a hanging the question of how to quickly draw a number of chain must add to a force directed along the chain, parabolas, leading to the passage cited above. since any component perpendicular to the chain The reason that one might mistake Galileo’s would cause it to move in that direction to gain intentions and conclusions is that elsewhere in equilibrium. Similarly, one wants the combined Two New Sciences ([4], p. 149) Galileo addresses the forces at each point of an arch to add up to a vector same problem and says simply that the “chain will tangent to the arch. In both cases, the horizontal Robert Osserman is professor emeritus of mathematics at component of the force is constant and simply Stanford University and special projects director at MSRI. transmitted along the arch or chain, while the His email address is [email protected]. vertical forces are mirror images. 220 Notices of the AMS Volume 57, Number 2 Figure 1. Left: 45◦ parabola and catenary; Right: 30◦ parabola and catenary. At least in part for these reasons, the shape of There is a certain amount of arbitrariness in the Gateway Arch is often described mistakenly as these definitions. The smoothness of the function a catenary (when not even more mistakenly as a f (x) is deliberately left vague, since one might parabola). In fact, the equation on which the arch well want to allow different classes of functions is based is in different situations. Similarly, one could allow the density function to have point masses or to be (1) y = A cosh Bx + C, even more general. which is a catenary only if A = 1/B. For the Gateway Nonetheless, conditions (i)–(iii) are natural ones Arch, A, B, and C are numerical constants, with A based on the physics underlying the problem. As approximately equal to .69(1/B). As a result, those noted earlier, for equilibrium we want the sum of who wish to be more accurate describe the shape the horizontal and vertical forces acting on the of the Arch as a “modified catenary” or, more often, chain to be a vector tangent to the curve. The a “weighted catenary”. The idea of the weighted horizontal force is constant and simply transmitted catenary is to use either a chain with literal weights along the curve, while the vertical force is due to attached at various points or else a continuous gravity acting on the part of the chain from the version, such as a cord of varying density. (Both of lowest point to the point that we are considering those methods, incidentally, were used by Saarinen on the curve and is equal to the total weight of to find a shape for the Gateway Arch that appealed that part of the chain. to him esthetically.) Note that we do not require that the endpoints Some natural questions to ask are: be at the same height—in other words, that 1. Is there a density function that yields the f (x1) = f (x2)—although that will be true in all the curve defined in equation (1) above, and if examples of interest to us. In that case, condition so, what is that function? (i) would follow from the physics of the situation. 2. How descriptive is the term “weighted However, condition (i) does guarantee the existence 0 = catenary”? In other words, what are the of an interior point x0 of the interval where f 0, curves that fall in this category? while condition (ii) implies that there is a unique such point. In order to answer these questions, we start Note also that the fact that there is a constant with some precise definitions. nonzero horizontal component to the force acting Definitions. (1) • A weighted chain C is a pair on the curve implies that the slope cannot be (f (x), ρ(s)) where f (x) is a suitably smooth func- infinite at the endpoints. For example, a weighted catenary cannot be in the form of a semicircle. tion on an interval x1 ≤ x ≤ x2, s is arclength along the curve y = f (x), and ρ(s) is a positive Note finally that if ρ(s) is multiplied by a positive continuous function of s. constant, then both H and V (x) are multiplied • The function ρ(s) is called the density function by the same constant. It follows that if the pair of C. (f (x), ρ(s)) defines a weighted catenary, then so • A weighted catenary is a weighted chain C in does the pair (f (x), cρ(s)) for any positive constant which c. 0 0 (i) −∞ < f (x1) < 0 < f (x2) < ∞, We are now able to state our first result. (ii) f 00(x) is continuous and positive, Theorem 1. If f (x) satisfies conditions (i) and (ii) (iii) for x ≤ X ≤ x , 1 2 above, then there is a function ρ(s), unique up (2) f 0(X) = V (X)/H, to a multiplicative constant, such that the pair (f (x), ρ(s)) is a weighted catenary. Namely with Hf 00(x) Z X ds (5) ρ(s(x)) = , (3) V (X) = ρ(s(x)) dx p1 + f 0(x)2 x0 dx where H is an arbitrary positive constant. The value where f 0(x ) = 0, and H is a constant that 0 of H is then given by depends on both f and ρ: W 0 (6) H = , (4) H = V (x2)/f (x2). 0 0 |f (x1)| + |f (x2)| February 2010 Notices of the AMS 221 with and Z x2 ds Z s(x2) W = ρ(s(x)) dx ρ(s)ds = 2aH(x2 − x1). x1 dx s(x1) equal to the total weight of the chain. Comment: This is in a sense the answer to Galileo’s original question of finding a mechanical Proof. Suppose first that there exists a function method for drawing a parabola. It tells us exactly ρ(s) such that the pair (f (x), ρ(s)) is a weighted how we have to weight a chain so that it will catenary. Then by equations (2) and (3), for X ≥ x0, hang in the form of a parabola. The key is that 00 0 Hf (X) = V (X) = ρ(s(X))(ds/dx)|x=X . the weight distribution has to be uniform in the p horizontal direction. For this reason, it is usually Since ds/dx = (1 + f 0(x)2), it follows that ρ(s) stated that the cables on a suspension bridge will must have the form given in equation (5). For X < hang in the shape of a parabola, since the weight x0, Z x0 of the cables themselves, together with that of the V (X) = − ρ(s(x))(ds/dx)dx vertical support cables holding up the roadway, X is small compared to the weight of the roadway, and which has (in essence) a uniform horizontal weight 00 0 Hf (X) = V (X) = ρ(s(X))(ds/dx)|x=X distribution. A curious by-product of the equation for the as before. Finally, the total weight W is density ρ in the case of a parabola is that, since Z x2 Z x0 Z x2 ds/dx ≥ 1, with equality only where f 0(x) = 0, it W = ρ(s)ds = + = V (x2) − V (x1). x1 x1 x0 follows that ρ(s(x)) is maximum at the vertex and 0 0 0 then decreases monotonically as |f (x)| increases. But V (x2) = Hf (x2) > 0, and V (x1) = Hf (x1) < 0, so that H takes the form indicated. In other words, the chain has to be weighted the This proves that the density function ρ is most near the vertex and then decrease as the uniquely determined by the function f (x) up to steepness of the curve increases. As a result, if the choice of the constant H. Saarinen had decided that he found a parabolic For the existence of a suitable density function arch most pleasing esthetically, he would have been ρ(s), given f (x), define ρ(s) by equation (5). Then faced with the paradox that in order to have the line of thrust be everywhere directed along the arch, the Z X ds Z X V (X) = ρ(s(x)) dx = Hf 00(x) dx arch would have to be thickest at the top and taper x0 dx x0 down toward the bottom, which would be both 0 0 0 = Hf (X) − Hf (x0) = Hf (X), ungainly esthetically and potentially disastrous structurally. so that the pair (f (x), ρ(s)) form a weighted cate- nary, and then the constant H must have the form Example 3 (circular arc). indicated in equation (6). x = R cos θ, y = R sin θ, π < θ1 ≤ θ ≤ θ2 < 2π. Example 1 (catenary). Then 1 0 dy . dx 00 1 3 f (x) = cosh Bx + C. f (x) = = − cot θ, f (x) = − csc θ, B dθ dθ R Then while q s = Rθ, ds/dx = − csc θ > 0. 0 0 2 f (x) = sinh Bx, 1 + f (x) = cosh Bx So H and f 00(x) = B cosh Bx, so ρ(s(x)) = csc2 θ = HR/y 2. R ρ(s(x)) = HB, constant. Comment: As noted earlier, a full semicircle can- Comment: This is a kind of inverse to the origi- not be a weighted catenary. We see explicitly that nal catenary problem of asking for the shape taken the density function would tend to infinity. How- by a uniformly weighted hanging chain. It tells us ever, any circular arc that is short of a semicircle that only for a uniform chain will the resulting at both ends can be realized (at least in theory) as curve be a catenary. a weighted catenary. The weighting is simply H Example 2 (parabola). ρ(s)= , πR 222 Notices of the AMS Volume 57, Number 2 of a semicircle, and the question of exactly how where j is the unit vector directed along the positive much it should be tapered was potentially critical. y-axis. (See [12] for more on this subject.) The proof of the above converse depends on the fact that equation (9) is the equation of equilibrium Note that this theorem answers the first part of Question 1 above about the existence of a for an arbitrary curve joining the points P1,P2 in density function yielding the particular “weighted the x, y-plane, with no special assumptions about catenary” shape of the Gateway Arch. It also its shape. (See [2], Chapter III, sections 1–3, for a answers Question 2 concerning the term “weighted detailed discussion of these questions in maximum catenary” itself, which turns out to have essentially generality. I would like to thank Joe Keller for no content beyond “convex curve”. Before turning providing this reference, as well as further helpful to the second part of Question 1 regarding the information. Note that in contrast to [2], we take exact form of the density function that yields the the density ρ to mean weight, rather than mass, curve underlying the Gateway Arch, we prove a per unit length, so that the acceleration of gravity strong form of the converse to the theorem. is absorbed into it.) Converse of Theorem 1. Let C be a curve with end- Proof of the converse. Let C be defined by points P1 = (x1, y1, 0) and P2 = (x2, y2, 0), where X(s) = (x(s), y(s), z(s)), s1 ≤ s ≤ s2, x1 < x2. Assume that C is parameterized by arc length s in the form where X(s)=(x(s), y(s), z(s)), P1 =X(s1), P2 =X(s2). X(s1) = (x1, y1, 0), X(s2) = (x2, y2, 0), x1 < x2, Let ρ(s) > 0 be a continuous function along C, taken as a density function. Assume gravity acts in the and let ρ(s) > 0 be a continuous density func- negative y-direction, and the pair (C, ρ(s)) is in tion along C. Then the pair (C, ρ(s)) will be in equilibrium. Then: equilibrium if and only if (9) holds; that is, 1. C lies entirely in the x, y-plane. d dx d dy (10) τ(s) = 0, τ(s) = ρ(s), 2. C is in the form of a graph y = f (x), x1 ≤ ds ds ds ds x ≤ x2. d dz 3. f 0(x) is finite and monotone increasing for τ(s) = 0. ds ds x1 ≤ x ≤ x2. Hence The idea underlying the proof is that if we know dx that C is a graph and (C, ρ(s)) is in equilibrium, (11) τ(s) (s) = c, ds so that equation (2) above holds, where V (X) is Z s defined by (3), then dy dy (12) τ(s) (s) = τ(s1) (s1) + ρ(σ )dσ , Z x0 ds ds s 0 1 1 −f (x1) = ρ(s(x))(ds/dx)dx < ∞, dz H x1 (13) τ(s) (s) = d, 1 Z x2 ds 0 = ∞ f (x2) ρ(s(x))(ds/dx)dx < , where c and d are constants. We claim first that C H x0 must lie completely in the x, y-plane. If not, there and for x1 ≤ x3 < x4 ≤ x2, must be a value s0, s1 < s0 < s2, where |z(s)| attains Z x4 0 0 1 its maximum |z(s )| > 0. Then (dz/ds)(s ) = 0, f (x4) − f (x3) = ρ(s(x))(ds/dx)dx > 0. 0 0 H x3 and the constant d in (13) must vanish. But if Hence condition 3 in the statement of the converse z(s) 6≡ 0, there must be some value s3 such that must hold. dz/ds is nonzero at s3 and hence in an interval In terms of the representation above for C as around s3. But then, since d = 0, it follows from X(s), the unit tangent vector to C will be given (13) that τ(s) ≡ 0 on that interval. But that contra- by T = dX/ds. We denote the tension at the point dicts equation (9), since ρ(s) > 0. Hence we must X(s) by τ, and for the case considered here, the have z(s) ≡ 0, and the curve lies entirely in the tension vector will be x, y-plane. (7) τT = (H, V , W ), Exactly analogous reasoning using equation (11) shows that there cannot be a value s where where H is constant and W = 0, while V is given 4 dx/ds = 0. In fact, since x(s ) = x > x = x(s ), by (3). Then 2 2 1 1 there must be a point where dx/ds > 0. It follows (8) d dV dV . ds that dx/ds > 0 on the whole interval [s1, s2], and τT= 0, , 0 = 0, , 0 =(0, ρ(s), 0) hence x is a strictly monotone increasing function ds ds dx dx of s on that interval. Thus we have a monotone by (3), or increasing inverse s(x), and we can set d (9) (τT) = ρ(s)j, ds y(s(x)) = f (x), x1 ≤ x ≤ x2, February 2010 Notices of the AMS 223 defining C as a graph. It follows from equations Conversely, suppose that p(x) = af (x) + b. Then (11) and (12) that f 00(x) = (af (x) + b)/H. Let (14) g(x) = f (x) + b/a. dy . dx 1 dy 1 Z s 0 = = + f (x) τ(s1) (s1) ρ(σ )dσ . Then g0(0) = f 0(0) = 0, and ds ds c ds c s1 Hence f 0(x) is a monotone increasing function of g00(x) = f 00(x) = ag(x)/H. s and therefore of x. This completes the proof of It follows that g(x) = A cosh Bx where B2 = a/H. the converse. Hence Note that it follows from the above that f (x) = g(x) − b/a = A cosh Bx + C, d . dx 1 ds (15) f 00(x) = f 0(x) = ρ(s(x)) > 0. where ds ds c dx p (19) A = f (0) + b/a, B = a/H, C = −b/a. Since c is the horizontal component of the tension vector, it is the quantity we denoted earlier by H, and equation (15) is the same as equation (5) above. Historical Note We now come to our principal example of the The flattened catenary was studied in the nineteenth theorem. century by a number of authors in France, England, Definition 1. A flattened catenary is a curve of the Ireland, and Scotland. The first may have been form y = f (x) = A cosh Bx + C, or Villarceau [19]; see Heyman [6], p. 48. Others include W. J. M. Rankine [13] and A. M. Howe [9]. (I y = Dg(x) + C,D = AB, thank William Thayer for these latter references where 0 < D < 1, and and for alerting me to the connection between the 1 equation for the Gateway Arch and the nineteenth- g(x) = cosh Bx century application discussed here.) In those works, B the flattened catenary is often referred to as a is a catenary. “transformed catenary”, and the term “two-nosed Example 4 (flattened catenary). We examine this catenary” is used for a highly flattened catenary, case in the following theorem. for reasons explained below. The context in which it arises is in answer to the question: What is the Theorem 2. Let C be a weighted catenary de- ideal shape of an arch that supports a horizontal fined by the pair (f (x), ρ(s)), and let p(x) = roadway held up by evenly packed dirt on top of ρ(s(x))(ds/dx). Assume that coordinates are the arch? The vertical load applied to each point on chosen so that f 0(0) = 0. Then p(x) will be of the the arch is then proportional to the distance from form that point up to the level of the roadway. If we p(x) = ay + b = af (x) + b write the equation of the arch as y = f (x), where y for some constants a > 0 and b if and only f (x) is is the distance from the roadway down to a given a flattened catenary: point on the arch, then the total vertical thrust f (x) = A cosh Bx + C, acting on the part of the arch from the apex to an arbitrary point of the arch will correspond to the for constants A, B, C. weight of a hanging chain from the minimum point 0 R Proof. We have for all X in [x1, x2] that f (X) = to a given point, and it will be of the form p(x)dx, V (X)/H, where where p(x) is a linear function of y. The same reasoning as in the above theorem then shows Z X Z x2 p(x) V (X) = p(x)dx, H = dx. that f (x) should be of the form (16): a flattened | 0 | + | 0 | x0 x1 f (x1) f (x2) catenary. (See, for example, Heyman [6], pp. 48-49.) Then We now come to the Gateway Arch itself. At f 00(X) = V 0(X)/H = p(X)/H. first sight it would appear to be a very different Suppose first that situation from that of an arch supporting a bridge, since it supports nothing except its own weight. (16) y = f (x) = A cosh Bx + C. On closer look, however, the fact that the Arch is Then f 0(x) = AB sinh Bx and tapered means that the weight supported at each level is not determined just by the length of the f 00(x) = AB2 cosh Bx = B2(f (x) − C). arch above it but by an integral with respect to Then arclength of a function exactly analogous to the (17) p(x) = Hf 00(x) = af (x) + b, density function for a weighted chain. The physical Gateway Arch is reasonably well modeled as one where of a class of geometric shapes that are of interest (18) a = HB2 > 0, b = −HB2C. in their own right. 224 Notices of the AMS Volume 57, Number 2 Tube-like Domains earlier approaches, Chern was able to provide an in- Two chance meetings in 1939 led to one of the major trinsic proof of the Gauss–Bonnet theorem without advances in differential geometry, the intrinsic the detour through embeddings. proof by Chern of an n-dimensional Gauss–Bonnet Our interest here is in a generalization of theorem. The first occurred when mathematician Hotelling’s result in a different direction. We go Hermann Weyl attended a lecture by statistician back to the case of a curve C in Euclidean space, Harold Hotelling in which Hotelling described a but are interested in much more general domains. new result of his and posed an open question. n Hotelling had been led by a statistical problem Notation 1. A curve C in R will be denoted by to pose the question: Given r > 0, what is the X(s) = (x1(s), . . . , xn(s)), 0 ≤ s ≤ L, probability that a point chosen at random on the n-sphere would lie within a distance r of a given s = arc length parameter, curve C? The answer amounts to computing the e1(s) = dX/ds volume of the domain consisting of all points whose distance to C is less than r and comparing the unit tangent vector, that to the total volume of the sphere. Hotelling e1(s), e2(s), . . . , en(s) was able to answer the question in both Euclidean n-space and the n-sphere for r sufficiently small a positively oriented orthonormal frame field along [8]. The answer in Euclidean space was that the C. volume is equal to the length of C times the volume Definition 2. A tube-like domain T with centerline of an (n − 1)-ball of radius r, with an analogous C is the image under a diffeomorphism F of a do- result for the n-sphere. The answer was perhaps main D in Rn, with coordinates u , . . . , u , such a surprise, since one might think that the volume 1 n that 0 < u < L, and for any t, 0 < t < L, the of the “tube” around the curve C would depend 1 intersection D of D with the hyperplane u = t somehow on the twists and turns in the curve, and t 1 contains the point (t, 0,..., 0). The map F satis- not simply on its total length. fies F(u , u , . . . , u ) = X(u ) + u e (u ) + · · · + What Hotelling wanted to know, and was un- 1 2 n 1 2 2 1 = = able to derive, was a formula for the case that the unen(u1), with u1 s arc length parameter along curve C was replaced by a surface of two or more C. dimensions. Hermann Weyl [22] provided the an- For 0 < s < L, let Ts be the intersection of T with swer in spectacular fashion: for a compact smooth the normal hyperplane to C at the point X(s). Then submanifold S of Euclidean space, the volume of F : Ds → Ts is a linear transformation mapping Ds the domain consisting of all points lying within isometrically to Ts . a fixed distance r of S is given, for r sufficiently Theorem 3. Let T be a tube-like domain in Rn with small, by a polynomial in r whose coefficients are centerline C. Then the volume V of T is given by intrinsic quantities associated with S , and do not depend on the way that S is immersed in space. Z L Z L Most strikingly, the top coefficient does not even (20) V = V (Ts )ds − 0 0 depend on the geometry of S, but only its topology; "Z # it is just a constant times the Euler characteristic k(s)~ · (F(s, u2, . . . , un) − X(s))du2 . . . dun ds, χ of S. Ds The other chance meeting in 1939 took place where k(s)~ = d2X/ds2 is the curvature vector, and between André Weil and Lars Ahlfors in Finland. for each fixed value of s, the vector Weil first learned about the classical Gauss–Bonnet theorem from Ahlfors, who expressed an interest F(s, u2, . . . , un) − X(s) = u2e2(s) + · · · + unen(s) in having a generalization to higher dimensions traces out the normal section T of T as the point in order to develop a higher-dimensional Nevan- s (s, u , . . . , u ) ranges over the parameter domain linna theory of value distribution ([20], p. 558, [21]. 2 n D . When Weil came to America several years later he s met Carl Allendoerfer and learned that both Allen- Proof. Let J = det dF be the Jacobian of the map doerfer and Werner Fenchel had independently F. If Y = (y1, . . . , yn) = F(u1, . . . , un), then proved a Gauss–Bonnet theorem for manifolds Z embedded in Euclidean space, using Weyl’s for- ∂(y1, . . . , yn) J = , V (T ) = J du1du2 . . . dun mulas. Remembering Ahlfors’s desire for a general ∂(u1, . . . , un) D Gauss–Bonnet theorem, Weil was able, together Z L "Z # with Allendoerfer, to prove the Gauss–Bonnet theo- = J du2 . . . dun ds. 0 D rem for all Riemannian manifolds ([1], [20], p. 299). s Their proof went via Weyl’s formulas together with Now write local embedding theorems. Shortly after, using the appropriate form of the integrand from these Y = X(u1) + u2e2 + · · · + unen, February 2010 Notices of the AMS 225 with ej = ej (u1), 0 ≤ u1 ≤ L. Then Corollary 1. If the point Xs is the centroid of Ts for each s ∈ [0, L], then ∂Y de2 den ∂Y Z L = e1 + u2 + · · · + un and = ei , ∂u1 du1 du1 ∂ui (21) V (T ) = V (Ts ) ds. i = 2, . . . , n, 0 Note that it suffices to consider points where and k(s)~ 6= 0. ∂Y ∂Y ∂Y ∧· · ·∧ = ∧e ∧· · ·∧e = Je ∧· · ·∧e . Proof. For each value of s, the inner integral in the ∂u ∂u ∂u 2 n 1 n 1 n 1 second term of the formula for V (T ) is a sum of Write terms consisting of integrals over Ds of terms of n ∂ei X the form cj uj , where cj is a constant. But if X(s) is = aij ej = ai1e1 + · · · , the centroid of Ts , then the origin is the centroid ∂u1 j=1 of Ds , and each of those integrals must vanish. where the coefficients are given by Definition 3. Under the hypotheses of Corollary 1, dei de1 the centerline C is called a centroid curve of T . ai1 = · e1 = − · ei = −k~ · ei , du1 du1 A given domain may have more than one cen- 2 d X de1 troid curve; for example, a Euclidean ball can be since the curvature vector k(s)~ = equals . ds2 ds viewed as a tube-like domain with respect to any So diameter as centerline, and that diameter will be a n n X X centroid curve for the ball. ai1ui = −k~ · ui ei = −k(u~ 1) · (Y − X), i=2 i=2 Corollary 2 (Hotelling’s theorem). If T is a tube domain in Rn of radius r around a curve C of length where a = a (u ), X = X(u ), Y = Y (u , . . . , u ). ij ij 1 1 1 n L, then Finally, n−1 V (T ) = ωn−1r L, n ∂Y X where ω is the volume of the ball of radius r in Rk. = 1 + a u e + · · · , k ∂u i1 i 1 1 i=2 Proof. In this case D is a cylinder of radius r and where the omitted terms involve the vectors height L. For 0 < s < L, the normal section Ts of T is an (n − 1)-ball of radius r centered at the point e2, . . . , en, and will vanish after taking the wedge X(s) of C. product with e2 through en. Hence, n A number of statisticians returned to questions X ~ J = 1 + ai1ui = 1 − k(u1) · (Y − X(u1)), related to Hotelling’s theorem after a lapse of many i=2 years. See the paper [11] and further references given there. as asserted. Note. Different versions of this theorem have Special Cases been known at least since 1978, when Michael n = 2. Then e2 is uniquely determined by e1 = Raugh [14] studied the case n = 3 in the context dX/ds. The orthogonal section Ts is an interval of the growth and structure of tendrils. Another Is , and the second term vanishes if X(s) is the discussion of the case n = 3 appears in Chapter midpoint of Is at all points where the curvature is 3 (Cálculo de várias variáveis) of [3], pp. 202–210. nonzero. Both of those discussions assume the existence Note. For n = 2, k(s)~ = κ(s)e2(s), where the of a Frenet frame field for the centerline of the curvature κ can be positive, negative, or zero. domain. A recent paper of Raugh [15] also treats n = 3. If k(s)~ 6= 0 at a point, then the same holds the n-dimensional case. in an interval around the point, and we may define Note. For the proof to be valid, one needs the e2 = k(s)/κ(s)~ , where κ(s) = |k(s)~ | is the curva- Jacobian to be positive. If it were to change sign, ture. Then e2(s) is a smooth unit vector field on the then there would be cancellations, and the inte- interval, and e1, e2, together determine e3. Since gral would not represent the actual volume. Since de1/ds = k~ = κe2, we have a21 = −κ, a31 = 0, and our definition of a tube-like domain includes the J = 1−κ(s)u2. As already noted, this equation also assumption that the defining map is a diffeomor- holds whenever κ = 0. Therefore, for n = 3, phism, the Jacobian cannot change sign. However, Z L "Z # if one starts with a domain D and tries to repre- V (T ) = (1 − κ(s)u2) du2 du3 ds sent it as a tube domain, then one must check the 0 Ts Z L Z L positivity of the Jacobian. We shall return to this = V (Ts ) ds − κ(s)Ms ds, point later on. 0 0 226 Notices of the AMS Volume 57, Number 2 where Ms is the moment of Ts about the line linearly in Y between the corresponding areas so through X(s) in the e3 (binormal) direction. It fol- defined: lows that for (21) to hold, one does not need X(s) Q(x, y) = Qv + (Qb − Qv )Y /Yb, to be the centroid of the orthogonal section, but only that the moment of that section be zero with where Qv is the cross-sectional area at the vertex, respect to the line through X(s) in the binormal and Qb is the corresponding area at the base. Since the area Q of an equilateral triangle is equal to direction, at all points where κ 6= 0. (See Raugh √ [15].) 3 Q = d2, An analogous statement holds for arbitrary n. 4 Another formulation is where d is the length of a side, and since d = 17 at n Z Z X the vertex v and d = 54 at the base, we find J du2 . . . dun = (1 + ai1ui ) du2 . . . dun Ds Ds i=2 Qv = 125.1406,Qb = 1262.6651, n X while Yb = Y |y=0 = 625.0925. It follows that = V (Ts ) + ai1Mi , i=2 Q(x, y) = 125.1406 + 1.81977Y where Mi is the i-th moment of Ds , and ai1 = ai1(s). = 1262.6651 − 1.81977y. Using the relation above between the area Q and Gateway Arch Specifications the side d of an equilateral triangle, we obtain The shape of the Gateway Arch is a polyhedral sur- the exact dimensions of the cross-sections at each face, the piecewise linear approximation by quadri- height y. It remains to specify that the positions of laterals to the surface S of the theoretical Gateway the triangles in the planes orthogonal to the curve Arch, with two of the sides of each quadrilateral C are symmetric with respect to the x, y-plane, with lying on designated straight-line segments of S. one vertex lying in that plane and interior to the Note. The surface just described is an abstract curve C, and that the centroid of each of those mathematical representation of the surface of the triangles lies on C, so that C is a centroid curve of physical Gateway Arch, that surface consisting of the Arch. the exterior of a set of stainless steel plates subject It follows from these specifications that the sur- to a variety of distortions. Among those are the (un- face of the theoretical Gateway Arch is the union even) expansion and contraction with changes of of three surfaces: a pair of ruled surfaces that are temperature, deflection due to wind forces, and symmetric images with respect to the x, y-plane deformation under the moving weight distribu- and meet in a curve lying in the x, y-plane, called tion of the internal transportation system carrying the intrados of the Arch, together with a cylindri- passengers to the top. cal surface orthogonal to that plane: the extrados. The theoretical Gateway Arch is a tube-like do- The maximum height of the extrados occurs at the main T whose centerline C is the reflection in a point directly above the vertex of C where x = 0 horizontal plane of a flattened catenary. The curve and y = yv = 625.0925. At this point the orthog- C is given by the equation onal plane is vertical, and the distance from the (22) y = 693.8597 − A cosh Bx, vertex of C to the base of the triangle√ is one third of the length of the median, or 17 3/6 = 4.90748. where A and B are the numerical constants Adding this to yv gives for the maximum height of (23) A = 68.7672,B = .0100333, the Arch h = 629.99998. The width of the Arch is a bit more subtle. We take the points where y = 0 on the centroid curve (24) − 299.2239 ≤ x ≤ 299.2239, C to represent ground level. At those points, x = and x, y represent distances measured in feet. We 299.2239 and −299.2239 so that the width of the may also write the equation of C as curve at ground level is 598.4478. The equilateral triangle representing the cross-section at those (25) y = 625.0925 − Y points has side 54, and hence the distance to the where outer curve of the Arch is √ √ (26) Y = A cosh Bx + C = A(cosh Bx − 1) 54 3/6 = 9 3 = 15.58846. represents the vertical distance down from the Adding this on each side to the width of C at ground vertex of C. level gives a total width for the Arch of 629.6247. The orthogonal section T (x, y) at the point However, since the curve C, although steep, is not (x, y) of C is in the form of an equilateral triangle vertical at ground level, the cross-section of the of area Q(x, y), where the sides of T (x, y) when Arch is not horizontal, and the actual outer width x = 0 (at the vertex of C) have length 17 feet and is slightly larger. However, one sees that the dimen- when y = 0 (at the base of the arch) have length 54 sions of the centroid curve together with the size feet. The area Q(x, y) is defined by interpolating and shape of the cross-sections produce an arch February 2010 Notices of the AMS 227 that for all practical purposes has exactly the same 3, which is equivalent, up to a similarity, to com- total height as width. It may be worth noting, how- pressing it vertically by a factor of 3. ever, since it is sometimes a source of confusion, It may be worth observing that when illustrating that the centroid curve is distinctly taller than wide, geometric graphs, one often sees different scales and the same is even more true of the inner curve used on the two axes. For a parabola, the resulting of the Arch, whose height is 615.3 feet, and width curve will still be a parabola, but for a catenary, the is 536.1. For more on this and related matters, see result will be a flattened or elongated catenary. As [12]. the figure, as well as the above calculation, makes clear, the shape may be very different geometrically Curvature Computations from a true catenary. Such differences are obvious = = = 1 when different scales in the horizontal and vertical Let y f (x) A cosh Bx D( B cosh Bx) define a flattened catenary, where A, B are positive con- directions turn a circle into an ellipse, but they are stants, and D = AB. Then f 00(x) = B2y > 0 so that often overlooked for noncompact curves such as f is convex and the curvature κ > 0. Specifically, the catenary. Finally, we note that for the centroid curve of B2y κ = . the Gateway Arch, we have D = AB = .69, well 2 2 + − 2 3/2 √ (B y 1 D ) above the critical value D = 3/3 = .577 ... , so For a catenary, D = 1 and κ = 1/By 2, with a that the curvature will take its maximum value at maximum of B at the vertex (0, 1/B) and decreasing the vertex, where it is equal to DB = .0069 so that monotonically to zero as y → ∞. the radius of curvature at that point is R = 145 feet, In general, let g(y) = (B2/κ)2 = B4R2,R = 1/κ. and it grows monotonically from there. Since all Then dR/dy ≥ 0 a g0(y) ≥ 0. But one finds points of the cross-sectional equilateral triangles 2 that define the Gateway√ Arch are less than 54 feet g0(y) = (B2y 2 + 1 − D2)2(2B2y 2 − 1 + D2) y 3 (and in fact, < 54 3/3 < 32 feet) from the centroid curve, it follows that the Jacobians entering into ≥ 2 2 + − 2 ≥ and since y A, B y 1 D 1, and the computations leading up to equation (21) are g0(y) > 0 a 2B2y 2 > 1 − D2. all positive, where the Gateway Arch is considered as a tube-like domain. Case 1: D > 1. An “elongated” catenary, obtained by stretching a catenary uniformly in the verti- Note. This paper is an expanded version of the cal direction. Then the right-hand side is negative, Roever Lecture given by the author at Washington g0(y) > 0 along the whole curve, so that R has a University in St. Louis on March 24, 2009. William H. minimum and κ a maximum at the vertex, where Roever was born in St. Louis, and after receiving x = 0, y = D/B, and κ = BD. his Ph.D. at Harvard University and teaching at MIT, he returned to Washington University, where he Case 2: D = 1. A catenary, for which, as we have served as chairman of the mathematics department. seen, R has a minimum value of 1/B at the vertex He wrote two papers on the curves of light formed and R → ∞ as y → ∞. by reflection under particular circumstances [16], Case 3: 0 < D < 1. Then there are two possibilities. [17]. He would undoubtedly have been fascinated by the light curve that forms on the Gateway Arch Case 3a: D2 ≥ 1/3. Since y takes its minimum when the sun is at a certain angle. value A at the vertex, we have 2B2y 2 + D2 ≥ 2B2A2 + D2 = 3D2 ≥ 1 Final Note so that g0(y) ≥ 0 along the whole curve, and as Although the essence of Hooke’s dictum relating in cases 1 and 2, the curvature has its maximum the standing arch to the hanging chain seems clear, value of BD at the vertex. there are subtle issues that arise, some involving the fact that a standing arch is a three-dimensional Case 3b: D2 < 1/3. Then 1 − D2 > 2D2 = 2B2A2, body, whereas the hanging chain is modeled just and we have g0(y) < 0 in the interval as a curve. In fact, a careful analysis of the stability 1 − D2 1/2 of an arch and its relation to the geometry of the y|x=0 = A ≤ y < . 2B2 arch is a subject about which whole books could In this case, the vertex of the curve will be a lo- be (and have been) written. (See, for example, [6].) cal minimum of the curvature, which will increase Here is an example of a seemingly paradoxical monotonically until y reaches the value indicated fact that does not seem to have been mentioned on the right, where the curvature will reach its max- anywhere. As we noted at the outset, the fact that imum. The curve is in this case sometimes referred the shape of a hanging chain is a catenary is often to as a two-nosed catenary. derived by using the calculus of variations, finding Figure 2 shows a two-nosed catenary for which the curve of given length that has the lowest center D = 1/3, obtained by stretching a catenary uni- of gravity. It follows then, according to Hooke, that formly in the horizontal direction by a factor of the ideal shape of an arch would be an upside-down 228 Notices of the AMS Volume 57, Number 2 Figure 2. Left: catenary; right: 2-nosed catenary. catenary, which would have the highest center of [9] A. Malverd Howe, A Treatise on Arches, New York, gravity among all arches of the same length. But John Wiley & Sons, 1897. [10] Charles Inglis, Applied Mechanics for Engineers, that would appear to make it maximally unstable. Cambridge University Press, 1951. Some of these and related issues are discussed [11] Mark Knowles and David Siegmund, On in the paper [12]. We hope to return to the subject Hotelling’s approach to testing for a nonlin- for a fuller discussion at a later date. ear parameter in regression, International Statistical Review, 57 (1989), 205–20. Acknowledgment. My thanks to Silvio Levy and [12] Robert Osserman, How the Gateway Arch got its Paul Shields for valuable TEXnical assistance. shape, Nexus Network Journal, 12 (2010). [13] W. J. Macquorn Rankine, Miscellaneous Scientific References Papers, London, Charles Griffith and Company, 1881. [1] Carl Allendoerfer and André Weil, The Gauss– [14] Michael R. Raugh, Some geometry problems sug- Bonnet theorem for Riemannian polyhedra, Trans. gested by the shapes of tendrils, Stanford University Amer. Math. Soc. 53 (1943), 101-29. thesis, 1978. [2] Stuart S. Antman, Nonlinear Problems of Elasticity, [15] , Some differential geometry moti- New York, Springer-Verlag 1995. vated by coiling tendrils, http://mikeraugh. [3] Vera L. X. Figueiredo, Margarida P. Mello, interconnect.com/tg2008.html. and Sandra A. Santos, Cálculo com Aplicações: [16] William H. Roever, Brilliant points and loci of Atividades Computacionais e Projetos, Coleç ao brilliant points, Annals of Math. 3 (1902), 113-28. IMECC, Textos Didáticos 3, Campinas, Instituto de [17] , The curve of light on a corrugated dome, Matemática, Estátistica, e Computação Cientifica Amer. Math. Monthly 20 (1913), 299-303. 2005. [18] Clifford Truesdell, The Rational Mechanics of [4] Galileo Galilei, Two New Sciences, Translated Flexible or Elastic Bodies, 1638-1788, Introduction to by Henry Crew and Alfonso de Salvio, New York, Leonhard Euler Opera Omnia Ser. 2, Vol. X and XI, Dover Publications 1954, originally published by 1960. The Macmillan Company, 1914. [19] Antoine J. F. Yvon Villarceau, Sur l’établissement [5] Herman H. Goldstine, A History of the Calculus of des arches de pont, Paris, Imprimerie Impériale, 1853. Variations from the 17th through the 19th Century, [20] André Weil, Collected Papers, Vol. 1, Springer- New York, Springer-Verlag 1980. Verlag 1979. [6] Jacques Heyman, The Masonry Arch, Chichester, [21] , Letter to the author, March 21, 1990. Ellis Horwood Limited, 1982. [22] Hermann Weyl, On the volume of tubes, Amer. J. [7] , The Science of Structural Engineering, Math. 61 (1939) 461–72. London, Imperial College Press, 1999. [8] Harold Hotelling, Tubes and spheres in n-space and a class of statistical problems, Amer. J. Math. 61 (1939), 440–60. February 2010 Notices of the AMS 229 Model Theory and Complex Geometry Rahim Moosa odel theory is a branch of mathe- powers of M to M, but by replacing them with their matical logic whose techniques have graphs I will, without loss of generality, restrict proven to be useful in several dis- myself to relational structures. For example, a ring ciplines, including algebra, algebraic can be viewed as a structure where the underlying geometry, and number theory. The set is the set of elements of the ring and there are, Mlast fifteen years have also seen the application of besides equality, two basic relations: the ternary model theory to bimeromorphic geometry, which relations given by the graphs of addition and is the study of compact complex manifolds up to multiplication. If the ring also admits an ordering bimeromorphic equivalence. In this article I will try that we are interested in, then we can consider to explain why logic should have anything to say the new structure where we add the ordering as about compact complex manifolds. My primary another basic binary relation. The definable sets of focus will be on the results in bimeromorphic ge- a structure are those subsets of cartesian powers ometry obtained by model-theoretic methods and of M that are obtained from the basic relations in the questions about compact complex manifolds finitely many steps using the following operations: that model theory poses. • intersection, • union, Structures and Definable Sets • complement, Besides being a discipline in its own right, model • cartesian product, theory is also a way of doing mathematics. Given a • image under a coordinate projection, and mathematical object, such as a ring or a manifold, • fibre of a coordinate projection. we begin by stating explicitly what structure on I am avoiding talking about the logical syntax here, that object we wish to investigate. We then study but the reader familiar with some first-order logic those sets that can be described using formal ex- will recognize that the basic relations form the lan- pressions that refer only to the declared structure guage of the structure and the various operations and whose syntax is dictated by first-order logic. correspond to logical symbols such as conjunction, Let me give a few details. disjunction, and negation. The operation of taking A structure consists of an underlying set M the image under a coordinate projection corre- together with a set of distinguished subsets of sponds to existential quantification, while that of various cartesian powers of M called the basic taking a fibre of a coordinate projection corre- relations. It is assumed that equality is a basic sponds to substituting parameters for variables (binary) relation in every structure. One could (that is, specialisation). The definable sets are then also allow basic functions from various cartesian the sets described by first-order formulae. This way of viewing definable sets is an essential feature Rahim Moosa is professor of mathematics at the Univer- sity of Waterloo, Canada. His email address is rmoosa@ of model theory, even though many expositions math.uwaterloo.ca. (including this one) avoid formulae by introducing The author was supported by an NSERC Discovery definability set-theoretically, just as I have done Grant. here. 230 Notices of the AMS Volume 57, Number 2 In any case, given a structure we have an asso- algebraically independent set over Q, then T will ciated collection of definable sets. When (R, +, ×) not contain any proper infinite analytic subsets. In is a commutative unitary ring, for example, it is particular, such tori, which are called generic com- not hard to see that if f1, . . . , fℓ are polynomials plex tori, cannot be projective varieties if n > 1. in R[x1, . . . , xn], then the algebraic set they define, Another example of a nonalgebraic compact com- namely their set of common zeros in Rn, is de- plex manifold is the following Hopf surface: fix finable. Hence the finite boolean combinations of a pair of complex numbers α = (α , α ) with such sets, that is, the Zariski constructible sets, are 1 2 | | ≤ | | all definable. It is an important fact that if R is an 0 < α1 α2 < 1, and consider the infinite 2 algebraically closed field, then these are the only cyclic group of automorphisms of C \{(0, 0)} 1 Γ definable sets. generated by (u, v) ֏ (α1u, α2v). The quotient 2 But we are interested in a somewhat different Hα = C \{(0, 0)}/ is a compact complex surface sort of example. Fix a compact complex manifold that is never algebraic.Γ Indeed, this can already be X and consider the structure A(X) where the deduced from its underlying differentiable struc- 1 3 basic relations are the complex analytic subsets of ture; Hα is diffeomorphic to S ×S , something that Xn, for all n > 0. By a complex analytic subset, or is never the case for a projective surface. Unlike just analytic subset for short, I mean a subset A tori, Hopf surfaces always contain proper infinite ∈ n such that for all p X there is a neighbourhood analytic subsets: the images of the punctured axes U of p and finitely many holomorphic functions in C2 \{(0, 0)} give us at least two irreducible f1, . . . , fℓ on U such that A ∩ U is the common curves on Hα. If α is chosen sufficiently generally, zero set of {f1, . . . , fℓ}. Note that the local data then these will be the only curves. In that case, like of U and f1, . . . , fℓ are not part of our structure; only the global set A is named as a basic relation. generic complex tori, Hα will have no nonconstant The model theory of compact complex manifolds meromorphic functions. was begun by Zilber’s [14] observation in the In model theory there are several notions, of early 1990s that A(X) is “tame”. In particular, varying resolution, that describe the possible inter- as a consequence of Remmert’s proper mapping action between two definable sets. These notions, theorem, Zilber shows that every definable set in specialised to the structure A, have relevant coun- A(X) is a finite boolean combination of analytic terparts in bimeromorphic geometry. Suppose that subsets. But the tameness goes much further, X and Y are irreducible complex analytic sets in making the geometry of analytic sets susceptible A. The strongest possible interaction is if X and to a vast array of model-theoretic techniques. Y are biholomorphic. This can be weakened to Zilber’s analysis of individual compact complex bimeromorphic equivalence: there exists an irre- manifolds extends to the many-sorted structure A, which includes all compact complex manifolds ducible analytic subset G ⊂ X × Y such that both at once, and where all complex analytic subsets projections G → X and G → Y are generically are named as basic relations. Note that algebraic bijective. By “generically bijective” I mean that off geometry is part of this structure; amongst the a countable union of proper analytic subsets of X compact complex manifolds in A are the complex the fibres of G → X are singletons, and similarly projective spaces, and so all quasi-projective alge- for G → Y . If G → X and G → Y are only assumed braic varieties are definable in A. Butthere arealso to be generically finite-to-one, then we say that nonalgebraic compact complex manifolds, and in G is a generically finite-to-finite correspondence some sense the model-theoretic analysis focuses between X and Y . Finally, we can weaken the con- on those. Let me discuss two examples that we dition much further by merely asking that there will see again later. n exist some proper analytic subset of X × Y that Suppose {α1, . . . , α2n} is an R-basis for C , and projects onto both X and Y . In that case we say = Zα1 + · · · + Zα2n is the real 2n-dimensional latticeΛ it generates. Then the quotient T = Cn/ that X and Y are nonorthogonal. For example, any is a compact complex manifold of dimension nΛ, two projective varieties are nonorthogonal. On the called a complex torus, that inherits a holomorphic other hand, any compact complex manifold with- group structure from (Cn, +). While some com- out nonconstant meromorphic functions—such plex tori can be embedded into projective space, as a generic complex torus or a generic Hopf if is chosen sufficiently generally, then the re- surface—is orthogonal to any projective variety. sultingΛ torus is nonalgebraic. For example, if the Indeed, nonorthogonality to some projective va- real and imaginary parts of {α1, . . . , α2n} form an riety implies nonorthogonality to the projective line P1, and if G ⊂ X × P1 witnesses this, then G 1This is quantifier elimination for algebraically closed fields, or equivalently Chevellay’s theorem that over an has nonconstant meromorphic functions, and as algebraically closed field the projection of a constructible G → X will necessarily be generically finite-to-one, set is constructible. so does X. February 2010 Notices of the AMS 231 Classifying Simple Compact Complex or its cartesian powers have no “rich” definable Manifolds families of analytic subsets. More precisely, if X is One way for a structure to be considered “tame” a simple compact complex manifold, then exactly is if the definable sets have a dimension theory, one of the following holds: that is, if a certain intrinsic model-theoretically I. X is a projective curve, or defined dimension function, called the rank, takes II. X is modular: whenever Y is a compact on ordinal values on all definable sets. Zilber’s complex manifold with dim Y > 0, and initial analysis of A showed that every analytic A ⊆ Y × X2 is an analytic subset whose subset of a compact complex manifold is of finite generic fibres over Y are distinct proper rank. In fact, this rank is bounded by, but typically infinite irreducible analytic subsets of X2 not equal to, the complex dimension. I will not that project onto each coordinate, then Y define rank here, but I will in a moment explain is simple. what it means for a compact complex manifold While the condition of modularity only explicitly to have rank 1. This reticence is partially justified mentions X2, it actually restricts the rank of fami- by the fact that there is general model-theoretic lies of analytic subsets of Xn, for all n. All projective machinery available, due mostly to Shelah, that curves are nonmodular. For example, the family analyses arbitrary finite-rank definable sets in of lines y = ax + b is a two-parameter algebraic terms of rank 1 sets. In particular, and this is family; it gives rise to a family of subvarieties of only the starting point of such an analysis in A, P ×P that is parameterised by a two-dimensional every analytic set will be nonorthogonal to one 1 1 projective variety, and two-dimensional projective of rank 1. It follows that the study of rank 1 varieties are not simple. sets is central to understanding compact complex So, by the above dichotomy, every simple man- manifolds in general. Note that this is not true ifold of dimension at least two is modular. As of complex dimension: understanding compact we have seen, examples can be found among complex curves tells us essentially nothing about the complex tori and the Hopf surfaces. In fact compact complex surfaces that contain no curves. the model-theoretic analysis provides us with a So what does it mean for a compact complex further dichotomy which distinguishes sharply be- manifold X to be of “rank 1”? Here is a geometric tween these two examples. Very roughly speaking, characterisation: X is not covered by a definable family of proper infinite analytic subsets. Pillay [10] a modular manifold that is not a complex torus observed that such manifolds were already of admits only binary definable relations. More pre- interest to complex geometers and were called cisely, if X is a simple modular compact complex simple. More precisely: manifold, then exactly one of the following holds: I. X is in generically finite-to-finite corre- Definition. A compact complex manifold X is said spondence with a complex torus, or to be simple if dim X > 0, and whenever Y is a com- II. X is relationally trivial: if A ⊆ Xn is an irre- pact complex manifold, A ⊂ Y × X is an analytic ducible analytic subset that projects onto subset, and E ⊂ Y is a proper analytic subset such X in each coordinate, and if {π , . . . , π } that the fibres of A above Y \ E are proper infinite 1 ℓ is an enumeration of all the coordinate subsets of X, then the union of all the fibres of A projections from Xn to X2, then A is an above Y \E is contained in a proper analytic subset ℓ of X. −1 irreducible component of \ πi πi (A). Projective curves are simple; they are the only i=1 simple projective varieties. But so is a generic Note that complex tori are not relationally triv- complex torus, or indeed any compact complex ial: being complex Lie groups they admit a truly manifold without proper infinite analytic subsets. ternary analytic relation, namely the graph of the The generic Hopf surfaces, having precisely two group law. Simple Hopf surfaces are relationally curves on them, are also simple. It is not hard to trivial. Actually one can be more precise in case I: see that simplicity is a bimeromorphic invariant; there exists a generically finite-to-one meromor- in fact, it is preserved by generically finite-to-finite phic surjection from X to a Kummer manifold, correspondences. In a sense that I have hinted a manifold of the form T/ where T is a com- at above and will return to again later, simple plex torus and is a finite groupΓ of holomorphic manifolds are the building blocks for all compact automorphismsΓ of T . complex manifolds. The above characterisation of simple modu- The contribution of model theory to bimero- lar compact complex manifolds was suggested morphic geometry begins with the following di- by Hrushovski in his 1998 address to the Inter- chotomy for simple compact complex manifolds, national Congress of Mathematicians [4]. It was which is a consequence of the deep results of proved by Pillay and Scanlon in [12], building on Hrushovski and Zilber from [5]. It says that a sim- some unpublished work of Scanlon. The result is ple compact complex manifold is either algebraic obtained as a corollary to the main theorem in 232 Notices of the AMS Volume 57, Number 2 that paper which I would at least like to state in of analytic subsets of cartesian powers of X such brief. Meromorphic groups are a natural generali- that every analytic set is definable in the reduct sation of algebraic groups to the complex analytic where only the sets in L0 are named as basic category; they are complex Lie groups that are relations. Essentially saturated compact complex “compactifiable” in an appropriate sense. What manifolds are significantly more amenable to a Pillay and Scanlon actually prove, using model- model-theoretic analysis. Exactly why is described theoretic methods and building on work of Fujiki, in [7] and is somewhat beyond the scope of this is that every meromorphic group is the extension article. Instead, I would like to focus on a very of a complex torus by a linear algebraic group. suggestive geometric characterisation of essential In any case, putting the two dichotomies to- saturation. gether, we get: Recall that a k-cycle of a compact complex man- ifold X is a finite linear combination Z = n Z Theorem 1. If X is a simple compact complex man- Pi i i where the Z are distinct k-dimensional irreducible ifold, then i complex analytic subsets of X and each n is a • i X is a projective curve, or positive integer. In particular, every irreducible an- • X is in generically finite-to-finite correspon- alytic subset is itself a cycle. The set of all k-cycles dence with a simple complex torus of di- is denoted by B (X), and B(X) denotes the disjoint mension > 1, or k union of all the B (X). In the 1970s Barlet endowed • X is relationally trivial. k B(X) with a natural structure of a complex ana- It remains then to understand the relationally lytic space. If X is algebraic, then B(X) coincides trivial compact complex manifolds. In dimension with the Chow scheme. With this terminology in one there are none because all compact curves place, we can characterise essential saturation as are algebraic (this is by the Riemann existence follows: X is essentially saturated if and only if all theorem), and it is not hard to see that projective the irreducible components of B(Xn) are compact, curves are not relationally trivial. Besides Hopf for all n ≥ 0. This follows from [7], in which the surfaces, examples of relationally trivial surfaces universal family of complex analytic subspaces can also be found among the K3 and Inoue sur- (the Douady spaces) were used instead of Barlet’s faces. Relationally trivial manifolds remain quite cycle spaces. In any case, it is important here that elusive, and, except for the case of surfaces, there the condition is on all the cartesian powers; if H are only conjectural results. These conjectures are is a Hopf surface, then B(H) has only compact restricted to Kähler manifolds, which also play components, but B(H × H) has a noncompact a special role from the model-theoretic point of component. In particular, Hopf surfaces are not view, and to which I now turn. essentially saturated. While the property of B(X) having only compact components is one that ap- Compact Cycle Spaces and Kähler pears often in bimeromorphic geometry, it seems Manifolds that the extension of this condition to all cartesian One of the obstacles to the full application of powers of X has not been studied. For example, model-theoretic techniques to compact complex is essential saturation preserved under cartesian manifolds is the size of the language, the fact that products and bimeromorphic equivalence? every analytic set is a basic relation. Let me point How does one check if a component of the out that in the algebraic case there is a more eco- Barlet space is compact? In the late 1970s Lieber- nomical choice of structure. Consider, for example, man [6] gave the following differential geometric the compact complex manifold X = Pm, projec- criterion: a set of cycles is relatively compact in tive m-space over the complex numbers. Instead of the cycle space if and only if the volume of the A(X) we can consider the structure AQ(X), where cycles in that set (with respect to some hermitian we only include as basic relations the algebraic metric) is bounded. This gives us many examples subsets of Xn that are defined by polynomials with of essentially saturated manifolds. Recall that a rational coefficients. Then AQ(X) has only count- Kähler manifold is one that admits a (hermitian) ably many basic relations. But all analytic subsets metric which locally approximates to order 2 the n n of X are still definable in AQ(X). This is because standard euclidean metric on C . With respect to every complex analytic subset of projective space such a metric, the volume of the cycles in any given is algebraic (Chow’s theorem) and every algebraic component of the Barlet space is constant, and set is obtained by specialisation from an algebraic this remains true of cartesian powers because käh- set over Q. Hence A(X) and AQ(X) have the same lerianity is preserved under cartesian products. It definable sets, even though the latter is equipped follows from Lieberman’s theorem that all com- with a much reduced collection of basic relations. pact Kähler manifolds are essentially saturated. This motivates the following definition: A com- Moroever, any compact complex analytic space pact complex manifold X is called essentially that is bimeromorphic to a Kähler manifold—this saturated if there exists a countable collection L0 is Fujiki’s class C—is also essentially saturated. February 2010 Notices of the AMS 233 While C does not include all essentially satu- strictly less than that of X, such an analysis must rated manifolds (Inoue surfaces of type SM are stop after finitely many iterations. It is in this counterexamples, see [8]), it is a large class of sense, via sequences of meromorphic fibrations, manifolds that contains all projective varieties that simple manifolds control the structure A. and complex tori and that is preserved under var- Of course this does not reduce the classification ious operations including meromorphic images problem to the case of simple manifolds; at the and generically finite-to-finite correspondences. very least one needs to also understand how such Compact Kähler manifolds play a prominent role manifolds fit into meromorphic fibrations and how in bimeromorphic geometry largely because they they interact with each other. I want to state one are susceptible to many of the techniques of al- conjecture that is central to this question. gebraic geometry. Because of essential saturation, First of all, from the definition of simplicity it they are also important to the model-theoretic follows that two simple manifolds are nonorthogo- approach. nal to each other if and only if they are in generically Let us return to the classification problem for finite-to-finite correspondence. The three classes simple compact complex manifolds. In the previ- of compact complex manifolds appearing in The- ous section I explained how this reduces to under- orem 1—projective curves, simple complex tori standing the relationally trivial manifolds. We can of dimension > 1, and simple relationally triv- restrict the problem further and ask: What are the ial manifolds—are all mutually orthogonal in the relationally trivial simple Kähler manifolds? We sense that any two manifolds coming from dif- have already seen that there are none in dimension ferent classes will be orthogonal. Moreover, while one. In dimension two, inspecting the Enriques- all curves are nonorthogonal to each other, there Kodaira classification, we see that all relationally exist orthogonal pairs within each of the other two trivial simple Kähler surfaces are bimeromorphic classes. to K3 surfaces. These surfaces (introduced by Weil One fundamental question is whether or not and named in honour of Kummer, Kodaira, Kähler, there exist entire definable families of simple man- and the mountain K2) are by definition simply- ifolds any two of which are orthogonal. Actually, connected compact surfaces with trivial canonical it follows from observations of Pillay and Scanlon bundle. In higher dimensions the correct general- that such families do exist, but not, conjecturally, isation of K3 seems to be irreducible hyperkähler: among the Kähler manifolds. More precisely, the simply connected compact Kähler manifolds with conjecture is that if f : X → Y is a meromorphic the property that their space of holomorphic surjection between compact Kähler manifolds with 2-forms is spanned by an everywhere nondegener- simple generic fibres, then any two generic fibres ate form. Pillay has conjectured that all relationally are in generically finite-to-finite correspondence. In trivial simple Kähler manifolds are in generically 2005 Campana [2] proved an isotriviality result finite-to-finite correspondence with an irreducible which proves the conjecture, with the stronger hyperkähler manifold. Since irreducible hyperkäh- conclusion of isomorphism rather than correspon- ler manifolds are always even-dimensional, Pillay’s dence, in the following cases: when the generic conjecture, coupled with Theorem 1 above, would fibres are surfaces, when the generic fibres are imply that every odd-dimensional simple Kähler “general” complex tori, and when the generic manifold is in generically finite-to-finite corre- fibres are irreducible hyperkähler. spondence with a complex torus. In dimension three this is essentially a conjecture of Campana Analogies and Peternell, namely that every simple Kähler By way of conclusion, I want to discuss the spe- threefold is Kummer. cial role of model theory as a medium between different geometric contexts. Certain results from Variation in Families bimeromorphic geometry have informed advances In justifying our focus on simple manifolds I have in abstract model theory which can then be reap- already mentioned the fact that every compact plied in other areas. In this way model theory acts complex manifold is nonorthogonal to a simple as a conduit between bimeromorphic geometry one. From this one can deduce that for every and, in the case I want to discuss, differential- compact complex manifold X there exists a mero- algebraic geometry. I need to say a few words morphic surjection f : X → Y , where dim Y > 0 and about differential-algebraic geometry. Y is in generically finite-to-finite correspondence Like bimeromorphic geometry, differential- with some cartesian power of a simple manifold. algebraic geometry is an “expansion” of algebraic The same is then also true of each generic fibre geometry. A differential field is a field K equipped Xy of f . At least in the Kähler case, using essential with a derivation; an additive map δ : K → K saturation, one can show that the corresponding such that δ(xy) = xδ(y) + yδ(x). For example, C d meromorphic surjections on Xy vary uniformly ( (t), dt ) is a differential field. Differential algebra in the parameter y. Since the dimension of Xy is is then commutative algebra in the presence of 234 Notices of the AMS Volume 57, Number 2 this derivation. The role of polynomials is played the 1970s and continues to fuel the internal by differential polynomials: functions of the form development of the subject. r P x, δ(x), . . . , δ (x), where P ∈ K[X0,...,Xr ] is an ordinary polynomial. A differential field (K, δ) References is differentially closed if any system of differential [1] F. Campana, Algébricité et compacité dans l’espace polynomial equations and inequations having des cycles d’un espace analytique complexe, a solution in some differential field extension Mathematische Annalen 251(1) (1980), 7–18. already has a solution in K. The model theory of [2] , Isotrivialité de certaines familles Kähléri- differentially closed fields of characteristic zero ennes de varietés non projectives, Mathematisches 252 is also tame. In particular, every definable set Zeitschrift (1) (2006), 147–56. [3] A. Fujiki, On the Douady space of a compact com- in (K, +, ×, δ) is a finite boolean combination of plex space in the category C, Nagoya Mathematical differential-algebraic sets: zero sets of systems of Journal 85 (1982), 189–211. differential polynomial equations. These are the [4] E. Hrushovski, Geometric model theory, in Pro- objects of study in differential-algebraic geometry. ceedings of the International Congress of Math- Many of the model-theoretic techniques that ematicians, Vol. I (Berlin, 1998), pages 281–302, apply to the structure A also apply to (K, +, ×, δ). Documenta Mathematica, 1998. In fact, there is a fruitful analogy between these [5] E. Hrushovski and B. Zilber, Zariski geometries, J. structures whereby complex analytic sets cor- Amer. Math. Soc. 9(1) (1996), 1–56. respond to finite-rank differential-algebraic sets [6] D. Lieberman, Compactness of the Chow scheme: (see [10]). This analogy can lead to transferring applications to automorphisms and deformations of Kähler manifolds, in Fonctions de plusieurs results between the two disciplines that these variables complexes, III (Sém. François Norguet, structures represent. The example I have in mind 1975–1977), volume 670 of Lecture Notes in Math., is based on the following theorem of Campana [1], Springer, 1978, 140–86. due also independently to Fujiki [3], from the [7] R. Moosa, On saturation and the model theory of early 1980s. Suppose X is a compact complex compact Kähler manifolds, Journal für die reine und manifold and C is a compact analytic subset of angewandte Mathematik 586 (2005), 1–20. the cycle space B(X). Then for any a ∈ X, the [8] R. Moosa, R. Moraru, and M. Toma, An essentially set of cycles in C that pass through a is (up saturated surface not of Kähler-type, Bull. London 40 to bimeromorphism) an algebraic set. Moreover Mathematical Society (5) (2008), 845–54. R. Moosa A. Pillay this happens uniformly as a varies. In 2001 Pil- [9] and , On canonical bases and internality criteria, to appear in the Illinois Journal lay [11] observed that this theorem could be used of Mathematics. to give a more direct proof of the first dichotomy; [10] A. Pillay, Some model theory of compact com- the fact that every simple manifold is either a plex spaces, in Hilbert’s Tenth Problem: Relations curve or modular. Pillay’s argument involves for- with Arithmetic and Algebraic Geometry (Ghent, mulating an abstract model-theoretic counterpart 1999), American Mathematical Society, Providence, to Campana’s theorem, which replaces the diffi- RI, 2000, 323–38. cult and much more general results of Hrushov- [11] , Model-theoretic consequences of a theorem ski and Zilber [5]. Pillay and Ziegler [13] then of Campana and Fujiki, Fundamenta Mathematicae 174 show that this model-theoretic counterpart also (2) (2002), 187–92. A. Pillay T. Scanlon holds in differentially closed fields. As a result [12] and , Meromorphic groups, Trans. Amer. Math. Soc. 355(10) (2003), 3843–59. one has an analogue of Campana’s theorem in [13] A. Pillay and M. Ziegler, Jet spaces of varieties differential-algebraic geometry, as well as a di- over differential and difference fields, Selecta Math. rect proof of the corresponding dichotomy for (N.S.) 9(4) (2003), 579–99. rank 1 differential-algebraic sets. Another related [14] B. Zilber, Model theory and algebraic geometry, example of this phenomenon can be found in [9], in Proceedings of the 10th Easter Conference on in which Campana’s “algebraic connectedness” Model Theory (Wendisch Reitz, 1993), Humboldt is abstracted from bimeromorphic geometry to Universität, 1993, 93–117. model theory and then applied to differentially closed fields. The outcome in that case is a cri- terion for when a finite-rank differential-algebraic set is in generically finite-to-finite correspondence with the CK -points of an algebraic variety, where CK = {x ∈ K : δ(x) = 0}. The kind of transfer of ideas that we see in the above examples, and the role that model theory plays here of recognising, formalising, and facilitating analogies between different geometric settings, is not something new or unique to its interaction with bimeromorphic geometry. This has been a defining feature of model theory since February 2010 Notices of the AMS 235 WHATIS... ? a Halting Probability? Cristian S. Calude and G. J. Chaitin Turing’s famous 1936 paper “On computable continuum would have strange consequences, and numbers, with an application to the Entschei- B is a case in point. The number B is an oracle dungsproblem” defines a computable real number that “would give answers to all past, present, and and uses Cantor’s diagonal argument to exhibit an future enigmas of science, history, and curiosity.” uncomputable real. Let’s change this into something more mathe- Roughly speaking, a computable real is one that matical, more realistic: an oracle for the halting can be calculated digit by digit, one for which there problem H. is an algorithm for approximating as closely as one Systematically enumerate all programs with no may wish. All the reals one normally encounters input (say, lexicographically). The Nth bit hN of in analysis are computable, like π, √2, and e. the real number H tells us whether or not the Nth But they are much scarcer than the uncomputable computer program ever halts. H is an oracle for reals because, as Turing points out, the computable the halting problem, which would be extremely reals are countable, whilst the uncomputable reals useful to have, as we will show later. have the power of the continuum. Furthermore, However, this oracle H for the halting problem any countable set of reals has measure zero, so is extremely wasteful, because N instances of the the computable reals have measure zero. In other halting problem are not N bits of mathematical words, if one picks a real at random in the unit information, they are only log2 N bits of math- interval with uniform probability distribution, the ematical information. One only needs to know probability of obtaining an uncomputable real is how many of these N programs halt—a number unity. One may obtain a computable real, but that expressed in less than or equal to 1 log N bits— + 2 is infinitely improbable. to be able to determine which ones halt. Indeed, But how about individual examples of uncom- one just runs in parallel all programs till exactly putable reals? We will show two: H and the halting the known number of programs that halt have probability , both contained in the unit inter- stopped: all the remaining programs won’t halt. val. Their constructionΩ was anticipated in 1927 by Using a slightly more sophisticated version Émile Borel, who “defined” a real number 0 < B < 1 of this idea, we finally arrive at the halting in the following way. The Nth digit bN of B answers probability , which is defined by the following the Nth question in an enumeration of all possible formula: Ω yes/no questions that one can write in French. (size of p in bits) 0 < X 2− < 1. = Borel argued that knowledge of every point of the Ω program p halts Cristian S. Calude is professor of computer science at the This is the probability that a computer program University of Auckland. His email address is cscalude@ whose bits are generated one by one by indepen- gmail.com. dent tosses of a fair coin will eventually halt. There G. J. Chaitin is emeritus researcher at the IBM Watson Re- are actually many halting probabilities, not one, because the precise numerical value of L de- search Center and honorary professor at the University = of Buenos Aires. His email address is gjchaitin@gmail. pends on the choice L of programmingΩ languageΩ com. for the programs p. 236 Notices of the AMS Volume 57, Number 2 Nota Bene: For this definition of a halting these bits have actually been determined. For a probability to work properly, the computer pro- natural choice of the programming language L, 1 gramming language L that the programs p are the first 40 bits of L are = written in has to satisfy certain requirements: 0001000000010000101001110111000011111010Ω Ω . The programs p have to be self-delimiting 2 • If we knew the first 7,780 bits, which is less than (no extension of a valid program is a valid one quarter of this note’s size in bits, we would program) or the sum won’t converge. This know whether the Riemann hypothesis is correct. enables one to construct large programs Now for some more advanced results. by concatenating (abutting) subroutines. Although uncomputable, is the most “com- The language L has to be universal and putable” among all algorithmicallyΩ random reals: • concise. More precisely, if any other lan- any L is left-computable, i.e., the least upper guage L′ can program something in K bits, boundΩ of a computable sequence of rationals. The then L can do it in K cL bits. ≤ + ′ set L of all halting probabilities of univer- For historical reasons, such a programming lan- sal self-delimiting{Ω } Turing machines L coincides guage L is often referred to as a universal with the set of all left-computable algorithmically self-delimiting Turing machine, and we shall do random reals. so below. Can a halting probability be formally proved What are the key properties of a halting algorithmically random in Peano arithmetic (PA)? probability ? The answer depends on the representation: if By writingΩ in binary one can show that given L is givenbya self-delimiting Turing machine = the first N bitsΩω1 . . . ωN of LΩ whoseΩ universality (and conciseness) is proved K in PA, then PA proves that is algorithmically X ωK /2 , = random. Every can be writtenΩ as , for Ω K 1,2,3,... L = some L satisfyingΩ the above requirements,Ω = Ω so it is one can solve the halting problem for all programs 3 provably random in PA. up to N bits in size. This implies that the bits of are algorithmically and logically irreducible: Further Reading. For the intellectual history leading up to , including the role played by Ω The smallest program for computing the Leibniz, see theΩ second author’s Meta Math!. For • first N bits of has N O(1) bits. a technical treatment, see the first author’s In- (This algorithmicΩ irreducibility≥ − property is formation and Randomness, or An Introduction to normally called (algorithmic) randomness.) Kolmogorov Complexity and Its Applications by M. Any formal axiomatic theory, like ZFC, Li and P. Vitányi. For halting probabilities for ver- • that enables one to prove what are the sions of a real programming language, LISP, and a values of the first N bits of must have plethora of other nonstandard halting probabili- N O(1) bits of axioms. (ThisΩ is logical ties and their applications, see the second author’s irreducibility.≥ − ) Information-Theoretic Incompleteness. For some of In other words, the bits of are the most con- the latest results, see Computability and Random- cise oracle for solving the haltingΩ problem, the ness by A. Nies and Algorithmic Randomness and best possible compression of all the answers to Complexity by R. G. Downey and D. Hirschfeldt individual instances of the halting problem. (forthcoming). For discussions of the philosophical The philosophical and epistemological signif- impact and additional historical material, see the icance of is primarily due to the following second author’s Thinking about Gödel and Turing. surprising fact:Ω The bits ω1ω2ω3 ... of the halting probability provide a perfect simulation withinΩ pure mathemat- ics, where all truths are necessary, of an infinite stream of contingent, accidental yes/no facts. For instance, provides us with a natural 1C. S. Calude, M. J. Dinneen, Exact approximations of Ω example of what É. Borel termed an absolutely omega numbers, Internat. J. Bifur. Chaos 17(6) (2007), normal real number. This means that if is 1937–54. written in any base b 2, then all blocksΩ of K 2 ≥ C. S. Calude, E. Calude, M. J. Dinneen, A new measure base-b digits will occur with equal limiting relative of the difficulty of problems, J. Mult.-Valued Logic Soft K frequency b− . Comput. 12 (2006), 285–307. In fact, this number can be It is an immediate corollary of ’s algorithmic lowered to less than 5,000 bits. and logical irreducibility that isΩ uncomputable 3C. S. Calude, N. J. Hay, Every computably enumer- and that, in fact, no algorithmΩ can compute more able random real is provably computably enumerable than finitely many scattered bits of . Some of random, Logic Jnl. IGPL 17 (2009), 325–350. Ω February 2010 Notices of the AMS 237 A MERICAN MATHEMATICAL SOCIETY Headlines & Deadlines for Students, a service from the AMS Public Awareness Offi ce, provides email notifi cation of mathematics news and of upcoming deadlines. These email notifi cations are issued about once a month, and when there’s special news. Imminent deadlines are included in these emails, which link to a web page that’s a centralized source for information relevant to students and faculty advisors, at: www.ams.org/news-for-students/ Sign up for the email service at: www.ams.org/news-for-students/signup Check out the latest Headlines & Deadlines for Students for news and due dates for applications, registrations, proposals... Pi Mu Epsilon Student Paper Presentation Awards Fellowships and Grants New Mathematical Moments ∋ Marshall Scholarships Stipends for Study and Travel π Trjitzinsky Awards MathΩ in Moscow Semester - Call for Applications AWM Essay Contest ∋Special Book Sales on AMS Bookstore Poster Session Proposals Putnam Exam ResultsΩ π Employment Center Registration ClayCl Research R Fellowships σ www.ams.org/news-for-students Book Review What Then? Plato’s Ghost: The Modernist Transformation of Mathematics Reviewed by Yuri Manin Plato’s Ghost: The Modernist Transformation of Euclidean geometry, including a description of Mathematics Gauss’s role in its formation and the subsequent Jeremy Gray contributions of Riemann and Poincaré that led to a Princeton University Press, 2008 new vision of what geometry is and what it should US$45.00, 526 pages be. Similar essays on algebra, analysis, “British Al- ISBN-13: 978-0691136703 gebra and Logic”, and philosophy follow, all of this interspersed with information about professional The new book of the renowned historian of math- organizations, teaching, journals, and quotations ematics Jeremy Gray can be succinctly described as a from contemporary articles and letters. (I cannot rich and thorough study of Western ethnomathemat- resist the temptation to reproduce a delightfully ics in the period of five decades or so (1880–1930) preceding and following World War I. What distin- funny sentence from a letter by C. G. J. Jacobi to guishes such a project from more conventional A. von Humboldt, although in the book it appears Bourbaki–style enterprises is the stress on extra– only considerably later, on page 268: mathematical background: social and political If Gauss says he has proved something, structures of society, economic practices, forms of it seems very probable to me; if Cauchy professional self–organization, and culture. says so, it is about as likely as not; if In fact, culture (represented by visual arts, Dirichlet says so, it is certain.) music, philosophy) dominates the discourse at several key points, and culture furnished the basic Chapter 2, containing this wealth of informa- metaphor for the book. The time span in question tion, is called “Before Modernism”, and ends with is characterized as the period of modernist trans- a brief synopsis “Consensus in 1880”. formation in mathematics, and the first quotation The arrival of mathematical modernism is ad- in the Introduction is taken from Guillaume Apol- dressed in Chapter 3. Again, the discussion is linaire’s “The Beginning of Cubism” (1912). consecutively focused on geometry (in particular, The scope of the book is already ambitious. reemerging connections to physics and Schwarz- Chronologically, it starts well before the arrival schild’s paper of 1900); then analysis, algebra, of “modernism”: There are brief essays on Monge, and logic. With hindsight, the central modernizing Poncelet, and projective geometry, followed by figure appears in the person of Georg Cantor. Es- a discussion of Lobachevsky, Bolyai, and non- pecially interesting for me were pages discussing religious overtones of Cantor’s transcendental Yuri Manin holds the Trustee Chair and is professor of mathematics at Northwestern University and professor flights in the infinity of infinities and the related emeritus at the Max-Planck Institut für Mathematik in subsection on “Catholic Modernism”. (As I have Bonn, Germany. His email address is manin@mpim-bonn. written elsewhere, I discern a subtle mockery in mpg.de. the famous and often quoted sentence of Hilbert FEBRUARY 2010 NOTICES OF THE AMS 239 about “Cantor’s Paradise”, usually perceived only it turned out that Pappus’s Theorem, now taken as unconditional support for set theory). as an axiom, is a necessary and sufficient condi- The subsequent chapters are called “Modernism tion for the possibility of introducing projective Avowed”, “Faces of Mathematics”, “Mathematics, coordinates in P with values in an abstract field (at Language and Psychology”, and “After the War” least, if P is infinite). Several more decades elapsed, and are bursting with information, insight, and and it was discovered that this latter statement can scholarship. As far as I know, nobody before be vastly generalized in the theory of models of Jeremy Gray discussed in this context “Popular- formal languages: “Zariski geometries” of Zilber izing Mathematics around 1900” (section 5.4), or and Hrushovski describe when abstract sets with “Languages Natural and Artificial” (section 6.1). subsets satisfying certain combinatorial require- ments can be conceived as sets of solutions of Constrained by the restrictions of space, I will arbitrary polynomial equations over a field. end this brief survey of the detailed contents of A sine qua non condition of such permanency, this remarkable book and turn to the discussion continuity of preoccupation, is the creation and of several of its grand themes, some- maintenance of a very safe, reliable, times underlying the exposition and intergenerational flow of informa- sometimes explicitly invoked in it. tion. Linguistic resources of math- Mathematics is a product of civi- ematics by which this information lization: it is discovered or created is encoded and transferred are ar- (see discussion below) by a very guably much more variable, fickle, limited number of human beings in subject to sweeping winds of history, each generation, then encoded in a than the content of the information mixture of words, formulas, and pic- itself. tures (nowadays software and hard- The whole bulk of discoveries ware might be added to this list), and of one generation, or one of its ac- in this way transmitted to the next tive but geographically restricted generation, which continues this subgroups, might in a few decades process. To estimate the number of become almost incomprehensible persons creating mathematics these to another generation, unless new days, one can refer to the attendance expressive means, with their new of International Congresses of Math- intuitions and hygienic rules, come ematicians that are held every four to the rescue. The old arguments years: recently it was about three to five thousand. are translated into a new language, rewritten, Undoubtedly, this should be viewed as a great purged of perceived obscurities and errors, in the explosion in comparison with previous centuries. process of mastering old results and moving to In Newton’s time the respective number probably new discoveries. did not exceed a few dozen. My professional youth was spent in this exhila- One remarkable feature of mathematical knowl- rating process of assimilation of glorious Italian edge is this: we learn more and more about the algebraic geometry and recasting it using schemes, same objects that ancient mathematicians already coherent sheaves, and homological algebra of the started to see with their mental eyes: integers and emerging “new age” algebraic geometry, which in prime numbers, real numbers, polynomial equa- two to three decades was to become one of the tions in one or many variables, space and various powerful vehicles not only in pure mathematics space forms… To illustrate this statement, I will but also in theoretical physics and quantum field mention just one string of events. Some time theory. Perhaps for this personal reason, I became around 300 BC, the Hellenistic scholar Pappus dis- perceptive of similar events that took place in covered a remarkable and, in the Euclidean context, other times and locations. quite unusual theorem (see the modernized state- From this perspective, “the modernist transfor- ment and the picture on page 463 of Gray’s book, mation of mathematics”, which is the subtitle of and beware of a small misprint: B A must be B A ). Jeremy Gray’s book, was one such periodically oc- Pappus’s Theorem does not conform to the spirit of curring refurbishing of basic vocabulary, grammar, Euclidean geometry, because lengths, angles, and and, yes, aesthetic requirements for mathematical rigid plane motions are quite irrelevant here: only thought and mathematical texts. It took about points, lines, and incidence relations “a point x lies thirty years, and its outcomes dominated the next on a line X” play a role. In the nineteenth century it thirty years: all in all, just about the span of three was understood that Pappus’s Theorem character- generations. The fact that during the lives of these izes projective geometry of the real plane. When generations two world wars swept Europe (or, ac- an abstract projective plane—as a set of points P , cording to some accounts, one Thirty Years’ War with a set of subsets l ⊂ P called lines—was intro- 1914–1945) influenced mathematics and the wider duced at the beginning of the twentieth century, culture in many ways. 240 NOTICES OF THE AMS VOLUME 57, NUMBER 2 The Bourbaki group formed itself in the 1930s is first introduced as “a well established theme in in a conscious attempt to fill the void left by the writing about modernism”. Later this theme is de- “lost generation”: not in the somewhat egocentric veloped in the section 4.8, “Anxiety”, already fully sense of the phrase coined by Gertrude Stein, who in the context of the mathematics of the first half referred to bohemian American émigrés in Paris, of the twentieth century, against the background but a real and painful void. About 40 percent of of the “crisis of foundations”. French mathematicians died in the first world war. Is it conceivable that Gray’s whole discourse on Although my argument posits “the modernist modernism in mathematics is informed by exis- transformation” as one of the series of structur- tential anxiety and that the last section, 7.5, “The ally similar transformations that occurred in other Work Is Done”, is as self–referential as it sounds? times and places, both the choice of the time span Of course it is, as the present author willingly and the use of the adjective “modernist” for it testifies, relying upon his own experience, and seem to me very felicitous. After formulating what as Jeremy Gray himself hints in the last lines of he sees as characteristic of modernist aesthetics, page 4. creative psychology, the relationship with society, But before bidding farewell to the reader, let’s and self–perception, shared by, say, painting and brace ourselves and try to face the Ghost question mathematics of that period, Jeremy Gray himself “What then?”, which I will interpret in the most warns that this looks like a case of “convergent prosaic way, as a question about the direction evolution” (page 8) rather than an effect of “com- in which self–consciousness of mathematicians mon genes”. moved during the last few decades, separated In his own sober words: “I am not sure there by more than a half–century from the 1930s and is more than a resonance that links mathematical 1940s where the detailed analysis of Gray stops. modernism to the arrival of modernity, modern So far as I can judge, “Platonism” of working capitalism, and its horrific opponent, the Nazi mathematicians is based on a feeling that impor- state. We are far from knowing much about the tant mathematical facts are discoveries rather than societies we inhabit.” inventions. In a book largely dedicated to the history and The Bering Strait was named after Vitus Ber- philosophy of mathematics, Plato’s name might ing who discovered it, whereas the diesel engine have been invoked in many contexts, but the was named after Rudolf Diesel who invented the first that comes to mind, that of “Mathematical engine. What about Galois groups? If you feel that Platonism”, is relegated to the very end of the dis- they were discovered by Évariste Galois, rather cussion, section 7.3, pages 440–452. Besides this than invented by him, you are in a sense a Platonist. section 7.3, Jeremy Gray, according to the Index, I will call such an attitude emotional Platonism in mentions Plato or Platonism on twelve more pages order to stress that (in my view) it is intellectually scattered throughout the text. By contrast, more indefensible, but not to the least degree invalidated than fifty entries in the Index invoke Poincaré and by this fact, since our emotions happily resist ra- only slightly fewer, Bertrand Russell. tional arguments. Why, then, is this monograph called “Plato’s Being such an emotional Platonist myself, I do Ghost” ? The explanation is on the very first page, not want to say that all mathematics is a discovery which quotes in its entirety Yeats’s “What Then?”, of a Platonic world, whatever that could mean. Cer- a poem of four five-line stanzas about an accom- tainly, the history of mathematics is also marked plished life of an accomplished human being: by inventions of marvelous intellectual telescopes and efficient vehicles, allowing people to travel Everything he wrote was read, from one discovery to another. Moreover, there After certain years he won are mathematicians whose oeuvre deserves com- Sufficient money for his need; parison to the Odyssey rather than to Columbus’s Friends that have been friends indeed; travelogue, insofar as mathematics is exteriorized “What then?” sang Plato’s ghost, “What in texts forming a part of the much vaster general then?” culture of the written word. The most direct reading of these stanzas sug- Elaborating the latter metaphor, one can repre- gests to me a simple and universal human emo- sent the history of the “modernist transformation tion, existential anxiety, as beautifully expressed of mathematics” simply as a tale about the birth by Yeats as by many others before and after him. and development of a certain style of thought, Jeremy Gray himself hears in Plato’s voice denial expression, and teaching, starting with Georg Can- of a claim to perfection (page 14). This interpre- tor and culminating with Bourbaki. Much will have tation is readily accommodated in a chronicle of to be left out of such a tale, but it could serve as (para)philosophical controversies around math- a good starting point for guessing “What then?” ematics. But, significantly, the word “anxious” This style is formed around a system of pretty appears already in the first paragraph of the Intro- explicit prescriptions: how a mathematician is sup- duction, and “anxiety” pops up on page 4, where it posed to introduce his or her object of study (“what FEBRUARY 2010 NOTICES OF THE AMS 241 I am talking about”), how his or her results should mathematical logic and botched the respective be stated (“what I am saying”), and finally how one chapters of his Treatise. As a result, the basic new should write arguments convincing her/himself discoveries in logic, the Turing–Church Thesis and and potential readers that the stated results are the theories of computability and of complexity, correct (“proofs”). suffered because their ties to the mainstream Briefly, the object of study (group, space with mathematics were loosened. Fortunately, such measure, topological space…) is introduced by a researchers as A. N. Kolmogorov helped tighten definition, presenting it as a Bourbaki–style struc- these ties. ture, which in turn is a collection of Cantorian sets What we see now in the flow of mathematical satisfying certain relationships stated in terms research can be termed loosely as a “post-modern” of the basic relation “being an element of” and period, but the term can be taken literally only if standard logical means. The results are stated as we adopt Jeremy Gray’s concept of “modernism”. theorems, statements of the type “if a structure has Contemporary mathematics bears no traces or a certain property P , then it has another property connotations of Lyotard’s description of the post- Q as well” etc. Finally, proofs are texts written in a modern condition, whose main characteristic is mixture of natural language, formulas, and some- the loss of credibility of all grand narratives. The times pictures (although the latter are considered grand narrative of mathematics is steadily develop- “bad taste” in the Bourbaki aesthetics). Such a text ing, reaching new depths and new sophistication. can be considered as a valid proof only if in prin- One trend that visibly changes the face of set ciple it can be replaced by a sequence of statements theory as the foundational language of mathemat- such that each term of this sequence is valid either ics is the current popularity of categories (func- by definition, or can be obtained by an elementary tors, enriched categories, polycategories…) as the logical step from previously obtained statements. dialect in which basic definitions of mathematical There are several more or less hidden or explicit objects are formulated. Another related, but not sources of self–referentiality in this picture, of identical, trend is the evolution of homotopy which I will mention two. theory, which has gradually become a kind of new One is that the notion of mathematical reason- language for postmodern mathematics (for an ing invoked in the previous paragraphs can itself initiated reader, I can mention “brave new rings”). be rigidified to become a mathematical structure. Briefly, these two trends together revolutionize This was of course the main discovery of the our collective vision of both semantics and syntax formalist program; as soon as it had crystallized, of the language(s) of mathematics. Gödel’s and Tarski’s theorems formalizing the First, basic new objects “category” (up to equiva- “liar’s paradox” or Cantor’s diagonal argument lence) and “homotopy type” are not Bourbaki became inevitable. structures: they are formed by a class of Bourbaki Another source of self–referentiality is less for- structures that are related by certain equivalence mal. Namely, a “proof” presented in a mathemati- relations. Both “class” and “relation” in this state- cal paper must convince the reader not just that ment can be represented set–theoretically by “in- the stated theorem is true, but that it by itself is a definitely large” collections of sets. PROOF, the incarnation of an ideal object residing Accepting this, a working mathematician not in the Platonic territory of formalized mathemat- only sheds the primeval horror of large infinities ics. Severe rules of hygiene are imposed upon but opens his/her imagination to a radically new vi- mathematical exposition in order to ensure this. sion even of common objects. For example, natural The reader must be alerted even to the occurrence numbers “simply” count things (or cardinalities of of an “empty proof” (empty in the set–theoretical finite sets), but the history of mathematics teaches sense), as Edmund Landau used to practice with us how late zero and negative integers were in- his inimitable “Beweis: Klar. (Proof: Clear.)”. troduced and accepted. External references such The imposition of the hygienic restrictions was as the notion of “debt” in trade were necessary in a part of the reaction to the perceived crisis of order to legitimate negatives. Nowadays integers foundations, but in the real life of mathematics are homotopy classes of pointed loops in a real of the twentieth century it played another, and plane with a deleted point: this is a germ of the idea probably unexpected, role: that of unification of of “brave new rings”. Even more generally, “small many diverse fields of mathematical studies. Alge- sets” that in the Cantor/Bourbaki paradigms could braists, analysts, geometers, probabilists, number be turned into topological spaces by imposing theorists, logicians could now speak in the same an additional structure now become secondary/ language, even when speaking about different derived objects: say π0 of a homotopy type. The structures. traditional view “continuous from discrete” gives Logicists stubbornly struggled in the losing way to the inverted paradigm: “discrete from battle against this loss of their dominance as continuous”. Keepers of Foundations. Nicolas Bourbaki did not Second, the notion of formal language that really understand the structures emerging in new crystallized in the 1930s as a purified written 242 NOTICES OF THE AMS VOLUME 57, NUMBER 2 form of natural languages, after it became treated of knots”, from the physical intuition related to, as a Bourbaki structure, could be vastly extended. say, “topological quantum field theories” and the Seemingly, there are other Bourbaki structures (or respective formalism of path integrals. These better, categories) that have “language–like proper- facts could afterwards be investigated and put on ties”, such as categories of graphs. The intuition firm ground by mathematicians: we got “quantum behind considering, say, a directed graph as a topology”, “quantum cohomology” (from string potential linguistic entity is that of “flowcharts”. theory), and much more. Generally, computer science, that no–nonsense If you search in the arXiv, MathSciNet, and child of mathematical logic, will exert growing Google for the terms I put in quotation marks influence on our thinking about the languages by above, you will find oceans of information about which we express our vision of mathematics. this stuff. Finally, I want to discuss briefly the collabo- At this point, Plato’s apparition intervenes again ration of mathematicians and physicists, which and sings “What then?” became dormant during the several decades of And I respond: “For us, there is only the trying. 1 “mathematical modernism” but renewed with new The rest is not our business.” vigor after 1950s and 1960s. I will primarily stress References the benefits of this collaboration for mathematics Yuri Manin and will describe these benefits as the emergence [1] , Georg Cantor and his heritage, in Math- ematics as Metaphor, Amer. Math. Soc., Providence, of a vast research program that could be called RI, 2007, pp. 45–54. “quantization of classical mathematics”. [2] Vladimir Tasic´, Mathematics and the Roots of Post- This program is historically related to the fact modern Thought, Oxford University Press, 2001. that when physicists started to see quantum [3] Philippe Grotard, Géométrie Arithmétique et Géomé- phenomena as the basic natural causes underly- trie Quantique, de Gauss à Grothendieck et de Witten à ing observable classical behavior of matter and Kontsevich, manuscript dated May 22, 2002, 540 pp. fields, they had to gradually discover what kind of changes must be made in order to proceed from a known mathematical description of a classical 1 T. S. Eliot, Four Quartets. system to a new, quantum description (“quantiza- tion”), and back (“classical approximation”). Some of the discovered prescriptions, such as “deforma- tion quantization” involving Planck’s constant as a small parameter, turned out to be much more universal mathematically than suggested just by their initial uses. ESFMathematicsConferenceinPartnershipwithEMSandERCOM Another prescription stressing quantum observ- TeichmüllerTheoryanditsInteractions ables as operators in (generally infinite–dimen- sional) Hilbert spaces led to “non–commutative inMathematicsandPhysics geometry”, one germ of which was the Heisenberg 28JuneͲ3July2010,Bellaterra,Spain commutation relation, and later to “quantum www.esf.org/conferences/10321 groups”. This ESFͲEMSͲERCOM Conference aims to highlight some of the most Philosophical problems related to the changed important advances in Teichmüller theory. This theory will be considered both from the geometric point of view (Thurston's theory and its role of observation and measurement led to dis- ramifications)andfromtheanalyticpointofview(theAhlforsͲBerstheory cussions of “quantum logic” and later, in a more anditsramifications).Therelationwith physicswillalsobeemphasized. applied vein, “quantum computation”. Conferencetopicswillinclude:ͲWeilͲPeterssongeometryandothermetric In the 1940s a development started that pro- structures;Ͳmapping class groups and their duced some mathematical miracles. Richard Feyn- representation theory;Ͳrigidity theory;ͲinfiniteͲ dimensional Teichmüller spaces;Ͳrelations with man developed path integrals, a notion that is dynamicalsystems;Ͳrelationswithnumbertheory highly intuitively appealing though mathemati- and probability theory;Ͳinvariants of 3Ͳ and 4Ͳ cally vague, as well as the powerful machinery of manifolds;Ͳmoduli spaces of flat connections;Ͳ clusteralgebrasandquantization. Feynman diagrams, which are well understood Inadditiontospecializedtalks,therewillbeseveral mathematically but are motivated only by the survey talks given by leading experts in the field. idea that they somehow capture path integrals. It Youngresearchersareparticularlyencouragedtoparticipate,andgraduate was a great success in quantum field theory and studentsinthefieldarealsowelcome. elementary particles, but nobody could foresee Applicationform&programmeavailablefrom how, in mathematics, it would backfire. www.esf.org/conferences/10321 This became reality after many physics papers Closingdateforapplications:5April2010 of Witten and his collaborators, which math- EuropeanScienceFoundationͲResearchConferencesUnit ematicians could interpret as rich and powerful 149avenueLouise,Box14ͲBrussels,Belgium heuristic tools to guess precise and striking new Tel:+32(0)25332020ͲFax:+32(0)25388486 Email:[email protected]Ͳwww.esf.org/conferences mathematical facts, such as “quantum invariants FEBRUARY 2010 NOTICES OF THE AMS 243 Book Review Solving Mathematical Problems: A Personal Perspective Reviewed by Loren Larson Solving Mathematical Problems: A Personal “love and delight are Perspective better teachers than Terence Tao compulsion.” Oxford University Press, September 2006 One means of US$34.99, 128 pages promoting prob- ISBN-13:978-0199205608 lem solving is by organizing math In 1980 Paul Halmos concluded his Monthly article contests, and there “The Heart of Mathematics” [1], with this thought: are now a variety of contests and plenty I do believe that problems are the of opportunities heart of mathematics, and I hope that for participation. as teachers, in the classroom, in semi- These contests are nars, and in the books and articles we supported by a vast write, we will emphasize them more literature of prepa- and more, and that we will train our ratory supplements: students to be better problem posers anthologies, compi- and problem solvers than we are. lations, how-to-solve Now, thirty years later, I think he would be books, special topics, problem-solving strategies, pleased with what has transpired. Problem solv- online classes, forums, summer math camps, ing is central in current mathematics education, and videos (e.g., see http://www. from kindergarten through middle school, from artofproblemsolving.com). Contests have a high school through college and beyond. Learning proven record for fostering interest in mathemat- mathematics means doing mathematics, a mix of ics, but they aren’t for everyone. Some students are turned off by competition, or have mathematical practice exercises to develop skills and problems interests that aren’t particularly amenable to a for deeper understanding. The proportion of each contest format, or feel that contests favor speed depends on the context, anything from 100/0 for over power (not that they can’t coexist). But there crash courses where the problems will come later are other ways of promoting problem solving and (in more than one sense?) to 0/100 for Moore- personal involvement, such as problem seminars, method courses where mastering techniques will independent research, and interdisciplinary proj- come later. Both are necessary, but from a peda- ects. gogical point of view, appropriately chosen prob- lems under the right conditions are more fun and Finding suitable problems is a time-intensive offer more satisfaction, at least for mathematically task. Problem books can provide ideas and starting inclined students. To paraphrase Albrecht Dürer, points, for example nontrivial tweaks or special cases of published results. But this approach can Loren Larson is professor of mathematics at St. Olaf lead to stagnation and irrelevance. A better source College. His email address is [email protected]. for keeping ideas meaningful and up-to-date is the 244 NOTICES OF THE AMS VOLUME 57, NUMBER 2 research community—through personal communi- it can be safely put into a high school cation, networking, newsgroups, problem sections of curriculum. However, a few gems can professional journals, and journal articles. Prob- still be found here and there. lems for high-level math competitions not only need to be original, but they should also be pre- Number Theory: Unlike algebra, which sented with a certain flair. Composers of such has as its backbone the laws of manipu- problems have to indulge their feel for language, lating equations, number theory seems their artistic temperament, and even their sense of to derive its results from a source un- humor. It’s in the best interest of our community to known. Take, for example, Lagrange’s encourage this kind of effort, as getting students Theorem, … . Number theory is a fun- interested in current research through problems is damental cornerstone which supports an important means of renewal. Paul Erdo˝s was a a sizeable chunk of mathematics. master at introducing rich mathematical ideas by way of simply stated problems. Putnam problems Analysis: Analysis is the study of func- have stayed fresh over the years because of contri- tions and their properties. …Functional butions from researchers and inveterate problem equations form a sort of “pocket math- enthusiasts. A book by Paul and Judith Sally [4] ematics”, where instead of the three is helpful in showing how a single mathematical dozen or so axioms and countless concept can sometimes be adapted and framed thousands of theorems, one has only into an interesting problem at virtually every level a handful of “axioms” (i.e., data) to use of mathematical expertise. and there is a clear direction in which to Terence Tao, the author of the book under go. And yet, it still has surprises. review, is one of math’s luminaries, a winner of the Fields Medal in 2006. If you aren’t familiar Geometry: The true beauty of geometry with his prodigious accomplishments and life as is in how a non-obvious-looking fact a child prodigy, by all means check out his amaz- can be shown to be undeniably true ing story. This book was written when he was just by repeated application of obvious fifteen years old, but even by this age he was well- facts. As an example: the midpoints qualified for the task: he had already competed of the four sides of a quadrilateral al- in three International Mathematical Olympiads, ways make up a parallelogram. These winning a bronze medal, a silver medal, and then a gold medal just days after his thirteenth birthday. facts—they have a certain something In the narrowest sense, the book is aimed at about them. showing high school students how to solve Math- Statements like these have to be put into con- Olympiad-like problems. Almost all the problems text. Remember that his target reader is someone are taken from published collections of problem like himself at age nine, intensely eager and ca- sets for mathematics competitions. In the Preface pable and focused on learning as much as possible to this second edition, the mature Tao at age thirty about Olympiad-level math problems. A certain comments that if he were to write a problems book panache and dash, even some exaggeration and now it would be very different. But he resisted eccentricity, is expected at this age. I was all ears, a the temptation to tamper with it except for a few teenager again, fired-up and flattered to be treated organizational changes, noting that “his younger as an equal. His intentions for the book are laid self was almost certainly more attuned to the world out in the Preface: of the high-school problem solver.” Indeed, one of the most charming features of the book is the I will try to demonstrate some tricks of exuberant exposition, which the elder Tao points the trade when problem solving. Two out “[sometimes] has a certain innocence, or even of the main weapons—experience and naiveté .” Here are some delightful excerpts taken knowledge—are not easy to put into from chapter previews (and if you’ve worked with a book: they have to be acquired over precocious young people, you’ll recognize the time. But there are many simpler tricks voice): that take less time to learn. There are ways of looking at a problem that make Algebra: Algebra is the basic founda- it easier to find a feasible attack plan. tion of a large part of applied math- There are systematic ways of reduc- ematics. Problems of mechanics, ing a problem into successively easier economics, chemistry, electronics, op- subproblems. timization, and so on are answered by algebra and differential calculus, which Math “how-to” books usually focus on specific is an advanced form of algebra. In fact, applications of methods given by George Pólya [3], algebra is so important that most of and this book is no exception—the intention is to its secrets have been discovered—so motivate the solution. But the manner in which Tao FEBRUARY 2010 NOTICES OF THE AMS 245 does it is the book’s most distinguishing feature. problem are exchanged with more natu- For not only does Tao motivate the solution but ral, flexible, and cooperative methods. he also walks through his entire thinking process, including rejected ideas and false starts. He seems It is best to try elementary techniques to be thinking aloud, explaining as he goes—each first, as it may save a lot of dashing journey an apparently effortless flow of ideas. It’s about in circles later. a slim book, about 100 pages, but the problems are well chosen, only twenty-six of them, with As long as one always tries to simplify about the same number of instructive exercises, and connect, chances are that the solu- without solutions. The solutions to the problems tion will soon fall into place. (Assum- are almost incidental to Tao’s observations and ing, of course, that there is one—and reformulations along the way. He shows us how most problems are not trying to pull to think like mathematicians. For example, the your leg.) actual solution to Problem 2.2 (Is there a power of 2 whose digits could be rearranged and made into Here’s an example of how it plays out: One of the another power of 2?) takes only six lines, but it is geometry problems asks you to prove that either preceded by five pages of discussion. the given triangle is isosceles or a particular angle The defining characteristic of Tao’s perspective is 60◦ . Tao quickly rules out coordinate geometry: on problem solving is based on Pólya’s dictum, “Hack-and-slash coordinate geometry is one long nicely stated by Halmos [1]: and boring way that is prone to abysmal complica- Make it easier. In slightly greater detail: tions and huge errors. Let us try that as a last re- if you cannot solve a problem, then sort.” Realizing that the conclusion involves angles there is an easier problem that you can- and the given data involves equal line segments, not solve, and your first job is to find he needs results that relate length and angle. So it! Make it sharp. By that I mean: do not he writes down relevant facts about right triangles, insist immediately on asking the natu- isosceles triangles, the law of cosines, the law of ral question (“what is…?”, “when is…?”, sines. There aren’t a lot of right triangles in the “how much is…?”), but focus first on an figure so let’s not create any yet. The equal line easy (but nontrivial) yes-or-no question segments aren’t part of an isosceles triangle that (“is it…?”). directly connects to the conclusion, so we’ll pass on that for now. He continues: “The law of cosines Essentially, this approach is analogous to the usually complicates rather than simplifies, and it Bolzano-Weierstrass Method for catching a lion just creates more unknown lengths. This leaves in the desert [2]: Repeatedly bisect the area with only the law of sines as a feasible alternative.” This alternating vertical and horizontal lines, choosing leads to equations, then to further manipulations the half that contains the lion. At each stage, build and reformulations and simplifications driven by a fence. The lion is ultimately enclosed by a fence the overall objective, and finally to the relevant of arbitrarily small perimeter. angles. In other words, first you have to survey the Granted, this step-by-step reduction is particu- desert, that is, understand the problem and write larly effective for Olympiad-like problems, and Tao down everything you can think of that might be relevant. This is especially important in problems acknowledges this in the original Preface: at this level because the most obvious beginnings [Olympiad-like] mathematics problems probably won’t work. Then, with this list in hand, are “sanitized” mathematics, where you start bisecting, eliminating ideas that appear to an elegant solution has already been make things more difficult. Iterate on this process, found, the question is stripped of systematically asking questions and reducing it all superfluousness and posed in an into easier subproblems until the tricky parts are interesting and (hopefully) thought- no longer tricky. “Make it easy” is the theme of the provoking way. If mathematics is lik- book, which also makes it fun to read. Expressed ened to prospecting for gold, solving more flamboyantly by Tao: a good mathematics problem is akin So that is it. We keep reducing the equa- to a “hide-and-seek” course in gold- tion into simpler and simpler formulas, prospecting: you are given a nugget until it just collapses into nothing. A bit to find, and you know what it looks of a long haul, but sometimes it is the like, that it is out there somewhere, only way to resolve these very compli- that it is not too hard to reach, that its cated questions: step by step reduction. unearthing is within your capabilities, and you have conveniently been given Simplify repeatedly until the more the right equipment (i.e., data) to get unusable and unfriendly parts of the it. It may be hidden in a cunning place, 246 NOTICES OF THE AMS VOLUME 57, NUMBER 2 New textbooks from A K PETERS but it will require ingenuity rather than digging to get it. Understanding Real Analysis Contest problems provide a setting in which to Paul Zorn acquire the habit of thinking mathematically. Here the psychological advantage makes it ideally suited 978-1-56881-415-5; $49.00; 292 pp. for developing a way of thinking that will transfer This book introduces the to “unsanitized” mathematics, the habit of doing rigorous study of real what you can and thinking ahead about how the numbers and real-valued complicated parts might be simplified. The book assumes that the reader has a strong functions. Written in a background in basic arithmetic and algebra (modu- clear and approachable lar arithmetic, factorization formulas, elementary style, this text’s narrative, trig identities), standard Euclidean geometry, problems, and selected including elementary vector methods, and prop- solutions help students erties of polynomials and functions, as well as build both understanding and familiarity mathematical induction. If Tao were fifteen today his book would probably have included more top- with mathematical language and ways of ics from discrete math, such as the pigeonhole thinking. Intuitively familiar objects from principle, elementary graph theory (Euler’s formula elementary calculus—sets, functions, limits, V − E + F = 2, Euler and Hamiltonian circuits), derivatives, and integrals—are revisited and combinatorics (counting principles, recur- in the language of formal mathematics; rences). Nevertheless, the overall problem-solving approach would be the same, and it is largely this informally known facts and theorems are feature, along with the thorough, caring, and ener- proved rigorously and systematically. gizing delivery, that makes the book a noteworthy addition to the literature on problem solving and Differential Geometry of Curves how to teach it. and Surfaces References Thomas Banchoff, Steve Lovett [1] P. R. Halmos, The heart of mathematics, The Ameri- 978-1-56881-456-8; $49.00; approx. 300 pp. can Mathematical Monthly 87 (7, Aug.-Sept.) (1980), 519–24. This textbook covers a one-semester [2] H. Pétard, A contribution to the mathematical theory undergraduate course in the differential of big game hunting, The American Mathematical geometry of curves and surfaces, assuming Monthly, 45 (1938), 446–7. [3] G. Pólya, How to Solve It, 2nd Edition, Princeton Uni- only multivariable calculus and linear versity Press, 1957. algebra. The book culminates with the [4] J. D. Sally and P. J. Sally Jr., Roots to Research: A celebrated Gauss-Bonnet Theorem and Vertical Development of Mathematical Problems, AMS, applications to spherical and hyperbolic 2007. geometry. Online interactive computer graphics applets coordinated with each section form an integral part of the exposition. The applets allow teachers and students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena. Each section includes numerous interesting exercises that range from straightforward to challenging. Differential Geometry of Manifolds Steve Lovett COMING SOON Request Examination Copies: [email protected] A K PETERS 508.651.0887 www.akpeters.com FEBRUARY 2010 NOTICES OF THE AMS 247 Simons Foundation Launches US$40-Million Program for Theoretical Research Allyn Jackson The Simons Foundation has launched a program more time on his philanthropic work, which also that is now spending an estimated US$40 million a includes support for autism research. year to support research in mathematics and the- “The mission of the Simons Foundation is to oretical aspects of areas related to mathematics. support basic scientific research,” said James Si- The program began in fall 2009, creating seventy mons in an email message. “There is already an postdocs intended to ease the effects of the tight active program in the life sciences, and several im- academic job market. portant but isolated activities in mathematics and In mathematics, US$40 million is a serious the physical sciences, such as the newly founded chunk of change: For comparison, the yearly bud- Center for Geometry and Physics, but there is no get of the Division of Mathematical Sciences at the coherently articulated program in the latter area. National Science Foundation (NSF) was US$226 mil- David Eisenbud was brought on to establish such lion in fiscal year 2009. The Simons Foundation a program, and over the next several months he program will not be solely devoted to mathe- will be gathering ideas and advice to guide him in matics but will also fund “theoretical science carrying out this mission.” (For more information radiating from mathematics”, as David Eisenbud on the center, see “Major Gift to Stony Brook for put it. Eisenbud, a professor at the University Simons Center in Geometry and Physics”, Notices, of California Berkeley and former director of the June/July 2008.) Mathematical Sciences Research Institute (MSRI), is The new program does not have a name yet—nor heading up the program as the Simons Foundation has it been decided exactly how the US$40 million Vice President for Mathematics and the Physical will be used. But the driving idea is to “strengthen Sciences. basic research internationally and across science”, James Simons, of Chern-Simons invariant fame, Eisenbud said. The plan is to structure the pro- has made billions through his investment com- gram so that it complements rather than duplicates pany, Renaissance Technologies. In 1994 he and his modes of support already available through exist- wife Marilyn established the Simons Foundation, ing funders of theoretical research, such as the which supports basic science and mathematics. NSF. Private sources of support for mathemat- The Simons fortune also funds Math for America, ics include the Clay Mathematics Institute, which a separate entity from the Simons Foundation provides funding for individual mathematicians that focuses on attracting and retaining out- and specific projects like conferences, and the standing individuals to teach mathematics in electronics-chain-store magnate John Fry, who is public secondary schools. In 2009 James Si- the main funder of the American Institute of mons, seventy-one years old, announced he is Mathematics in Palo Alto, California. The Simons retiring from day-to-day management of the firm Foundation program will be “substantially larger” he has run for thirty-one years and will spend than what these other private funders provide, Eisenbud said. Allyn Jackson is senior writer and deputy editor of the As a first step in establishing the program, Notices. Her email address is [email protected]. the foundation decided to fund about seventy 248 Notices of the AMS Volume 57, Number 2 postdoctoral positions, called Simons Postdoc- continue to fund postdocs? That is not clear yet— toral Fellowships. Fifteen three-year positions in and indeed only a few aspects of the program have mathematics and ten three-year positions in the- been decided upon. One of these is the areas to be oretical physics will be funded this year and next; supported: mathematics and theoretical topics in in theoretical computer science, nine two-year po- areas connected to it, such as theoretical physics sitions will be offered for three consecutive years. and theoretical computer science (but not the bio- According to Eisenbud, the breakdown of posi- logical sciences, which are the focus of a different tions was dictated by a sense of the “size of the Simons Foundation program). Support for large enterprise” in each area, partly measured by the experimental facilities is not within the scope of number of new doctorates in each. With about the program, but Eisenbud said that it might be 1,400 new doctorates last year, mathematics had possible to fund some experimental projects “if by far the largest number. The funding will be the right project comes along”. Researchers from given to university departments, each of which anywhere in the world are eligible for support. will be able to hire one or two postdocs recruited Unlike grants from the NSF, which pay overhead from anywhere in the world. Recruitment began in rates set by the institutions receiving the grants, fall 2009, and the positions will start in fall 2010. the grants from the Simons Foundation program The Simons Foundation has decided not to will pay only the foundation’s standard overhead announce publicly the full list of departments rate of 20 percent of certain direct costs. receiving the postdoc funds. The selection of Exactly what kinds of activities the program will mathematics departments was made by a five- fund has yet to be determined. Asked whether the member committee of “distinguished mathemati- money might be used for a new institute, Eisen- cians”, Eisenbud said; parallel committees in bud said that he thought it unlikely, given that theoretical physics and theoretical computer sci- the foundation just recently launched the Simons Center. However, the foundation has been giving ence made the selections in those areas. The funds to enhance activities at some institutes, and awards were made to departments that can nur- this might continue. How about a small grants pro- ture postdocs well; they were “not seen as rewards gram, which many mathematicians in the United to great departments,” Eisenbud noted. The de- States have said is what they really need? That is a partments were chosen on the basis of “whether possibility. “Nothing is settled yet,” Eisenbud said. the committee felt that people in a reasonably “Figuring out something really useful and really broad range of subjects would advise their grad- effective to do, even with a lot of money, is, uate students to go to those departments,” he I think, not so easy,” Eisenbud remarked. To explained. this end the Simons Foundation will in coming These postdoctoral positions are sorely needed months hold several roundtable events in which in the job market facing new and recent Ph.D.s, mathematicians and scientists “will offer us, we a market that Eisenbud described as “grue- hope, sage advice about how to spend this money,” some”. “The number of postdoctoral positions Eisenbud said. “We will listen hard before we make is down at research universities” by as much any decisions.” The foundation will also work as one-half, he said. “This is something we will closely with organizations such as the NSF and feel for years.” A survey conducted in early the Clay Mathematics Institute to ensure that the 2009 by an AMS task force on employment new program complements rather than duplicates estimated that the number of academic positions existing ones. open to new doctorates had declined by 39 Eisenbud said he would be interested in hear- percent from summer 2008 to summer 2009 ing from researchers who have ideas for how (see http://www.ams.org/prof-services/ best to use the funds (his email address is employtaskforce/ETF.html for the task force [email protected]). It is not appropri- report). The NSF pitched in to help in 2009 with a ate at this stage to send proposals for specific set of temporary postdoctoral positions managed research projects; what is needed at this time are jointly by the mathematical sciences research ideas about what kind of grant mechanisms would institutes. These postdocs will probably not be be most effective. Input from a broad segment of repeated in 2010, absent the flexibility provided by the mathematical community can help ensure that funds from the Obama administration’s one-time this unusual program has a lasting and positive stimulus package. impact on the field. The Simons Foundation decided to give the funds for the postdocs to departments, rather than to individuals, because there was not time to set up a mechanism for reviewing individual applications. In the future, if it does fund more postdocs, it will have a mechanism for direct ap- plication by individuals. Will the program in fact February 2010 Notices of the AMS 249 2009 Annual Survey of the Mathematical Sciences (First Report) Preliminary Report on the 2008–2009 New Doctoral Recipients Polly Phipps, James W. Maxwell, and Colleen Rose This report presents a statistical profile of recipients of report. This year’s response rates were above 80% doctoral degrees awarded by departments in the mathematical for all groups except Group IV which was 74% sciences at universities in the United States during the period (up from 67% last year.) Overall, eighteen more July 1, 2008, through June 30, 2009. All information in the departments responded in time for the First Report report was provided over the summer and early fall of 2009 this year than responded in time for last year’s by the departments that awarded the degrees. The report First Report. Efforts continue to obtain data from includes a preliminary analysis of the fall 2009 employment as many of the non-responding departments as plans of 2008–09 doctoral recipients and a demographic possible. profile summarizing characteristics of citizenship status, gender, and racial/ethnic group. This preliminary report will Table 2: New Doctoral Degrees Awarded be updated in the Second Report of the 2009 Annual Survey by Group, Preliminary Count to reflect subsequent reports of additional 2008-2009 doctoral Group I (Pu) I (Pr) II III IV Va TOTAL recipients from the departments that did not respond in time for this report. No adjustments have been made to the 1999–00 256 157 223 132 284 67 1119 numbers in this report for the non-responding departments. 2000–01 233 129 203 125 237 81 1008 The Second Report, to appear in the August 2010 issue of 2001–02 218 139 164 124 222 81 948 Notices, will also reflect additional information provided 2002–03 258 138 170 121 239 91 1017 by the new doctoral recipients themselves, including their starting salaries. 2003–04 195 187 215 111 243 90 1041 Table 1 provides the number of departments responding 2004–05 243 146 203 153 285 86 1116 to the 2009 Survey of New Doctoral Recipients in time for this 2005–06 307 184 216 140 287 111 1245 2006–07 300 119 234 138 279 87 1157 Table 1: Number of Departments Responding to Doctorates Granted Survey 2007–08 234 172 290 142 291 106 1235 2008–09 326 211 282 171 352 88 1430 Group I (Pu) 23 of 25 including 0 with no degrees Group I (Pr) 21 of 23 including 0 with no degrees Group II 50 of 56 including 3 with no degrees Doctoral Degrees Granted in 2008–09 Group III 72 of 81 including 18 with no degrees Table 2 shows the number of new doctoral Group IV 68 of 92 including 4 with no degrees degrees granted by the different doctoral groups Statistics 45 of 57 including 2 with no degrees surveyed in the Annual Survey for the past ten years. Biostatistics 23 of 36 including 2 with no degrees The preliminary count of 1,430 new doctorates granted by these departments in 2008–09 is an Group Va 18 of 21 including 0 with no degrees increase of 195 from the preliminary count for See “Definitions of the Groups“ on page 258. 2007–08. From Table 2 we see that all groups except Polly Phipps is a senior research statistician with the Bureau of Labor Groups II and Va reported an increase in the number Statistics. James W. Maxwell is AMS associate executive director for special of doctoral recipients from the previous year. The projects. Colleen A. Rose is AMS survey analyst. 250 Notices of the AMS Volume 57, Number 2 2009 Annual Survey of the Mathematical Sciences in the U.S. decrease reported for Group II is almost certainly Highlights the result of the three departments in Group II that There were 1,430 new doctoral recipients reported for 2008-09 by responded in time for last year’s report but not this departments responding in time for the 2009 First Report. The year’s report. These departments have awarded number of departments responding in time for this year's report an average of 17 doctoral degrees each year over increased by 8 and in every group except Group IV. When one the past four years. The final count for 2008–09 is considers only the 208 departments that responded in time for likely to be very close to 1,500. the First Report in both years, the 2009 figure reported by these The 2008–09 numbers in Table 2 will be broken departments is up 9.7% over that reported for 2008. down in various ways, such as by gender, in later There were 669 U.S. citizens reported among this year's new sections of this report. The names of the 1,430 new doctoral recipients, 47% of the total. Last year's figure was 44%. doctoral recipients are found on pages 276–99 of The fall preliminary 2009 unemployment rate for the 1,231 new this issue of the Notices. doctoral recipients whose employment status is known is 7.9%, By way of background, additional information up from 5.4% for fall 2008. about various types of full-time graduate students Seventy-two new doctoral recipients hold positions at the is available in the Third Report of the 2008 Annual institution that granted their degree, although not necessarily in Survey (Notices, November 2008), Table 6B, page the same department. This is 6% of the new doctoral recipients 1299. who are currently known to have jobs and 10% of those who have academic positions in the U.S. Seventeen new doctoral recipients have part-time positions. Employment Plans of 2008–09 New Doctoral Recipients The number of new doctoral recipients employed in the U.S. is 987, up 101 from last year. The number of new doctoral recipients Tables 4A and 4B each provide a cross- employed in academic positions in the U.S. has increased to tabulation of the 1,430 new doctoral recipients in 741, compared to 650 last year. the mathematical sciences. These tables contain Of the 987 new doctoral recipients taking positions in the U.S., a wealth of information about these new doctoral 184 (19%) have jobs in business and industry, a decrease of 11% recipients, some of which will be discussed in this from last year's figure of 207. The fall 2009 number remains up report. Note that these tables give a breakdown by 69 (60%) from fall 2005. The number of new doctoral recipients gender for type of employer and type of degree- taking jobs in government is up 30 (107%) over fall 2008. granting department. Additional information is Among the 987 new doctoral recipients having employment in the available on the AMS website at www.ams.org/ U.S., 501 (51%) are U.S. citizens (up from 426 (48%) last year). employment/surveyreports.html. The number of non-U.S. citizens having employment in the U.S. The preliminary unemployment rate for these is 486; last year it was 460. data is 7.9%. This preliminary rate will be up- Among the 333 new doctoral recipients hired by U.S. doctoral- dated later with information gathered from the granting departments, 47% are U.S. citizens (down from 49% last year). Among the 408 having other academic positions in the Figure 1: New Doctoral Degrees Awarded U.S., 62% are U.S. citizens (up from 57% last year). by Combined Groups, Preliminary Count Of the 1,430 new doctoral recipients, 32% (462) are female, the same percentage reported in fall 2008. Of the 669 U.S. citizen I (Pu), I (Pr), II, III, & Va new doctoral recipients, 30% (202) are females, down from 31% I (Pu), I (Pr), & II in fall 2008. Among the 669 U.S. citizen new doctoral recipients, 4 are 1200 American Indian or Alaska Native, 44 are Asian, 19 are Black or 1100 African American, 19 are Hispanic or Latino, 1 is Native Hawaiian 1000 or Other Pacific Islander, 575 are White, and 7 are of unknown 900 race/ethnicity. 800 Group IV produced 352 new doctorates, of which 159 (45%) are 700 female, compared to all other groups combined, where 303 600 (28%) are female. In Group IV, 92 (26%) of the new doctoral recipients are U.S. citizens (while in the other groups 54% are 500 U.S. citizens). 400 Twenty-nine percent of the new doctorates had a dissertation in 300 Number of Degrees statistics/biostatistics (410). The next highest percentage was 200 in algebra and number theory with 14% (203). 100 0 The Faculty Salary Survey report, will appear in the March issue 1999-00 2007-08 2008-09 2000–01 2001–02 2002–03 2003–04 2004–05 2005–06 2006–07 of Notices. February 2010 Notices of the AMS 251 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 4A: Employment Status of 2008–09 New Doctoral Recipients in the Mathematical Sciences by Type of Degree-Granting Department TYPE OF DOCTORAL DEGREE-GRANTING DEPARTMENT Group I Group I Row (Public) (Private) Group II Group III Group IV Group Va Subtotals TYPE OF EMPLOYER Math. Math. Math. Math. Statistics Applied Math. TOTAL Male Female Group I (Public) 41 18 16 2 0 3 80 67 13 Group I (Private) 22 38 7 3 1 1 72 56 16 Group II 24 10 20 4 4 2 64 44 20 Group III 9 3 13 18 0 1 44 31 13 Group IV 0 0 3 2 54 1 60 37 23 Group Va 5 1 0 0 0 7 13 12 1 Master’s 11 3 19 10 5 2 50 30 20 Bachelor’s 34 16 59 30 16 5 160 92 68 Two-Year College 3 1 4 12 0 2 22 16 6 Other Academic Dept. 21 10 15 15 55 18 134 80 54 Research Institute/ 11 4 2 1 20 4 42 25 17 Other Nonprofit Government 7 1 10 13 27 4 62 35 27 Business and Industry 31 17 23 13 85 15 184 124 60 Non-U.S. Academic 38 37 22 9 17 6 129 99 30 Non-U.S. Nonacademic 4 3 2 1 3 2 15 10 5 Not Seeking Employment 3 4 2 1 5 0 15 11 4 Still Seeking Employment 22 13 18 18 9 5 85 64 21 Unknown (U.S.) 21 12 29 8 31 8 109 74 35 Unknown (non-U.S.)* 19 20 18 11 20 2 90 61 29 TOTAL 326 211 282 171 352 88 1430 968 462 Column Male 262 166 185 100 193 62 968 Subtotals Female 64 45 97 71 159 26 462 *Includes those whose status is reported as “unknown” or “still seeking employment”. Figure 2: Percentage of New Doctoral Recipients For further details, see the explanatory note on Unemployed in the U.S. unemployment rates at the end of the report. The unemployment rates, calculated by type of Preliminary Final doctoral degree-granting department using Table 20.0 4A, vary from group to group, with a high of 12.8% 18.0 for Group III and a low of 3.3% for Group IV. 16.0 There are 987 new doctoral recipients em- 14.0 ployed in the U.S. Table 5A gives a breakdown 12.0 10.0 of type of employer by type of degree-granting 8.0 department for these 987 new doctoral recipients. 6.0 Of these, 741 (75%) hold academic positions, 62 4.0 (6%) are employed by government, and 184 (19%) 2.0 Percent Unemployed hold positions in business and industry. In the 0.0 First Report for 2007–08, there were 886 new doctoral recipients employed in the U.S., of which 2006-07 2004-05 2008-09 2000–01 1996–97 1992–93 2002–03 1994–95 1998–99 649 (73%) held academic positions, 30 (3%) were in *Excludes those whose status is reported as “unknown” , “not seeking” or government, and 207 (23%) were in business and having employment outside the U.S. (Non-U.S. Academic or Non-U.S. industry. The number of new doctoral recipients Nonacademic). This is a change from prior reports which excluded only the unknown categories. employed in the U.S. increased in all categories except "Business and Industry" which decreased 11%. "Government" showed the largest percentage individual new doctoral recipients. The addi- increase, 107%. tional information from prior years is reflected Table 5B shows the number of new doctoral in the final unemployment rates displayed in recipients who took positions in business and Figure 2. The preliminary rate for fall 2008 was industry by the type of department granting 6.0%. The unemployment rates shown in Figure 2 their degree for fall 2005 to fall 2009. The num- differ from those given in previous Annual Survey ber of new doctoral recipients taking jobs in reports. The rates shown are now based on only business and industry, which had been steadily those 1,072 individuals in the U.S. labor market. increasing, decreased from 207 to 184, an 11% 252 Notices of the AMS Volume 57, Number 2 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 4B: Field of Thesis of 2008–09 New Doctoral Recipients by Type of Degree-Granting Department FIELD OF THESIS TYPE OF DOCTORAL Real, Comp., Discr. Math./ Numerical Linear Differential, DEGREE-GRANTING Algebra/ Funct., & Combin./ Analysis/ Nonlinear Integral, & Number Harmonic Geometry/ Logic/ Statistics/ Applied Approxi- Optim./ Difference Math. Other/ DEPARTMENT Theory Analysis Topology Comp. Sci. Probability Biostat. Math. mations Control Equations Educ. Unknown TOTAL Group I (Public) 77 29 56 34 23 11 40 21 2 31 1 1 326 Group I (Private) 52 10 53 24 19 3 22 7 1 16 0 4 211 Group II 57 30 28 32 19 12 42 27 6 28 1 0 282 Group III 16 22 9 20 4 32 16 16 4 21 11 0 171 Group IV 0 0 0 0 4 343 4 0 0 0 0 1 352 Group Va 1 1 0 12 8 9 36 12 5 4 0 0 88 Column Total 203 92 146 122 77 410 160 83 18 100 13 6 1430 Column Male 153 68 112 86 52 227 113 62 15 73 4 3 968 Subtotals Female 50 24 34 36 25 183 47 21 3 27 9 3 462 drop. Among the 987 new doctoral recipients department. This represents 6% of new doctoral known to have employment in the U.S. in fall 2009, recipients who are currently employed in the U.S. Group III has the smallest percentage taking jobs and 10% of the U.S. academic positions held by in business and industry at 11% and Group IV the new doctoral recipients. In fall 2008 there were 69 highest at 32%. such individuals making up 7% of the new doctoral Table 5C shows the number of new doctoral recipients who were employed at the time of the recipients who took academic positions in the U.S. First Report. Seventeen new doctoral recipients have by type of department granting their degree for fall taken part-time positions in fall 2009 compared 2005 to fall 2009. The total number of new doctoral with 18 in fall 2008. recipients taking academic employment in fall 2009 increased 14% to 741 from 649 last year. Among Information about 2008–09 Female the 987 new doctoral recipients employed in the New Doctoral Recipients U.S. in fall 2009, 75% have academic positions. This Tables 4A and 4B give the breakdown of the new percentage is highest for Group I (Pr) at 85% and doctoral recipients in 2008–09 by Field of Thesis, by lowest for Groups IV at 58%. Type of Degree-Granting Department and by Type Table 5D shows the number of positions filled of Employer. with new doctoral recipients for each type of Overall, 462 (32%) of the 1,430 new doctoral academic employer. Increases in positions filled by recipients in 2008–09 are female. In 2007–08, 388 new doctoral recipients were realized by all groups. (31%) of the new doctoral recipients were female. The biggest increase in hires of new doctorates into This percentage varies over the different groups, academic positions was in Groups IV and Va with and these percentages are given in the first row 43% and 44%, respectively. Hires of new doctorates of Table 5E. This year the percentage of females into positions at research institutes increased from produced is highest again for Group IV at 45%, 29 in fall 2008 to 42 in fall 2009. compared with 52% last year. The second row of In fall 2009, 72 new doctoral recipients held Table 5E gives the percentage of the new doctoral positions in the institution that granted their recipients hired who are female for each of the degree, although not necessarily in the same Groups I, II, III, IV, and Va. In addition, 40% of the Table 5A: 2008–09 New Doctoral Recipients Employed in Table 5B: Number of New Doctoral the U.S. by Type of Degree-Granting Department Recipients Taking Positions in Business and Group Industry in the U.S. by Type of Degree- Granting Department, Fall 2005 to Fall 2009 Type of Employer in U.S. I (Pu) I (Pr) II III IV Va TOTAL Group Groups I, II, III, IV, and Va 101 70 59 29 59 15 333 Master’s, Bachelor’s, and Year I (Pu) I (Pr) II III IV Va TOTAL 2-Year Colleges 48 20 82 52 21 9 232 Fall 2005 5 9 14 15 64 8 115 Other Academic and Fall 2006 27 14 19 9 80 18 167 Research Institutes 32 14 17 16 75 22 176 Fall 2007 39 10 16 19 88 15 187 Government 7 1 10 13 27 4 62 Fall 2008 24 19 32 22 87 23 207 Business and Industry 31 17 23 13 85 15 184 Fall 2009 31 17 23 13 85 15 184 TOTAL 219 122 191 123 267 65 987 February 2010 Notices of the AMS 253 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 5C: Number of New Doctoral Table 5D: Academic Positions in U.S. Filled Recipients Taking U.S. Academic Positions by by New Doctoral Recipients by Type of Type of Degree-Granting Hiring Department, Fall 2005 to Fall 2009 Department, Fall 2005 to Fall 2009 Group Group Year I–III IV Va M&B Other* TOTAL Year I (Pu) I (Pr) II III IV Va TOTAL Fall 2005 231 45 12 188 126 602 Fall 2005 131 88 130 83 131 39 602 Fall 2006 262 69 12 185 143 671 Fall 2006 167 108 123 86 137 50 671 Fall 2007 264 39 17 186 144 650 Fall 2007 178 76 146 87 120 43 650 Fall 2008 256 42 9 180 162 649 Fall 2008 126 90 174 77 133 49 649 Fall 2009 260 60 13 210 198 741 Fall 2009 181 104 158 97 155 46 741 *Includes other academic and research institutes/nonprofit. Table 5E: Females as a Percentage of 2008–09 in linear, nonlinear optimization/control, to 69% in New Doctoral Recipients Produced by and mathematics education and 45% in statistics. Hired by Doctoral-Granting Groups Later sections in this First Report give more information about the female new doctoral Group recipients by citizenship and the female new Percent I (Pu) I (Pr) II III IV Va TOTAL doctoral recipients in Group IV. Produced 20% 21% 34% 42% 45% 30% 32% Hired 16% 22% 31% 30% 38% 8% 26% Employment Information about 2008–09 New Doctoral Recipients by Citizenship new doctoral recipients hired in Group M, Master’s and Type of Employer departments, are female; 43% of the new doctoral Table 5F shows the pattern of employment recipients hired in Group B, Bachelor’s departments, within employer categories broken down by citi- are female, up from 40% last year. This year, zenship status of the new doctoral recipients. Group IV hired the highest percentage of women The unemployment rate for the U.S. citizens is with 38%, while Groups I, II, III ranged from 16% 8.6% compared to 5.8% in fall 2008. The unemploy- to 31%. ment rate for non-U.S. citizens is 7.3%. This varies The unemployment rate for female new doctoral by type of visa. The unemployment rate for non- recipients is 5.8%, compared to 9.0% for males and U.S. citizens with a permanent visa is 8.3%, while 7.9% overall. that for non-U.S. citizens with a temporary visa The percentage of female new doctoral is 7.2%. Among U.S. citizens whose employment recipients within fields of thesis ranged from 19% status is known, 84% are employed in the U.S. Table 5F: Employment Status of 2008–09 New Doctoral Recipients by Citizenship Status CITIZENSHIP NON-U.S. CITIZENS U.S. CITIZENS TYPE OF EMPLOYER Permanent Visa Temporary Visa Unknown Visa TOTAL U.S. Employer 501 66 414 6 987 U.S. Academic 412 45 280 4 741 Groups I, II, III, and Va 135 14 122 2 273 Group IV 23 6 31 0 60 Non-Ph.D. Department 238 20 107 1 366 Research Institute/Other Nonprofit 16 5 20 1 42 U.S. Nonacademic 89 21 134 2 246 Non-U.S. Employer 39 3 99 3 144 Non-U.S. Academic 36 3 87 3 129 Non-U.S. Nonacademic 3 0 12 0 15 Not Seeking Employment 8 0 7 0 15 Still Seeking Employment 47 6 32 0 85 SUBTOTAL 595 75 552 9 1231 Unknown (U.S.) 72 8 29 0 109 Unknown (non-U.S.)* 2 1 80 7 90 TOTAL 669 84 661 16 1430 *Includes those who left the U.S. and whose employment status is reported as “unknown” or “still seeking employment”. 254 Notices of the AMS Volume 57, Number 2 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 5G: 2008–09 New Doctoral Gender, Race/Ethnicity, and Citizenship Status Recipients Having Employment in the U.S. of 2008–09 New Doctoral Recipients by Type of Employer and Citizenship Table 6 presents a breakdown of new doctoral CITIZENSHIP recipients according to gender, racial/ethnic group, U.S. EMPLOYER U.S. Non-U.S. TOTAL and citizenship status. The information reported in this table was obtained in summary form from Academic 412 329 741 Groups I–Va 158 175 333 the departments granting the degrees. Additional M, B, & 2-Year 166 66 232 reports on gender, race/ethncity, and citizen- Other Acad. & Research Inst. 88 88 176 ship is available on the Web at www.ams.org/ Government, Business & Industry 89 157 246 employment/specialreports.html. TOTAL 501 486 987 There were 669 (47%) U.S. citizens among the 1,430 new doctoral recipients in 2008–09. Among U.S. citizens, 4 are American Indian or Alaska Native, Among non-U.S. citizens with a permanent visa 44 are Asian, 19 are Black or African American, 19 whose employment status is known, 88% have jobs are Hispanic or Latino, 1 is Native Hawaiian or in the U.S. (last year this percentage was 90%), while Other Pacific Islander, 575 are White, and 7 are of the similar percentage for non-U.S. citizens with a unknown race/ethnicity. Among non-U.S. citizens, temporary visa is 75% (last year the percentage was there are no American Indians or Alaska Natives, 76%). The number of non-U.S. citizens having em- 471 Asians, 18 Blacks or African Americans, 49 ployment in the U.S. is 486, up from 459 last year. Hispanics or Latinos, 217 Whites, and 6 of unknown Table 5G is a cross-tabulation of the 987 new race/ethnicity. doctoral recipients who have employment in the Table 7 gives the number of new U.S. doctoral U.S. by citizenship and broad employment catego- recipients and the number of U.S. citizens back ries, using numbers from Table 5F. Of the 987 new to 1998-99. The 669 U.S. citizen new doctoral doctoral recipients having jobs in the U.S., 51% are recipients is up by 236 (55%) from its low in U.S. citizens (up from 48% last year). Of the 333 2004–05. new doctoral recipients who took jobs in U.S. doc- Females make up 30% of the 669 U.S. citizens toral-granting departments, 47% are U.S. citizens receiving doctoral degrees in the mathematical sciences in 2008–09. Last year this percentage was (down from 49% last year). Of the 408 who took 31%. Among the 761 non-U.S. citizen new doctoral other academic positions, 62% are U.S. citizens (up recipients, 34% (260) are female, up from last year’s from 57% last year). Of the 246 who took nonaca- 32%. Figure 3 gives the historical record of U.S. citizen demic positions, 36% are U.S. citizens. Of the 501 female new doctoral recipients and the percentage U.S. citizens employed in the U.S., 32% have jobs in of females among U.S. citizen (full-time) graduate a doctoral-granting department, 51% are in other students beginning in fall 1989. The number of academic positions, and 18% are in nonacademic female U.S. citizen new doctoral recipients is up positions. For the 486 non-U.S. citizens employed 15 (8%) from 187 in 1998–99; reaching an all time in the U.S., the analogous percentages are 36%, 32%, new high. Additional historical information on U.S. and 32% respectively. citizen doctoral recipients is available on the Web Table 6: Gender, Race/Ethnicity, and Citizenship of 2008–09 New Doctoral Recipients MALE FEMALE NON-U.S. CITIZENS NON-U.S. CITIZENS U.S. Permanent Temporary Unknown Total U.S. Permanent Temporary Unknown Total RACIAL/ETHNIC GROUP CITIZENS Visa Visa Visa Male CITIZENS Visa Visa Visa Female TOTAL American Indian or 0 0 0 0 0 4 0 0 0 4 4 Alaska Native Asian 31 14 263 4 312 13 26 161 3 203 515 Black or African 8 3 12 1 24 11 1 1 0 13 37 American Hispanic or Latino 10 4 35 1 50 9 0 9 0 18 68 Native Hawaiian or 0 0 0 0 0 1 0 0 0 1 1 Other Pacific Islander White 413 24 131 3 571 162 12 43 4 221 792 Unknown 5 0 6 0 11 2 0 0 0 2 13 TOTAL 467 45 447 9 968 202 39 214 7 462 1430 February 2010 Notices of the AMS 255 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 7: U.S. Citizen Doctoral Recipients, Figure 3: Females as a Percentage of U.S. Citizen Doctoral Preliminary Counts Recipients and Graduate Students, Preliminary Counts Total Doctorates Total U.S. % Female U.S. Doctoral Recipients Granted by U.S. Citizen Doctoral Year Institutions Total % % Female U.S. Graduate Students 1998–99 1133 554 49% 40% 1999–00 1119 537 48% 2000–01 1008 494 49% 35% 2002–03 1017 489 48% 30% 2003–04 1041 441 42% 2004–05 1116 433 39% 25% 2005–06 1245 522 42% 2006–07 1157 500 43% 20% 2007–08 1235 540 44% 2008-09 1430 669 47% 15% 10% at www.ams.org/employment/specialreports. 5% html. 0% 2008–09 New Doctoral Recipients with Dissertations in Statistics/Biostatistics and Probability 1999-00 2001-02 2003-04 2005-06 2007-08 1989–90 1991–92 1993–94 1995–96 1997–98 Group IV contains U.S. departments (or programs) of statistics, biostatistics, and biometrics reporting and the 68 Group IV departments responding a doctoral program. In the Annual Survey Reports, to the Annual Survey reported 352 new doctoral Group IV is referred to as the Statistics Group. recipients, 25% of all new doctoral recipients in In addition, other groups in the Annual Survey 2008–09. The number of new doctoral recipients produce new doctoral recipients with dissertations in Group IV is up 61 from the number reported at in statistics/biostatistics or probability. The other this time last year, while the number of departments groups produced 140 new doctoral recipients responding is up 8 from the number responding by with dissertations in statistics/biostatistics or this time last year. probability in 2008–09 and have averaged 90 per Because of its size, the data from Group IV have a year over the ten-year period reported in Table large effect on the results when all doctoral groups 8. Information about these 140 new doctoral are combined. Furthermore, Group IV results are recipients and the 347 new doctoral recipients in often quite different from those for Groups I (Pu), Group IV is found in this section of the report. I (Pr), II, III, and Va. Group IV results can mask Table 8 contains information about new important changes in the other doctoral groups. In doctoral recipients in Group IV as well as those with dissertations in statistics/biostatistics and the following paragraphs some of these differences probability in other groups for this year as well as are presented. for the past nine years. In addition, the last two Group IV is producing a larger percentage of rows of Table 8 give a split of the 2008–09 results female doctorates than the other doctoral groups. between the 57 statistics departments and the Females accounted for 45% of the 352 new doctoral 35 biostatistics and biometrics departments in recipients in Group IV, while 28% of 1,078 are female Group IV. This year 487 new doctorates had a in the other doctoral groups. Among U.S. citizens, dissertation in statistics/biostatistics (410) or females accounted for 59% of the 92 Group IV new probability (77), a 27% increase from last year’s doctoral recipients, while for the other groups 26% number. Those with dissertations in statistics/ of 577 were female. Overall, 30% of the 669 U.S. biostatistics and probability accounted for 34% citizen new doctoral recipients were female. of new doctorates in 2008–09. Quite a bit of the Group IV is producing a smaller percentage of year-to-year variation in these numbers is due to U.S. citizen new doctorates than the other doctoral the changes made in the departments included in groups. In Group IV, 36% of the 352 new doctoral Group IV over the ten years and to the response rate recipients are U.S. citizens, while in other groups variation in this group. 50% of the 1,078 are U.S. citizens. In Group IV, 66% Group IV has 92 departments for 2008–09, of the 159 females were not U.S. citizens. 11 more than the next largest doctoral group. It Group IV doctorates are more likely to take jobs in contains 31% of all doctoral departments surveyed, business and industry than those in other doctoral 256 Notices of the AMS Volume 57, Number 2 2009 Annual Survey of the Mathematical Sciences in the U.S. Table 8: New Doctoral Recipients with Dissertations in Statistics/Biostatistics and Probability New Doctoral Recipients in Statistics/Biostatistics New Doctoral Recipients Group IV New Doctoral Recipients in Group IV only and Probability, Group IV and Other* Groups Hired by Group IV Group IV Depts Depts Responding Female Jobs in Percentage Other Percentage Year Surveyed (percent) Total (percent) Bus & Ind Unemployed Total Group IV Groups Unemployed Male Female 1999–00 89 75 (84%) 284 110 (39%) 79 2.6% 351 278 73 2.2% 24 22 2000–01 86 70 (81%) 237 98 (41%) 59 5.9% 289 221 68 6.1% 27 14 2001–02 86 72 (84%) 222 92 (41%) 56 6.6% 288 221 67 6.1% 31 15 2002–03 86 74 (86%) 239 98 (41%) 45 2.3% 302 234 68 3.6% 20 19 2003–04 87 65 (75%) 243 97 (40%) 50 3.4% 318 241 77 4.2% 48 15 2004–05 87 63 (72%) 285 126 (44%) 64 5.1% 374 283 91 6.1% 26 19 2005–06 88 60 (68%) 287 134 (47%) 80 1.8% 396 278 118 3.2% 41 28 2006–07 86 50 (58%) 279 127 (46%) 88 3.1% 380 279 101 4.2% 24 15 2007–08 89 60 (67%) 291 151 (52%) 87 1.3% 382 281 101 3.2% 21 21 2008–09 92 68 (74%) 352 159 (45%) 85 3.3% 487 347** 140*** 5.0% 37 23 Statistics 57 45 (79%) 245 103 (42%) 74 3.7% 25 12 Biostatistics 35 23 (66%) 107 56 (52%) 11 1.3% 12 11 * Includes other academic departments and research institutes/other nonprofits. ** Of 347, there were 343 in statistics/biostatistics and 4 in probability. For complete details, see Table 4B. *** Of 140, there were 67 in statistics/biostatistics and 73 in probability. For complete details, see Table 4B. groups. Of the 267 new doctoral recipients from Changes in Reporting of Unemployment Group IV who found employment in the U.S., 85 Rate (32%) took jobs in business or industry. From the In the unemployment calculations provided other groups, 720 new doctoral recipients found in this report the individuals employed employment in the U.S., of which 99 (14%) took jobs outside the U.S. have been removed from in business or industry. the denominator used in the calculation of the rate, in addition to the routine removal Group IV doctorates have a lower unemployment of all individuals whose employment status rate than the other doctoral groups. The employment is unknown. This is a change from prior status for 301 Group IV new doctoral recipients is Annual Survey Reports. As a consequence, the unemployment rate now being reported known, and 9 (3.3%) are unemployed. For the other more accurately reflects the U.S. labor market groups, the employment status of 930 is known, experienced by the new doctoral recipients. and 66 (9.5%) are unemployed. Group IV is hiring This change tends to increase the rate of unemployment over that produced in prior a bigger percentage of females than the other years. doctoral groups. Twenty-three of 60 (38%) new In a further small change from prior years, doctoral recipients hired by Group IV departments those individuals reported as not seeking employment have also been removed from the were female, down from last year’s 50%. The other denominator. The number of individuals so doctoral groups reported that 63 of 273 (23%) new designated is small each year, and the impact doctoral recipients hired were female, up from last of this change is to produce a slight increase in the rate over that reported in prior years. year’s 22%. The unemployment rates for years prior Group IV had 347 new doctoral recipients with to 2009 shown in this report have been fields of thesis in statistics/biostatistics (343) recalculated using this new method. One can view a comparison of the unemployment rates and the other doctoral departments had 140 using the traditional method and the new with fields of thesis in statistics/biostatistics (67) method by visiting the AMS website at www. and probability (73) (last year the other doctoral ams.org/employment/surveyreports.html. departments had 57 new doctorates in statistics and 44 in probability). The distribution of these degrees Previous Annual Survey Reports among the various groups can be found in Table 4B. The 2008 First, Second, and Third Annual Survey The number of new doctoral recipients with theses Reports were published in the Notices of the AMS in in statistics/biostatistics and probability (487) the February, August, and November 2009 issues respectively. These reports and earlier reports, as is substantially larger than any other field, with well as a wealth of other information from these algebra and number theory next with 203. surveys, are available on the AMS website at www. ams.org/employment/surveyreports.html. February 2010 Notices of the AMS 257 2009 Annual Survey of the Mathematical Sciences in the U.S. Acknowledgments Definitions of the Groups The Annual Survey attempts to provide an accurate appraisal and analysis of various aspects of the As has been the case for a number of years, much of the data in these academic mathematical sciences scene for the use reports is presented for departments divided into groups according to several characteristics, the principal one being the highest degree and benefit of the community and for filling the offered in the mathematical sciences. Doctoral-granting departments information needs of the professional organizations. of mathematics are further subdivided according to their ranking Every year, college and university departments of “scholarly quality of program faculty” as reported in the 1995 in the United States are invited to respond. The publication Research-Doctorate Programs in the United States: Annual Survey relies heavily on the conscientious Continuity and Change.1 These rankings update those reported in a 2 efforts of the dedicated staff members of these previous study published in 1982. Consequently, the departments which now compose Groups I, II, and III differ significantly from those departments for the quality of its information. used prior to the 1996 survey. On behalf of the Annual Survey Data Committee The subdivision of the Group I institutions into Group I Public and the Annual Survey Staff, we thank the many and Group I Private was new for the 1996 survey. With the increase secretarial and administrative staff members in in number of the Group I departments from 39 to 48, the Annual the mathematical sciences departments for their Survey Data Committee judged that a further subdivision of public cooperation and assistance in responding to the and private would provide more meaningful reporting of the data for these departments. survey questionnaires. Brief descriptions of the groupings are as follows: Group I is composed of 48 departments with scores in the 3.00–5.00 Other Sources of Data range. Group I Public and Group I Private are Group I departments Vist the AMS website at www.ams.org/ at public institutions and private institutions respectively. employment/specialreports.html for a listing Group II is composed of 56 departments with scores in the 2.00–2.99 of additional sources of data on the Mathematical range. Sciences. Group III contains the remaining U.S. departments reporting a doctoral program, including a number of departments not included in the 1995 ranking of program faculty. The Annual Survey series begun in 1957 Group IV contains U.S. departments (or programs) of statistics, biostatistics, and biometrics reporting a doctoral program. by the American Mathematical Society is currently under the direction of the Data Group V contains U.S. departments (or programs) in applied math- Committee, a joint committee of the American ematics/applied science, operations research, and management Mathematical Society, the American Statistical science which report a doctoral program. Association, the Institute of Mathematical Group Va is applied mathematics/applied science; Group Vb, which Statistics, the Mathematical Association of was no longer surveyed as of 1998–99, was operations research America, and the Society of Industrial and and management science. Applied Mathematics. The current members Group M contains U.S. departments granting a master’s degree as of this committee are Richard Cleary, Richard the highest graduate degree. M. Dudley, Susan Geller, John W. Hagood, Group B contains U.S. departments granting a baccalaureate degree Abbe H. Herzig, Ellen Kirkman, Joanna Mitro, only. James W. Maxwell (ex officio), Bart Ng, Polly Listings of the actual departments which compose these groups are Phipps (chair), Douglas Ravanel, Jianguo available on the AMS website at www.ams.org/employment/ (Tony) Sun, and Marie Vitulli. The committee groups_des.html. is assisted by AMS survey analyst Colleen A. 1Research-Doctorate Programs in the United States: Continuity and Rose. Comments or suggestions regarding Change, edited by Marvin L. Goldberger, Brendan A. Maher, and Pamela this Survey Report may be directed to the Ebert Flattau, National Academy Press, Washington, DC, 1995. committee. 2These findings were published in An Assessment of Research- Doctorate Programs in the United States: Mathematical and Physical Sciences, edited by Lyle V. Jones, Gardner Lindzey, and Porter E. Coggeshall, National Academy Press, Washington, DC, 1982. The information on mathematics, statistics, and computer science was presented in digest form in the April 1983 issue of the Notices of the AMS, pages 257–67, and an analysis of the classifications was given in the June 1983 Notices of the AMS, pages 392–3. 258 Notices of the AMS Volume 57, Number 2 Gerrit van Dijk Ivanka Stamova Andrei Khrennikov ADVANCED INTRODUCTION STABILITY INTERPRETA- FINANCIAL TO HARMONIC ANALYSIS OF TIONS OF MODELLING ANALYSIS AND IMPULSIVE PROBABILITY Ed. by GENERALIZED FUNCTIONAL 2nd rev. and ext. ed. 2009. Hc. 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Prices are subject to change. Prices do not include postage and handling. WWW.DEGRUYTER.COM Mathematics People symmetric spaces, L²-cohomology, arithmetic groups, and MacPherson Awarded Hopf the Langlands program. The list is by far not complete, and Prize we try only to give a representative selection of his contri- bution to mathematics. He influenced a whole generation Robert MacPherson of the Institute for Advanced Study of mathematicians by giving them new tools to attack has been chosen the first winner of the Heinz Hopf Prize difficult problems and teaching them novel geometric, given by ETH Zurich for outstanding scientific work in the topological, and algebraic ways of thinking.” field of pure mathematics. MacPherson, a leading expert Robert MacPherson was born in 1944 in Lakewood, in singularities, delivered the Heinz Hopf Lectures, titled Ohio. He received his B.A. from Swarthmore College in “How Nature Tiles Space”, in October 2009. The prize also 1966 and his Ph.D. from Harvard University in 1970. He carries a cash award of 30,000 Swiss francs, approximately taught at Brown University from 1970 to 1987 and at equal to US$30,000. the Massachusetts Institute of Technology from 1987 to The following quotation was taken from a tribute to 1994. He has been at the Institute for Advanced Study in MacPherson by Gisbert Wüstholz of ETH Zurich: “Singu- Princeton since 1994. His work has introduced radically larities can be studied in different ways using analysis, new approaches to the topology of singular spaces and or you can regard them as geometric phenomena. For the promoted investigations across a great spectrum of math- latter, their study demands a deep geometric intuition ematics. MacPherson works in several fields of geometry- and profound geometric insight; this is what MacPherson masters in a most striking and extremely artful way. He topology, algebraic geometry, differential geometry, combines geometric visions with algebraic rigidity. If you and singularity theory, and he is especially interested study his work you see that it is glowing with elegance in aspects of geometry that interact with other areas of and profound in its depth. mathematics. In 1992 he received the National Academy of “He consistently was ahead of his time, developing new Sciences Award in Mathematics, and in 2002 he received ideas and new approaches—ones often not shaped by the the Leroy P. Steele Prize of the AMS. main streams of mathematical thought of the day, but The Heinz Hopf Prize at ETH Zurich was established rather characterized by great vision. Repeatedly the math- through a donation made by Dorothee and Alfred Aep- ematical community came to embrace, extend, and apply pli and will be awarded every two years for outstanding his ideas and results as they caught up with that vision. scientific work in the field of pure mathematics. “MacPherson’s connections with the Swiss mathemati- cal community date back to 1983, when he participated —From an ETH announcement in and significantly contributed to the famous Borel semi- nar, a joint seminar organized by several Swiss universi- ties, including ETH Zurich, Lausanne, Geneva, Bern, and Lindenstrauss and Villani Basel. The seminar was initiated by Armand Borel, one of the most distinguished Swiss mathematicians of the Awarded Fermat Prize last century. The topic of the seminar was the Goresky- MacPherson intersection homology and its use for the co- Elon Lindenstrauss of Princeton University and Cédric homology of arithmetic groups, one of the main research Villani of École Normale Supérieure de Lyon have been areas of Armand Borel. awarded the 2009 Fermat Prize by the Institut de Mathé- “This illustrates only a small part of the work of MacPher- matiques de Toulouse. Lindenstrauss was honored for his son, for it spans a wide spectrum of contributions in many contributions to ergodic theory and their applications in very different areas: algebraic geometry and topology; number theory. Villani was selected for his contributions algebraic groups, group actions, and representation to the theory of optimal transport and his studies of non- theory; enumerative geometry and combinatorics; locally linear evolution equations. 260 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Mathematics People The Fermat Prize, given every two years, recognizes out- mathematicians, and engineers who are committed to the standing research in the fields in which Pierre de Fermat integration of research and education. The grants provide made significant contributions: statements of variational funding of at least US$400,000 over a five-year period. principles, foundations of probability and analytical The 2009 CAREER grant awardees and the titles of their geometry, and number theory. The prize is intended to grant projects follow. reward research that is accessible to the greatest number Miklos Abert, University of Chicago, Asymptotic In- of professional mathematicians within these fields. variants of Residually Finite Groups; Rafail Abramov, The prize carries a cash award of €20,000 (approxi- University of Illinois, Chicago, Predicting Global Climate mately US$29,700). Change through Fluctuation-Dissipation: A Practical Com- putational Strategy for Complex Multiscale Dynamics; —Elaine Kehoe Orly Alter, University of Texas, Austin, Integrative and Comparative Tensor Algebra Models of DNA Microarray Data from Different Studies of the Cell Cycle; Benjamin Pujals Awarded TWAS Prize in Brubaker, Massachusetts Institute of Technology, Mul- tiple Dirichlet Series, Automorphic Forms, and Combi- Mathematics natorial Representation Theory; Francesco Calegari, Northwestern University, Arithmetic of Cohomological Enrique Pujals of the Institute of Pure and Applied Automorphic Forms; Gautam Chinta, City College, City Mathematics (IMPA) in Rio de Janeiro, Brazil, has been University of New York, Multiple Dirichlet Series and Meta- named the winner of the 2009 TWAS Prize in mathemat- plectic Groups; Tommaso de Fernex, University of Utah, ics, awarded by the Academy of Sciences for the Develop- Singularities in the Minimal Model Program and Birational ing World (TWAS). He was honored for his contributions Geometry; Ioana Dumitriu, University of Washington, toward developing “a theory about robust dynamics and Synergistic Interactions between Numerical Linear Algebra about the role of homoclinic bifurcation as a universal and Stochastic Eigenanalysis (Random Matrix Theory); mechanism to describe the way to produce very rich Noureddine El Karoui, University of California Berke- and complex dynamics.” Pujals will receive a cash prize ley, Random Matrices and High-Dimensional Statistics; of US$15,000 and will deliver a lecture at the academy’s Yongtao Guan, Yale University, New Statistical Methods general meeting in 2010. for Massive Spatial, Temporal, and Spatial-Temporal Pro- cesses; Jeffrey Humpherys, Brigham Young University, —From a TWAS announcement Interdisciplinary Mentoring Program in Analysis, Compu- tation, and Theory (IMPACT); Marta Lewicka, University of Minnesota, Twin Cities, Thin Shells: Problems in Non- Dereich Receives 2009 linear Elasticity and Fluid Dynamics; Fengyan Li, Rensse- Information-Based Complexity laer Polytechnic Institute, Development and Applications of Discontinuous Galerkin Methods; Di Liu, Michigan Young Researcher Award State University, Modeling, Analysis, and Computation of Stochastic Intracellular Reactions; Gregory Lyng, Uni- Steffen Dereich of the Technische Universität Berlin, versity of Wyoming, Behavior of Solutions of Nonlinear Germany, has been awarded the Information-Based Com- Partial Differential Equations; Mauro Maggioni, Duke plexity Award for 2009. The award is given every year for University, Multiscale Methods for High-Dimensional Data, significant contributions to information-based complex- Graphs, and Dynamical Systems; Dionisios Margetis, ity by a young researcher who has not reached his or her University of Maryland, College Park, Thermodynamic thirty-fifth birthday by September 30 of the year of the and Kinetic Approaches for Epitaxial Material Systems; award. The prize consists of US$1,000 and a plaque. Lenhard Ng, Duke University, Symplectic Field Theory The award committee this year consisted of Dirk Nuy- and Low-Dimensional Topology; Jiawang Nie, University ens, Katholieke Universiteit, Leuven; Andreas Neuenkirch, of California San Diego, Linear Matrix Inequality Repre- University of Frankfurt; Jan Vybiral, University of Jena; sentations in Optimization; Duane Nykamp, University of Joseph F. Traub, Columbia University; and Henryk Woz- Minnesota, Twin Cities, Toward a Second-Order Descrip- niakowski, Columbia University and University of Warsaw. tion of Neuronal Networks; Jeffrey Schenker, Michigan State University, Analysis of Disordered Systems; Jian —Joseph Traub, Columbia University Song, Rutgers University, Canonical Metrics, Complex Monge-Ampère Equations, and Geometric Flows; Jason Starr, State University of New York, Stony Brook, Higher NSF CAREER Awards Made Rational Connectedness, Higher Fano Manifolds, and Ap- plications; Katrin Wehrheim, Massachusetts Institute of The Division of Mathematical Sciences (DMS) of the Technology, The Symplectic Category, Floer Field Theory, National Science Foundation (NSF) has honored twenty- and Relations to Gauge Theory and Topology; Anna nine mathematicians in fiscal year 2009 with Faculty Wienhard, Princeton University, Higher Teichmüller Early Career Development (CAREER) awards. The NSF Theory; Lexing Ying, University of Texas, Austin, Fast established the awards to support promising scientists, Algorithms for Oscillatory Integrals; Ming Yuan, Georgia FEBRUARY 2010 NOTICES OF THE AMS 261 Mathematics People Institute of Technology, Sparse Modeling and Estimation topics. His lectures were regarded by students and faculty with High-Dimensional Data; Aleksey Zinger, State Uni- as a model of mathematical clarity and precision. Both versity of New York, Stony Brook, Holomorphic Curves in within the mathematics department and among his circle Algebraic Geometry and Symplectic Topology; and Hui of friends and relatives, he was considered to be a kind, Zou, University of Minnesota, Twin Cities, New Statistical gentle, and compassionate man. Although he suffered a Methodology and Theory for Mining High-Dimensional serious heart attack in 1979, he made a remarkable re- Data. covery, aided by an intensive rehabilitation and exercise program that led to his becoming an avid runner; indeed, —Elaine Kehoe he won a medal in a 1983 race. He retired from Washington University in 1995. In 1989 Eddie and his wife visited his hometown of Memories of Eddie Nussbaum Rheydt in the Moenchen-Gladbach-Rheydt region and were very well received there; remarkably, he said he harbored A. Edward (Eddie) Nussbaum died of congestive heart no bitterness over his family’s experiences. He is survived failure on October 31, 2009, at the age of eighty-four. He by Anne, his wife of fifty-two years, and their children Karl, was a faculty member in the Department of Mathematics who teaches film studies at Montclair State University and at Washington University in St. Louis for thirty-seven years. has produced a number of films, and Franziska, who works Eddie Nussbaum was born in 1925 in the region with photographers and serves as a stylist for advertising Moenchen-Gladbach-Rheydt (three adjacent towns), Ger- agencies in the St. Louis area. many, where his parents operated a department store. His elder brother was arrested on Kristallnacht in 1938. —Guido Weiss and Edward Wilson, Washington University Soon thereafter, Eddie and his sister were sent by the Kindertransport train to live in Belgium. However, condi- tions in Belgium were not safe, and Eddie and his sister were soon separated. Eddie fled to southern France, and when conditions there also became unsafe, he crossed into Switzerland with the help of two local women and their woodsman father. When he was quickly put in jail by the Swiss authorities, he invented a story that led to his release, and he lived for several years with a spinster and her nephew in Switzerland while studying mathematics at the University of Zurich. Although Eddie’s sister also survived the Holocaust, sadly both his parents and his elder brother died in the Nazi death camp at Auschwitz. In 1947 Eddie arrived penniless in New York and began taking courses at Brooklyn College while supporting him- self by rolling clay tennis courts. Soon thereafter he was admitted to Columbia University for graduate studies in mathematics, receiving his M.A. from Columbia in 1950. The high regard in which he was held by the Columbia mathematics department is attested to by his appointment as a lecturer for the academic year 1951–52. For 1952–53 he was a staff member of the electronic computer project headed by John von Neumann at the Institute for Ad- vanced Study in Princeton. After serving as an instructor at the University of Connecticut (1953–55) and an instructor at Rensselaer Polytechnic Institute (1955–57), he received his Ph.D. from Columbia in 1957 with a dissertation titled “The Hausdorff-Bernstein-Widder Theorem for Semi- groups in Locally Compact Abelian Groups”. Following a year of service as an assistant professor at RPI, he was ap- pointed assistant professor of mathematics at Washington University in 1958; he was quickly promoted to associate professor in 1961 and to full professor in 1965. In 1955 Eddie, with Allen Devinatz and John von Neu- mann, coauthored a paper published by the Annals of Mathematics titled “On the permutability of self-adjoint operators”. This led to a distinguished career in functional analysis, with numerous important papers on unbounded operators on Hilbert spaces and on a variety of related 262 NOTICES OF THE AMS VOLUME 57, NUMBER 2 AMERICAN MATHEMATICAL SOCIETY Fan China Exchange Program Grants to support collaborations between Chinese and U.S./Canadian researchers are made possible through the generosity of Ky and Yu-Fen Fan. The Fan China Exchange Program is intended to send eminent mathematicians from the U.S. and Canada to make a positive impact on the mathematical research community in China and to bring Chinese scientists in the early stages of their research to the U.S. and Canada to help further their careers. The program encourages host institutions to provide some type of additional support for the travel or living expenses of the visitor and to ensure a suitable length of stay. Applications received before March 15 will be considered for the following academic year. For more information on the Fan China Exchange Program and application process see www.ams.org/employment/chinaexchange.html or contact the AMS Membership and Programs Department by telephone at 800-321- 4267, ext. 4170 (U.S. and Canada), or 401-455-4170 (worldwide), or by email at [email protected]. Mathematics Opportunities to enhance undergraduate education and training at the Summer Program for Women intersection of the biological and mathematical sciences Undergraduates and to better prepare undergraduate biology or mathemat- ics students to pursue graduate study and careers in fields The 2010 Summer Program for Women in Mathematics that integrate the mathematical and biological sciences. (SPWM2010) will take place at George Washington Uni- The program will provide support for jointly conducted versity in Washington, DC, from June 26 to July 31, 2010. long-term research experiences for interdisciplinary This is a five-week intensive program for mathematically teams of at least two undergraduates from departments in talented undergraduate women who are completing their the biological and mathematical sciences. Projects should junior years and may be contemplating graduate study in focus on research at the intersection of the mathematical mathematical sciences. The goals of this program are to and biological sciences and should provide students expo- communicate an enthusiasm for mathematics, to develop sure to contemporary mathematics and biology addressed research skills, to cultivate mathematical self-confidence with modern research tools and methods. Projects must and independence, and to promote success in graduate involve students from both areas in collaborative research school. experiences and include joint mentorship by faculty in Applicants must be U.S. citizens or permanent residents both fields. studying at a U.S. university or college who are completing Between six and nine awards are expected to be made their junior years or the equivalent and have mathematical in 2010. The deadline for full proposals is February 11, experience beyond the typical first courses in calculus and 2010. For more information, see http://www.nsf.gov/ linear algebra. Sixteen women will be selected. Each will pubs/2008/nsf08510/nsf08510.htm#awd_info. The receive a travel allowance, campus room and board, and UBM program is a joint effort of the Education and Human a stipend of US$1,750. The deadline for applications is Resources (EHR), Biological Sciences (BIO), and Mathemati- March 1, 2010. Early applications are encouraged. Applica- cal and Physical Sciences (MPS) directorates of the NSF. tions are accepted only by mail. For further information, please contact the director, Murli M. Gupta, email: mmg@ —From an NSF announcement gwu.edu; telephone: 202-994-4857; or visit the program’s website at http://www.gwu.edu/~spwm/. Application material is available on the website. Monroe H. Martin Prize —From an SPWM announcement The Institute for Physical Science and Technology at the University of Maryland, College Park, is seeking applica- tions and nominations for the eighth Monroe H. Martin NSF Support for Undergraduate Prize. The prize is awarded for an outstanding paper in applied mathematics (including numerical analysis) by a Training in Biological and young researcher who is a resident of North America and Mathematical Sciences who has not reached his or her thirty-sixth birthday by July 31, 2010. Submitted papers must be written by single The National Science Foundation (NSF) offers opportuni- authors and must have been published or accepted for ties for support through its Undergraduate Biology and publication. The work must not have been performed in Mathematics (UBM) program. The goal of the program is connection with the completion of requirements for an 264 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Mathematics Opportunities academic degree. The candidate must neither be nor have Foundational Courses: The foundational courses include been affiliated with the University of Maryland. “Large random planar maps and their scaling limits” Applications and nominations should include a copy of (Jean-Francois LeGall and Gregory Miermont), “SLE and the paper or contribution with a cover letter. The deadline other conformally invariant objects” (Vincent Beffara), for submissions is July 31, 2010. Submissions should and “Noise sensitivity and percolation” (Jeffrey Steif and be sent to R. Roy, Director, Institute for Physical Science Christophe Garban). and Technology, University of Maryland, College Park, Minicourses: These courses include “Random geometry Maryland 20742-2431. The award will be announced by and Gaussian free field” (Scott Sheffield), “Conformal November 1, 2010. The recipient will be asked to present invariance of lattice models” (Stanislav Smirnov), “Inte- his or her work at the Monroe H. Martin Lecture at the Uni- grable combinatorics” (Philippe Di Francesco), “Fractal versity of Maryland in December 2010 and will be awarded and multifractal properties of SLE” (Gregory Lawler), “The a prize of US$5,000 plus travel expenses. double dimer model” (Rick Kenyon), “Random polymers” The Monroe H. Martin Prize was established to com- (Frank den Hollander), and “Self-avoiding walks” (Gordon memorate the achievements of the late Monroe H. Martin, Slade). The latter two will be held jointly with the Brazil- former director of the Institute for Fluid Dynamics and ian School on Probability. Another possible minicourse, Applied Mathematics and chair of the Department of “Supersymmetric methods in disordered systems and Mathematics at the University of Maryland. Previous prize two-dimensional critical behavior” (John Cardy), is await- winners are Neil Berger (1975), Marshall Slemrod (1980), ing confirmation. See the website http://www.claymath. Jonathan Goodman (1985), Marek Rychlik (1990), A. M. Stu- org/summerschool for updated information. art (1995), Z. Xia (1995), R. J. McCann (2000), Y. Grabovsky Financial Support: Graduate students and postdoctoral (2000), C. Sinan Gunturk (2005), and Jared Tanner (2005). fellows who are within five years of receipt of the Ph.D. degree can apply for financial support. Support is decided —Institute for Physical Science and on a competitive basis and may include accommodations Technology announcement plus funds toward the cost of economy travel. The scientific committee consists of David Alexandre Ellwood, Charles Newman, Vladas Sidoravicius, and Wen- Clay Mathematics Institute delin Werner. The deadline for applications is March 1, 2010. More 2010 Summer School information and online application forms are available at The 2010 Clay Mathematics Institute (CMI) Summer School http://www.claymath.org/summerschool or by send- on Probability and Statistical Physics in Two (and More) ing email to [email protected]. Dimensions will be held in Buzios, Brazil, from July 11 to August 7, 2010. —From a CMI announcement In the past ten to fifteen years, various areas of prob- ability theory related to rigorous statistical mechanics, disordered systems, and combinatorics have enjoyed an News from CRM intensive development. A number of these developments deal with two-dimensional random structures. The ques- The Centro di Ricerca Matematica Ennio De Giorgi (CRM) tions related to critical systems are twofold: understand- invites applications for four one-year junior visiting posi- ing large-scale properties of lattice-based models (on a tions for the academic year 2010–11. Applicants should periodic deterministic lattice or in the case where the be new or recent Ph.D. recipients in mathematics and lattice is itself random) and, on the other hand, being should have exceptional potential in research. The annual able to construct and manipulate a continuous object stipend is 25,000 euros (approximately US$36,800), along that describes directly their scaling limits. In the case of with a research allowance of 2,500 euros (approximately a fixed planar lattice, a number of conjectures originating US$3,600) intended for support of other researchers in the physics literature have now been proved, but many invited to CRM by the junior visitors. Junior visitors will questions remain open. In the case of statistical physics on participate in a variety of scientific activities, including random planar graphs, sometimes referred to as quantum intensive research periods, workshops, and seminars, and gravity, many results have been recently understood, and will interact with prominent scientists who participate in a relation between discrete and continuous structures is the senior visiting program at CRM. now emerging. The aim of the Summer School is to provide The application deadline is January 28, 2010, and ju- a complete picture of the current state of the art in these nior visitors are expected to begin their research activities and related topics. at CRM no later than October 2010. For further informa- Organization: During the first two weeks of the school, tion and application details, see the website http://www. three foundational courses will be combined with after- crm.sns.it/hpp/grants.html?year=2010. noon activities in thematic working groups and evening The 2010 scientific program includes the following seminars. Weeks three and four will be dedicated to workshops: shorter advanced courses (the fourth week will run in par- January 25–February 5, 2010: Periodic Approximation allel with the Fourteenth Brazilian School of Probability). in Dynamics FEBRUARY 2010 NOTICES OF THE AMS 265 Mathematics Opportunities February 9–12, 2010: Linear and Nonlinear Hyperbolic About the Cover Equations March 8–14, 2010: Italy-India Conference on Diophan- Differential geometry issue tine and Analytic Number Theory In 2009 geometer Robert Osserman visited St. Louis March 1–April 30, 2010: Euclidean Harmonic Analysis, to observe and to lecture at Washington University Nilpotent Lie Groups and PDEs on the mathematical properties of the Gateway Arch. May 2–June 30, 2010: Configuration Spaces: Geometry, An article based on Osserman’s lecture is included in Combinatorics and Topology this issue. The cover photograph by Geir Arne Hjelle September 6–10, 2010: Geometric Evolutions and Mini- attests to the awesomeness of a marriage between mal Surfaces in Lorentzian Manifolds mathematics and architecture on a grand scale. September 13–17, 2010: On the Contested Expanding The cover photograph was taken on November Role of Applied Mathematics from the Renaissance to the 23, 2008, from the south leg of the St. Louis Arch, Enlightenment looking north and slightly up. For this magnificent October 12–16, 2010: Optimal Transportation and wide angle, unusual shot, the photographer used a Applications Canon EOS 40D camera fitted with a Canon EF-S 17- For further information see the website http://www. 85mm f4-5.6 IS USM lens. crm.sns.it. “I was down at the arch,” Geir Arne comments, “looking for a cover picture for my St. Louis photo —From a CRM announcement book, and I guess this was one of the candidates.” The photographer later favored a different shot. You can see the other shot at http://www.flickr.com/ photos/hjelle/3439055921/. Everett Pitcher Lectures The cover image was composed using GIMP (GNU The next series of Everett Pitcher Lectures will be held Image Manipulation Program) freeware. Find it at March 22–25, 2010, on the campus of Lehigh University in http://www.gimp.org/. Bethlehem, Pennsylvania. The speaker will be Rick Durrett —Steven G. Krantz of Cornell University, who will speak on “Three Faces of Probability”, with three lectures on improbabilities, life on a random graph, and cancer models. Durrett’s research focuses on probability problems that come from ecology and genetics. He is a member of the National Academy of Sciences. The lectures, which are free and open to the public, are held in honor of Everett Pitcher, who was secretary of the AMS from 1967 until 1988. Pitcher served in the mathematics department at Lehigh University from 1938 until 1978, when he retired as Distinguished Professor of Mathematics. He passed away in December 2006 at the age of ninety-four. For further information, contact the Everett Pitcher Lecture Series, Department of Mathemat- ics, Lehigh University, Bethlehem, PA, 18015; telephone 610-758-3731; or see the website http://www.lehigh. edu/~math/pitcher.html. —From a Lehigh University announcement 266 NOTICES OF THE AMS VOLUME 57, NUMBER 2 A MERICAN MATHEMATICAL SOCIETY AMS Sectional Meetings–Spring 2010 March 27–28 April 10–11 University of Kentucky Macalester College, Lexington, KY St. Paul, MN Invited Addresses by Percy A. Deift , Courant Institute–New Invited Addresses by Charles York University; Irina Mitrea, Doering, University of Worcester Polytechnic Institute; Michigan; Matthew James Bruce Reznick, University of Emerton, Northwestern Illinois at Urbana-Champaign; University; Vladimir Bernd Ulrich, Purdue University; Touraev, Indiana University; and Doron Zeilberger, Rutgers and Peter Webb, University University (Erdős Memorial of Minnesota MAR 27-28 Lecture) APR 10-11 University of Kentucky Macalester College, Lexington, KY St. Paul, MN April 17–18 May 22–23 University of New Mexico New Jersey Institute of Albuquerque, NM Technology Newark, NJ Invited Addresses by Kenneth Bromberg, University of Utah; Invited Addresses by Simon Danny Calegari, California Brendle, Stanford University; Institute of Technology; Ioana Konstantin M. Mischaikow, Dumitriu, University of Rutgers University; Ricardo Washington; and Steff en Rhode, H. Nochetto, University of University of Washington Maryland; and Richard E. Schwartz, Brown University APR 17-18 MAY 22-23 University of New New Jersey Institute Mexico of Technology Albuquerque, NM Newark, NJ See the AMS website for the most up-to-date lists of Invited Addresses and Special Sessions. www.ams.org/amsmtgs/sectional.html Come to hear presentations, connect with researchers in your field, and meet colleagues, mentors, and students. While at the meetings visit the AMS exhibit to see the book sale and receive information on AMS membership and programs. Inside the AMS [email protected] for information on how to submit From the AMS Public abstracts for AMS meetings and MAA sessions at January Awareness Office Joint Mathematics Meetings. Type HELP as the subject line. [email protected] to contact the AMS Acquisitions •Mathematical Moment. Recent topics include “Knowing Department. 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Linda Barkley, Boeing Communications, died on Au- [email protected] to request general information about gust 15, 2009. Born on December 12, 1951, she was a Society meetings and conferences. member of the Society for 31 years. [email protected] to request meeting registra- Enrique Bayo, professor, from San Juan, PR, died on tion forms be emailed. May 28, 2002. Born on April 20, 1922, he was a member [email protected] to submit completed regis- of the Society for 51 years. tration forms. Leonard D. Berkovitz, professor emeritus, Purdue [email protected] for information or questions about University, died on October 13, 2009. Born on January 24, registration and housing for the Joint Mathematics Meet- 1924, he was a member of the Society for 59 years. ings (Mathematics Meetings Service Bureau). Thomas Dietmair, ERGO Insurance Group, Germany, [email protected] for technical questions regard- died on September 6, 2009. 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Born on August [email protected] to send correspondence about AMS 10, 1934, he was a member of the Society for 17 years. professional programs and services. Alan A. Meyerhoff, professor, Rutgers University, [email protected] to send correspondence to the AMS New Brunswick, died on May 18, 2009. Born on March 20, Publication Division. 1926, he was a member of the Society for 35 years. [email protected] to submit accepted electronic Tetsuro Miyakawa, professor, Kanazawa University, manuscripts to AMS publications (other than Abstracts). died on February 11, 2009. Born on March 10, 1948, he See http://www.ams.org/submit-book-journal to was a member of the Society for 21 years. electronically submit accepted manuscripts to the AMS Charles T. Molloy, Falls Church, VA, died on May 5, book and journal programs. 2009. Born on November 22, 1914, he was a member of [email protected] to request permission to the Society for 58 years. reprint material from Society publications. A. 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[email protected] to request examination copies Marian Pour-El, professor emeritus, School of Math- or inquire about using AMS publications as course texts. ematics, University of Minnesota, died on June 10, 2009. [email protected] for general information or for Born on April 19, 1928, she was a member of the Society assistance in accessing and using the AMS website. for 56 years. FEBRUARY 2010 NOTICES OF THE AMS 269 Inside the AMS AMERICAN MATHEMATICAL SOCIETY Calvin R. Putnam, professor emeritus, Purdue Univer- sity, died on April 24, 2008. Born on May 25, 1924, he was a member of the Society for 63 years. The AMS Bettina Richmond, professor, Western Kentucky Uni- versity, died on November 22, 2009. Born on January 30, 1958, she was a member of the Society for 30 years. Bookstore Pierre Samuel, professor emeritus, University of Paris- www.ams.org/bookstore Sud, died on August 23, 2009. Born on September 12, 1921, he was a member of the Society for 63 years. Alice T. Schafer, former AWM president, died on September 27, 2009. Born on June 18, 1915, she was a Where you can … member of the Society for 68 years. Stefan Schwabik, professor emeritus, Institute of Mathematics of the Academy of Sciences of the Czech Republic, died on November 4, 2009. Born on March 15, 1941, he was a member of the Society for 16 years. Peter Seibert Get the best , from Mexico City, died on August 13, 2009. Born on April 13, 1927, he was a member of the deals Society for 49 years. s 3PECIAL -EMBER Nobuo Shimada, from Nagoya, Japan, died on Decem- 0RICING ber 17, 2007. Born on October 13, 1925, he was a member UP TO OFF of the Society for 47 years. Edsel F. Stiel, professor emeritus, California State s /NLINE SALES EVERY University, Fullerton, died on January 18, 2008. Born on MONTH DISCOUNTS December 19, 1933, he was a member of the Society for UP TO 48 years. Takayuki Tamura, professor emeritus, University of Browse new and California, Davis, died on June 1, 2009. Born on March 12, fforthcoming books 1919, he was a member of the Society for 51 years. Lincoln H. Turner, from Lakewood, CO, died on No- s 6IEW THE 4ABLE OF #ONTENTS vember 25, 2007. Born on June 30, 1928, he was a member s 2EAD A SAMPLE CHAPTER of the Society for 50 years. John H. Ursell, professor emeritus, Queen’s Univer- s "ROWSE THE h7HATS .EWv sity, died on July 30, 2009. Born on June 9, 1938, he was SECTION FOR UPCOMING TITLES a member of the Society for 24 years. J. Richard VandeVelde, professor, Arrupe House, Loyola University of Chicago, died on August 11, 2009. Born on February 21, 1935, he was a member of the Soci- FindF the right ety for 43 years. ttextbook Alfred G. Vassalotti, associate professor emeritus, ffor your course Hofstra University, died on October 21, 2009. Born on January 11, 1929, he was a member of the Society for 45 s s 3PECIlCALLY DESIGNED FOR years. UNDERGRADUATE OR GRADUATE Theresa P. Vaughan COURSES , retired professor, University of North Carolina at Greensboro, died on June 13, 2009. Born on October 13, 1941, she was a member of the Society for 37 years. Frederick N. Webb, from Littleton, MA, died on July 12, 2009. Born on July 22, 1944, he was a member of the Society for 8 years. Alvin M. White, professor emeritus, Harvey Mudd College, died on June 2, 2009. Born on June 21, 1925, he Go to the was a member of the Society for 49 years. AMS Online Bookstorekstore Kathleen B. Whitehead, retired professor, Tufts Uni- www.ams.org/bookstore versity, died on April 18, 2009. Born on November 9, 1920, she was a member of the Society for 66 years. Joseph Zelle, from Euclid, OH, died on July 11, 2009. Born on February 25, 1912, he was a member of the Society 1-800-321-4267 for 51 years. 270 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Reference and Book List The Reference section of the Notices MassMedia/, or contact Stacey Pasco, Program. 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StipendsNovember for 2009, Study p. 1313 and Travel—September 2006, p. 913 aaas.org/programs/education/ FEBRUARY 2010 NOTICES OF THE AMS 271 Reference and Book List February 1, 2010: Applications for Dimensions. See “Mathematics Op- July 31, 2010: Nominations and February review for National Acade- portunities” in this issue. applications for the 2010 Monroe H. mies Postdoctoral and Senior Research April 15, 2010: Applications for Martin Prize. See “Mathematics Associateship Program. See http:// fall 2010 semester of Math in Mos- Opportunities” in this issue. sites.nationalacademies.org/ cow. See http://www.mccme.ru/ August 1, 2010: Applications PGA/RAP/PGA_050491 or contact mathinmoscow or write to: Math in for August review for National Research Associateship Programs, Moscow, P.O. Box 524, Wynnewood, Academies Postdoctoral and National Research Council, Keck 568, PA 19096; fax: +7095-291-65-01; Senior Research Associateship 500 Fifth Street, NW, Washington, DC email: [email protected]. For infor- Program. See http://sites. 20001; telephone 202-334-2760; fax mation on AMS scholarships see nationalacademies.org/PGA/ 202-334-2759; email: [email protected]. http://www.ams.org/outreach/ RAP/PGA_050491 or contact Re- February 1, 2010: Applications mimoscow.html or write to: Math search Associateship Programs, for AWM Travel Grants and Mentor- in Moscow Program, Membership National Research Council, Keck ing Travel Grants. See http://www. and Programs Department, American 568, 500 Fifth Street, NW, Wash- awm-math.org/travelgrants. Mathematical Society, 201 Charles ington, DC 20001; telephone 202- html; telephone: 703-934-0163; Street, Providence RI 02904-2294; 334-2760; fax 202-334-2759; email: email: [email protected]; or con- email: [email protected]. [email protected]. tact Association for Women in May 1, 2010: Applications for October 1, 2010: Applications Mathematics, 11240 Waples Mill May review for National Academies for AWM Travel Grants. See Road, Suite 200, Fairfax, VA 22030. Postdoctoral and Senior Research http://www.awm-math.org/ February 5, 2010: Applica- Associateship Program. See http:// travelgrants.html; telephone: tions for Math for America Fellow- sites.nationalacademies.org/ 703-934-0163; email: awm@awm- ship Program. See http://www. PGA/RAP/PGA_050491 or contact math.org; or contact Association mathforamerica.org/. Research Associateship Programs, for Women in Mathematics, 11240 February 11, 2010: Full propos- National Research Council, Keck 568, Waples Mill Road, Suite 200, Fairfax, als for NSF Undergraduate Biology 500 Fifth Street, NW, Washington, DC VA 22030. and Mathematics (UBM) program. 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See http:// February 15, 2010: Applications Programs, National Research sites.nationalacademies.org/ for Institute for Pure and Applied Council, Keck 568, 500 Fifth PGA/policyfellows/index.htm Mathematics (IPAM) Research in In- Street, NW, Washington, DC 20001; or contact The National Academies dustrial Projects for Students (RIPS) telephone 202-334-2760; fax 202- Christine Mirzayan Science and undergraduate summer research pro- 334-2759; email: [email protected]. gram. See http://www.ipam.ucla. Technology Policy Graduate Fellow- ship Program, 500 Fifth Street, NW, edu. MPS Advisory Committee February 15, 2010: Applications Room 508, Washington, DC 20001; Following are the names and affilia- for AMS-AAAS Congressional Fel- telephone: 202-334-2455; fax: 202- tions of the members of the Advisory lowship. See http://www.ams.org/ 334-1667; email: policyfellows@ Committee for Mathematical and government/congressfellowann. nas.edu. Physical Sciences (MPS) of the Na- html or contact the AMS Washington May 1, 2010: Applications for tional Science Foundation. The date Office, telephone: 202-588-1100; AWM Travel Grants. See http:// of the expiration of each member’s email: [email protected]. www.awm-math.org/travelgrants. term is given after his or her name. February 27, 2010: Entries for html; telephone: 703-934-0163; The website for the MPS directorate AWM Essay Contest. See http:// or email: [email protected]. The may be found at http://www.nsf. www.awm-math.org/biographies/ postal address is: Association for gov/home/mps/. The postal address contest.html. Women in Mathematics, 11240 is Directorate for the Mathematical March 1, 2010: Applications for Waples Mill Road, Suite 200, Fairfax, and Physical Sciences, National Sci- Summer Program for Women in VA 22030. ence Foundation, 4201 Wilson Bou- Mathematics (SPWM2010). See “Mathe- June 1, 2010: Applications for levard, Arlington, VA 22230. matics Opportunities” in this issue. NSF’s Enhancing the Mathematical March 1, 2010: Applications for Sciences Workforce in the Twenty- Clay Mathematics Institute (CMI) First Century (EMSW21) program. Hector D. Abruna (10/10) Summer School on Probability and See http://www.nsf.gov/pubs/ Department of Chemistry and Statistical Physics in Two (and More) 2005/nsf05595/nsf05595.htm. Cornell University 272 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Reference and Book List Taft Armandroff (10/12) Jerzy Leszczynski (10/12) Book List W. M. Keck Observatory Department of Chemistry and The Book List highlights books that Kamuela, Hawaii Biochemistry have mathematical themes and are Jackson State University aimed at a broad audience potentially James Berger (10/11) including mathematicians, students, Department of Statistical Science Theresa A. Maldonado (10/10) and the general public. When a book Duke University (MPSAC/CEOSE Liaison has been reviewed in the Notices, a Department of Electrical and reference is given to the review. Gen- Daniela Bortoletto (10/11) Computer Engineering erally the list will contain only books Department of Physics Texas A&M University published within the last two years, Purdue University though exceptions may be made in Dennis L. Matthews (10/10) cases where current events (e.g., the Kevin Corlette (10/12) College of Engineering and School death of a prominent mathematician, Department of Mathematics of Medicine coverage of a certain piece of math- University of Chicago University of California Davis ematics in the news) warrant drawing readers’ attention to older books. Sug- Eric A. Cornell (10/10) James W. Mitchell (10/12) gestions for books to include on the list JILA Department of Chemical may be sent to notices-booklist@ University of Colorado Engineering ams.org. Howard University *Added to “Book List” since the Juan J. de Pablo (10/12) list’s last appearance. Department of Chemical Ramesh Narayan (10/11) and Biological Engineering Harvard University and The Archimedes Codex: How a University of Wisconsin-Madison Harvard-Smithsonian Center Medieval Prayer Book Is Revealing the for Astrophysics True Genius of Antiquity’s Greatest Joseph M. DeSimone (10/12) Scientist, by Reviel Netz and William Department of Chemistry Sharon L. Neal (10/11) Noel. Da Capo Press, October 2007. University of North Carolina Department of Chemistry and ISBN 978-03068-1580-5. (Reviewed at Chapel Hill Biochemistry September 2008.) University of Delaware The Best of All Possible Worlds: Barbara J. Finlayson-Pitts (10/11) Mathematics and Destiny, by Ivar Department of Chemistry Luis Orozco (10/12) Ekeland. University of Chicago Press, University of California, Department of Physics October 2006. ISBN-13: 978-0-226- Irvine University of Maryland, 19994-8. (Reviewed March 2009.) College Park Irene Fonseca (10/11) The Calculus of Friendship: What a Teacher and Student Learned About Department of Mathematical John Peoples Jr. (10/11) Life While Corresponding About Math, Sciences Fermilab by Steven Strogatz. Princeton Uni- Carnegie Mellon University Batavia, IL versity Press, August 2009. ISBN-13: 978-06911-349-32. Sharon C. Glotzer (10/12) Elsa Reichmanis (10/11) The Calculus Wars: Newton, Leib- Department of Chemical School of Chemical and Engineering Biomolecular Engineering niz, and the Greatest Mathematical University of Michigan Georgia Institute of Technology Clash of All Time, by Jason Socrates Bardi. Thunder’s Mouth Press, April Suzanne Hawley (10/11) Fred S. Roberts (10/12) 2007. ISBN-13: 978-15602-5992-3. Astronomy Department DIMACS (Reviewed May 2009.) University of Washington Rutgers University Chez les Weils (French), by Sylvie Weil. Buchet-Chastel, January 2009. Iain M. Johnstone (chair) (10/10) Joel E. Tohline (10/10) ISBN-13: 978-22830-236-93. Department of Statistics Department of Physics and As- Crossing the Equal Sign, by Stanford University tronomy Marion D. Cohen. Plain View Press, Jan- Louisiana State University uary 2007. ISBN-13: 978-18913-866-95. David E. Keyes (10/10) Crocheting Adventures with Hy- Department of Applied Physics & Geoffrey West (10/11) perbolic Planes, by Daina Taimina. Applied Mathematics Santa Fe Institute A K Peters, March 2009. ISBN-13: 978- Columbia University Santa Fe, NM 15688-145-20. Decoding the Heavens: A 2,000- Year-Old Computer—and the Cen- tury-Long Search to Discover Its Secrets, by Jo Marchant. Da Capo FEBRUARY 2010 NOTICES OF THE AMS 273 Reference and Book List Press, February 2009. ISBN-13: 978- Mathematicians of the World, Unite!: Out of the Labyrinth: Setting Math- 03068-174-27. The International Congress of Math- ematics Free, by Robert Kaplan and The Drunkard’s Walk: How Ran- ematicians: A Human Endeavor, by Ellen Kaplan. Oxford University Press, domness Rules Our Lives, by Leonard Guillermo P. Curbera. A K Peters, March January 2007. ISBN-13: 978-0-19514- Mlodinow. Pantheon, May 2008. ISBN- 2009. ISBN-13: 978-15688-133-01. 744-5. (Reviewed June/July 2009.) 13: 978-03754-240-45. Mathematics and Common Sense: A Passion for Discovery, by Peter Embracing the Wide Sky: A Tour A Case of Creative Tension, by Philip J. Freund. World Scientific, August Across the Horizons of the Human Davis. A K Peters, October 2006. ISBN 2007. ISBN-13: 978-9-8127-7214-5. Mind, by Daniel Tammet. Free Press, 1-568-81270-1. (Reviewed June/July *Perfect Rigor: A Genius and the January 2009. ISBN-13: 978-14165-696- 2009.) Mathematical Breakthrough of the 95. Mathematics Emerging: A Source- Century, by Masha Gessen. Houghton Ernst Zermelo: An Approach to His book 1540–1900, by Jacqueline Stedall. Mifflin Harcourt, November 2009. Life and Work, by Heinz-Dieter Ebb- Oxford University Press, November ISBN-13: 978-01510-140-64. inghaus. Springer, April 2007. ISBN-13 2008. ISBN-13: 978-01992-269-00. Picturing the Uncertain World: How 978-3-540-49551-2. (Reviewed August Mathematics in Ancient Iraq: A to Understand, Communicate, and 2009.) Social History, by Eleanor Robson. Control Uncertainty Through Graphi- Gaming the Vote (Why Elections Princeton University Press, August cal Display, by Howard Wainer, Prince- Aren’t Fair and What We Can Do About 2008. ISBN-13: 978-06910-918-22. ton University Press, April 2009. It), by William Poundstone. Hill and Mathematics in India, by Kim ISBN-13: 978-06911-375-99. Wang, February 2009. ISBN-13: 978- Plofker. Princeton University Press, Plato’s Ghost: The Modernist Trans- 08090-489-22. January 2009. ISBN-13: 978-06911- formation of Mathematics, by Jeremy The Housekeeper and the Profes- 206-76. Gray. Princeton University Press, Sep- sor, by Yoko Ogawa. Picador, February Mathematics in 10 Lessons: The tember 2008. ISBN-13: 978-06911- 2009. ISBN-13: 978-03124-278-01. Grand Tour, by Jerry P. King. Pro- 361-03. (Reviewed in this issue.) How Math Explains the World: A metheus Books, May 2009. ISBN: 978- The Princeton Companion to Math- Guide to the Power of Numbers, from 1-59102-686-0. ematics, edited by Timothy Gowers Car Repair to Modern Physics, by The Mathematics of Egypt, Meso- (June Barrow-Green and Imre Leader, James D. Stein. Collins, April 2008. potamia, China, India, and Islam: A associate editors). Princeton University ISBN-13: 978-00612-417-65. Sourcebook, by Victor J. Katz et al. Press, November 2008. ISBN-13: 978- How to Think Like a Mathemati- Princeton University Press, July 2007. 06911-188-02. (Reviewed November cian: A Companion to Undergradu- ISBN-13: 978-0-6911-2745-3. 2009.) ate Mathematics, by Kevin Houston. The Millennium Prize Problems, *Proofs from THE BOOK, by Martin Cambridge University Press, March edited by James Carlson, Arthur Aigner and Günter Ziegler. Expanded 2009. ISBN-13: 978-05217-197-80. Jaffe, and Andrew Wiles. AMS, June fourth edition, Springer, October Leonhard Euler and His Friends: 2006. ISBN-13: 978-08218-3679-8. 2009. ISBN-13: 978-3-642-00855-9 Switzerland’s Great Scientific Expatri- (Reviewed December 2009.) Pythagoras’ Revenge: A Mathe- ate, by Luis-Gustave du Pasquier (trans- More Mathematical Astronomy matical Mystery, by Arturo Sangalli. lated by John S. D. Glaus). CreateSpace, Morsels, by Jean Meeus. Willmann- Princeton University Press, May 2009. July 2008. ISBN: 978-14348-332-73. Bell, 2002. ISBN 0-943396743. ISBN-13: 978-06910-495-57. Lewis Carroll in Numberland: His The Numbers Behind NUMB3RS: Recountings: Conversations with Fantastical Mathematical Logical Life: Solving Crime with Mathematics, by MIT Mathematicians, edited by Joel An Agony in Eight Fits, by Robin Wil- Keith Devlin and Gary Lorden. Plume, Segel. A K Peters, January 2009. ISBN- son. W. W. Norton & Company. ISBN-13: August 2007. ISBN-13: 978-04522- 13: 978-15688-144-90. 978-03930-602-70. 8857-7. (Reviewed March 2009.) Rock, Paper, Scissors: Game Theory Logic’s Lost Genius: The Life of The Numbers Game: The Common- in Everyday Life, by Len Fisher. Basic Gerhard Gentzen, by Eckart Menzler- sense Guide to Understanding Numbers Books, November 2008. ISBN-13: 978- Trott, Craig Smorynski (translator), in the News, in Politics, and in Life, by 04650-093-81. Edward R. Griffor (translator). AMS- Michael Blastland and Andrew Dilnot. *Roger Boscovich, by Radoslav LMS, November 2007. ISBN-13: 978-0- Gotham, December 2008. ISBN-13: 978- Dimitric (Serbian). Helios Publishing 8218-3550-0. 15924-042-30. Company, September 2006. ISBN-13: The Mathematical Mechanic: Using The Numerati, by Stephen Baker. 978-09788-256-21. Physical Reason to Solve Problems, by Houghton Mifflin, August 2008. ISBN- Sacred Mathematics: Japanese Tem- Mark Levi. Princeton University Press. 13: 978-06187-846-08. (Reviewed ple Geometry, by Fukagawa Hidetoshi ISBN: 978-0691140209. October 2009.) and Tony Rothman. Princeton University Mathematicans Fleeing from Nazi Our Days Are Numbered: How Press, July 2008. ISBN-13: 978-0-6911- Germany: Individual Fates and Global Mathematics Orders Our Lives, by 2745-3. Impact, by Reinhard Siegmund- Jason Brown. McClelland and Stewart, *Solving Mathematical Problems: A Schultze. Princeton University Press, April 2009. ISBN-13: 978-07710-169- Personal Perspective, by Terence Tao. July 2009. ISBN 978-0-691-14041-4. 67. Oxford University Press, September 274 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Reference and Book List 2006. ISBN-13: 978-0-199-20560-8. (Reviewed in this issue.) Sphere Packing, Lewis Carroll, and Reversi, by Martin Gardner. Cambridge University Press, July 2009. ISBN: 978- 0521756075. Strange Attractors: Poems of Love and Mathematics, edited by Sarah Glaz and JoAnne Growney. A K Peters, November 2008. ISBN-13: 978- 15688-134-17. (Reviewed September 2009.) *The Strangest Man, by Graham Farmelo. Basic Books, August 2009. ISBN-13: 978-04650-182-77. Super Crunchers: Why Thinking- by-Numbers Is the New Way to Be Smart, by Ian Ayres. Bantam, August 2007. ISBN-13: 978-05538-054-06. (Reviewed April 2009.) Tools of American Math Teach- ing, 1800–2000, by Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts. Johns Hopkins University Press, July 2008. ISBN-13: 978-0801888144. (Reviewed January 2010.) The Unfinished Game: Pascal, Fer- mat, and the Seventeenth-Century Letter That Made the World Modern, by Keith Devlin. Basic Books, Sep- tember 2008. ISBN-13: 978-0-4650- 0910-7. What Is a Number?: Mathematical Concepts and Their Origins, by Rob- ert Tubbs. Johns Hopkins University Press, December 2008. ISBN-13: 978- 08018-901-85. What’s Happening in the Mathe- matical Sciences, by Dana Mackenzie. AMS, 2009. ISBN-13: 978-08218-447- 86. Why Does E=mc2? (And Why Should We Care?), by Brian Cox and Jeff Forshaw. Da Capo Press, July 2009. ISBN-13: 978-03068-175-88. Zeno’s Paradox: Unraveling the Ancient Mystery Behind the Science of Space and Time, by Joseph Mazur. Plume, March 2008 (reprint edition). ISBN-13: 978-0-4522-8917-8. FEBRUARY 2010 NOTICES OF THE AMS 275 Doctoral Degrees Conferred 2008–2009 ALABAMA Dube, Tina, Assessing and correcting the Liu, Congxiao, Mathematical study for the effects of measurement error among switch-initiating field of 2d ferroanti- Auburn University (10) correlated covariates in a proportional ferromagnet exchange coupled systems hazards setting Department of Mathematics and Wu, Leina, Grid refinement method for partial differential equations Statistics Keith, Scott, Free-knot splines and boot- strapping for nonlinear modeling in Baker, Charla, The intersection problem complex samples for Latin squares with holes of size 2 Li, Qing, Interim monitoring efficacy, ARIZONA and 3 safety and futility in Phase III clinical Bobga, Benkam, Some necessary condi- trials Arizona State tions for list colorability of graphs and Nair, Nitin, Adaptive procedures to detect University (14) a conjecture on completing partial Latin treatment effects under unexpected School of Mathematical and squares covariate interactions Statistical Sciences Chidume, Chukwudi, Iteration methods Prucka, William, Wavelet-based regres- for approximation of solutions of non- sion and classification for longitudinal Alvarado, Alejandra, Arithmetic progres- linear equations in Banach spaces diffusion tensor imaging data sions on curves Divavahi, Venkata, Graph decomposi- You, Zhiying, Power and sample size of Fortney, Jon Pierre, Dirac structures in tions cluster randomized trials pseudo-gradient systems with an em- Gammon, Kevin, Factorwise rigidity in- phasis on electrical networks volving hereditarily indecomposable Department of Mathematics Kang, Yun, The dynamics of plant- spaces herbivore interactions and their impli- Kulkarni, Mandar,Multi-coefficient cations for spatial expansion Jin, Kang, On the lattice Boltzmann Dirichlet-Neumann type elliptic inverse method: Implementation and applica- problems with application to reflection Kelkar, Ashwini, A study of the subgraphs tions seismology and the conjecture of the middle two layers graph, using modular matchings Newman, Nicolas, Enclosings of small Mavinga, Nsoki, Nonlinear second order cycle systems parabolic and elliptic equations with Kupresanin, Ana Maria, Topics in func- Nudurupati, Sai, Robust nonparametric nonlinear boundary conditions tional canonical correlation and regres- discriminant analysis procedures son Peng, Man, Palm measure invariance University of La Marca, Michael Benedetto, Control and exchangeability for marked point of continuum models of production (2) processes Alabama-Huntsville systems Turkmen, Asuman, Robust partial least Department of Mathematics Park, Russell, Optimal compression and squares for regression and classifica- numerical stability for Gegenbauer re- tion Bowie, Miranda, Liar’s domination and constructions with applications the domination continuum Saxena, Rishu, High order methods for University of Puckett, Matthew, Minimum wave speed edge detection and applications and uniqueness of monotone traveling (11) Spiriti, Steven Mark,Randomsearch Alabama-Birmingham waves optimization for free-knot splines and Department of Biostatistics University of P-splines Azuero, Andres, Comparisons of sequen- Stefan, Wolfgang, Total variation regu- tial testing approaches for detection Alabama-Tuscaloosa (4) larization for linear ill-posed inverse of association between disease and Department of Mathematics problems: Extensions and applications haplotype blocks Unver, Ali Kemal, Observation based Banerjee, Samprit, Bayesian genome-wide Fayoumi, Hiba,Astudyofbinarysystems PDE models for stochastic production QTL mapping for multiple traits on sets and applications to several systems Chen, Lang, Microarray data analysis for classes of such systems Wallington, Rachel,Numberfieldswith SNPs effects and inferring alternative Kang, Hyuna, Stochastic and key rate solvable Galois groups and small Galois splicing duration for value at risk root discriminants The above list contains the names and thesis titles of recipients of in parentheses following the name of the university is the number of doctoral degrees in the mathematical sciences (July 1, 2008, to June 30, degrees listed for that university. A supplementary list containing names 2009) reported in the 2009 Annual Survey of the Mathematical Sciences received since compilation of this list will appear in a summer 2010 issue by 223 departments in 158 universities in the United States. Each entry of the Notices. contains the name of the recipient and the thesis title. The number 276 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Yang, Hongling, A study of additive Claremont Graduate Cotton-Clay, Andrew, Symplectic Floer coefficient models University (7) homology of area-preserving surface Zhang, Guoyi, Smoothing splines using diffeomorphisms and sharp fixed point compactly supported, positive definite, School of Mathematical Sciences bounds radial basis functions Bazan, Carlos, PDE-based image and Dutter, Seth, Logarithmic jet spaces and structure enhancement for electron to- intersection multiplicities University of Arizona (9) mography of mitochondria Flenner, Joseph, The relative structure of Department of Mathematics Hui, Kwok, Risks analysis of soft- Henselian valued fields ware development using Bayesian belief Judson, Zachary, Connectivity in the Bhattacharya, Abhishek,Nonparametric network and non-linear programming Higson corona of the positive real statistics on manifolds with applica- methods access tions to shape spaces Limon, Alfonso, A multilevel framework Kalaycioglu, Selin, Computing the projec- Long, Yun, Mixing time of the Swendsen- for PDEs whose solution exhibits fast tive indecomposable modules of large Wang dynamics in complete graphs and transitions finite groups trees Muhi El-Ddin, Imad,Watermarking Mkrtchyan, Sevak, Scaling limits of ran- Program in Applied Mathematics schemes robust against affine attacks: dom skew plane partitions An application of digital image process- Arciero, Julia, Theoretical models of Peters, Emily, A planar algebra construc- ing in information technology blood flow regulation tion of the Haagerup subfactor Nguyen, James, A hardware implementa- Beauregard, Matthew, Nonlinear dynam- Prat-Waldron, Arturo, Pfaffian line bun- tion of the level set method for robotic ics of elastic filaments conveying a fluid dles over loop spaces, spin structures path finding with multiple obstacle and numerical applications to the static and the index theory avoidance Kirchhoff equations Ramakrishnan, Janak,Typeofo-minimal Park, Jeho, An empirical approach to Rael, Rosalyn, Comparing theory and theories communication and performance mod- data on multi-species interactions using 2 eling for message passing parallel ap- Rothbach, Brian, Borel orbits of X =0in evolutionary game theory plications on cluster systems Gln Reich, Daniel, Stochastic networks: Tracta- Zhu, Bing, Computational modeling and Sain, Jeremy, Berezin quantization from ble approaches for identifying strategic bifurcation analysis of bubbling flu- ergodic actions of compact quan- paths idized processes tum groups and quantum Gromov- Robertson, Suzanne, Spatial patterns Hausdorff distance in stage-structured populations with Naval Postgraduate Sammis, Ian, Implicit and fourth-order density-dependent dispersal School (1) semi-Lagrangian contouring for geo- Stover, Joseph, A general theory of mono- metric moving interface problems Department of Applied Mathematics tonicity for interaction map particle Schommer-Pries, Christopher,Theclas- systems and a stochastic spatial model sification of two-dimensional extended for biological invasions Dea, John, High-order non-reflecting boundary conditions for the linearized topological field theories Vandiver, Rebecca, Morphoelasticity: The Euler equations Snyder, Noah, Quantum groups, tensor mechanics and mathematics of elastic categories and knot invariants growth Stanford University (5) Somersille, Stephanie,Biasedtugofwar, Department of Statistics the biased infinity Laplacian and com- CALIFORNIA parison with exponential cones Hoefling, Holger, Topics in statistical Stovall, Lindsay, Lp inequalities for cer- California Institute of learning tain generalized Radon transforms Kapur, Karen, Modelling background and Technology (8) Tohaneanu, Mihai, Local energy esti- cross-hybridization effects of microar- mates and structure estimates on Department of Applied and ray probes Schwarzschild and Kerr black hole Computational Mathematics Khare, Kshitij, Constructions of Gaussian backgrounds fields from Markov processes, and Lyon, Mark, High-order unconditionally- Tolland, Andrew, Gromov-Witten gauge related topics stable FC-AD PDE solvers for general theory domains She, Yiyuan, Sparse regression with exact clustering Trokhimchouk, Maxim,Regularityfor Othmer, Jonathan, Mapping nucleic acid certain parabolic diffusion systems free energy landscapes Zhou, Hua, Topics on Markov chains with polynomial eigenfunctions Varilly-Alvarado, Anthony, Arithmetic of Stern, Ari, Geometric discretization of del Pezzo surfaces of degree 1 Lagrangian mechanics and field theories University of California, Wu, Guoliang, Impulse control of multidi- Department of Mathematics mensional diffusion and jump diffusion (41) Berkeley processes Pragel, Daniel, Embeddings of one- Department of Mathematics factorizations of hypergraphs and de- Zhu, Xinwen, Gerbal representations of double loop groups compositions of partitions Armstrong, Scott, Principal half-eigen- Saha, Abhishek,Oncriticalvaluesof values of fully nonlinear homogeneous Department of Statistics L-functions for holormorphic forms on elliptic operators GSp(4) × GL(2) Baginski, Paul,Stableℵ0-categorical alge- Ahn, Soyeon, Statistical topics in gene Vuletic, Mirjana, The Pfaffian Schur pro- braic structures regulation cess Blasiak, Jonah, Cyclage, catabolism, and Bae, Chongsoon, Analyzing random forests Wong, Manwah, Orthogonal polynomials, theaffineHeckealgebra Hather, Greg, Statistical analysis of DNA paraorthogonal polynomials, and point Brown, Jeffrey, Gromov-Witten invariants sequence motifs and microarray data perturbation of toric fibrations Liang, Richard, Two continuum-sites Zhuang, Dongping, A geometric study of Chen, Qingtao,Somemathematicalas- stepping stone models in population commutator subgroups pects of quantum field theory genetics with delayed coalescence FEBRUARY 2010 NOTICES OF THE AMS 277 Doctoral Degrees Conferred Obozinski, Guillaume, Simultaneous vari- Chen, Kun, Functional approaches for Gautam, Sushrut Zubin,Twogeometric able selection and simultaneous sub- high dimensional and gene expression obstruction results in harmonic analy- space selection for multitask learning data sis Ralph, Peter, Most recent common an- Chen, Ying, Statistical approaches for Herman Jr., Paul Edward, Applications of cestors, genetic inheritance, stochastic detection of relevant genes and pathway beyond endoscopy gene transcription and Brownian mo- in analysis of gene expression data Kim, Yunho, Variational methods: Theory tion on disconnected sets: A probabilis- Liu, Bitao, Functional data analysis for and its applications to image deblurring tic analysis of a few models online auction data and denosing problems Rosenberg, David, Semi-supervised learn- Liu, Jingyi, A shrinkage-average method Lie, Victor Daniel, Relational (quadratic) ing with multiple views for fitting high dimensional linear mod- time-frequency analysis Sly, Allan, Spatial and temporal mixing els Liu, Jian, Controlling the dynamics of of Gibbs measures Liu, Ziqi, Nonparametric bootstrap method recurrent neural networks with synaptic Vu, Vince, High dimensional data analy- on Stiefel manifold and GPAV algorithm learning rules sis: Entropy and regression for ASP fit Lu, Steve, New constructions in pairing- Xu, Na, Transcriptome detection by mul- Norris, Ann (Michelle), Parametric and based cryptography tiple RNA tiling array analysis and semiparametric joint modeling for lon- Mathews, Bryant, Canonical dimension identifying functional conserved non- gitudinal diagnostic outcomes coding elements by statistical testing of projective homogeneous varieties of Weng, Qian, Modeling progression of inner Type A and Type B Yan, Donghui, Some issues with dimen- neurodegenerative disease with longi- sionality in statistical inference Nguyen, Dinh Huu,OnP-generic splitting tudinal neuroimaging data varieties for Milnor K-symbols mod P Group in Biostatistics Xu, Qiuyan, Estimation of integrated Roy, Tristan, Global analysis of the covolatility for asynchronous assets in Gilbert, Houston, Multiple hypothesis defocusing cubic wave equation in the presence of microstructure noise testing: Methodology, software imple- dimension 3 mentation, and applications to ge- University of California, Shao, Shuanglin, Restriction of estimates nomics for paraboloids and cones in the cylin- Moore, Kelly, Targeted maximum likeli- Irvine (8) drically symmetric case hood estimation of treatment effects in Department of Mathematics Shiber, Dan, Tracial and non-tracial ran- randomized controlled trials and drug dom matrix models in free probability Bjork, Andrew-David, Criteria for rank- safety analysis Thompson, Alexander, A metamathemati- one transformations to be weakly mix- Rubin, Daniel, Applications of double cal condition equivalent to the existence ing, and the generating property robustness of a complete left invariant metric for a Cao, Yanping,Ontheα-regularization of Polish group University of California, nonlinear partial differential equations- Vanvalkenburgh, Michael,Hermite/ analysis and error estimates Davis (20) Laguerre-Gaussian modes and lower Cox, Sean, Covering theorems for the Department of Mathematics bounds for quasimodes of semiclassi- core model, with applications cal operators Baba, Shinpei, Decomposition theorems Hu, Zhengzheng, Phase-field modeling of Virtanen, Jukka Tapio, Structure of ele- for complex projective structures thin film epitaxial growth mentary particles in non-Archimedean Berg, Christopher,Combinatoricsof(l, 0)- Kypriotakis, Kyriakos,Combinatorialprin- spacetime JM partitions, l-cores, the ladder crystal ciples in Jensen type extender models and the finite Hecke algebra of the form L[E] Department of Statistics Hallowell, Karl, Higher spin approaches Le, Phong, Coherent decomposition of p- Anderson, Ariana, Machine learning for to quantum field theory and (pseudo)- adic Newton polygons for L-functions classification and diagnosis of func- Riemannian geometrics of exponential sums tional magnetic resonance image scans Haws, David, Matroid polytopes: Algo- Luo, Songting,Numericalmethodsfor rithms, theory, and applications Zhu, Long, Recursive composition in static Hamilton-Jacobi equations computer vision: Modeling, inference, Miller, Adam, How disturbance creates Matayoshi, Jeffrey,Thezerosofrandom and learning the storage effect and promotes main- polynomials tenance of species diversity University of California, Mulherkar, Jaideep, Some properties of University of California, the XXZ spin chain and their applica- Riverside (7) Los Angeles (22) tions to quantum computing Department of Statistics Needell, Deanna, Topics in compressed Department of Mathematics sensing Bi, Yingtao, Theoretical analysis of classi- Blunk, Mark, Del Pezzo surfaces of degree Raz, Hillel, Lieb-Robinson bounds in the fication under CCC-noise and its appli- 6 over an arbitrary field anharmonic lattice cation to semi-supervised text mining Chen, Lin, Chern-Simons theory of knot Sershen, Cheryl, A dynamic model of Kim, Sungsu, Inverse circular regression invariants DNA structure and function with possibly asymmetric error distri- Stevens, Alice, Knots in Heegaard surfaces Chifan, Ionut, Deformation spectral gap bution rigidity in von Neumann algebras Strawbridge, Eva, Mechanics, dynamics, Montes de Oca, Veronica,Nonparametric and structures of DNA Citro, Craig, L-invariants of adjoint cusum algorithms with applications to square Galois representations coming network surveillance Department of Statistics from modular forms Pal, Rupam Ranjan, Efficient designs for Barkauskas, Donald, Statistical analy- Conley, Clinton, Some applications of discriminating between two competing sis of matrix-assisted laser desorp- combinatorics in descriptive set theory linear regression models tion/ionization Fourier transform ion Dong, Bin, Applications of variational Shah, Payal, Sequential sampling meth- cyclotron resonance mass spectrome- models and partial differential equa- ods using generalized linear models try data with applications to cancer tions on medical image and surface with applications to pest density esti- biomarker detection processing mation 278 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Suarez Espinoza, Javier, Inference for the Department of Statistics and Applied COLORADO multiparameter skew normal distribu- Probability tion Colorado State Yu, Hua, Neutral zone classifiers within a Eli, Kollman, Calibrating market and asset University (13) decision-theoretic framework returns from stochastic volatility skew effects Department of Mathematics University of California, Jang, Homin, Shrinkage estimators of Bikowski, Jutta, Electrical impedance to- San Diego (16) variances and their applications to mography reconstructions in two and three dimensions; form Calderon to Department of Mathematics microarray data analysis direct methods Marick, Sinay, Bayesian inference for Armel, Jonathan, Holomorphic extension linear and generalized linear models Frederick, Christopher, Persistence ho- of solutions to homogeneous analytic with flexible prior structure on the mology of sequences of neighborhood partial differential equations covariance matrix complexes for graphs Averett, Maia Christine, On real Johnson- Jamshidi, Arthur, Modeling spatio-temp- Wilson theories Tung, David, Some contributions to in- oral systems with skew radial basis ference based on spacing Beeson, Amanda, Groups of special units functions: Theory, algorithms and ap- Comstock, Jana, A finite presentation of Wignall, Brian, Structural modeling and plications knotted trivalent graphs top-down reduced-form modeling for Jonsd´ ottir,´ Margr´et, Automorphism tow- Eldredge, Nathaniel, Holomorphic ex- multi-name credit derivatives ers of general linear groups tension of solutions to homogeneous Ladd, Joshua, Large-scale computational analytic partial differential equations University of California, analysis of national animal identifica- tion system mock data, including trace Fuller, Evan, Generating functions for Santa Cruz (1) composition/word statistics back and trace forward Grice, Jon, Discrete quantum control Department of Applied Mathematics Lee, Sheldon, An adaptive algorithm Gurvich, Michael, Some results on the and Statistics for an elliptic optimization problem, topology of quasitoric manifolds and and stochastic-deterministic coupling: A mathematical framework their equivalent mapping spaces Schuresko, Michael, Controlling global Horn, Paul, Random subgraphs of a given network connectivity of robot swarms Department of Statistics graph with local interactions Delorey, Mark, Penalized estimation for Jehring, Kristin, Harmonic functions on sample surveys in the presence of Walsh’s Brownian motion University of Southern auxiliary variables Kinnally, Michael, Stational distributions California (12) E, Lidong, Application of generalized for stochastic delay differential equa- fiducial inference tions with non-negativity contraints Department of Mathematics Hancock, Stacey, Estimating the number Lee, Nam Heon, A sufficient condition of structural breaks in nonstationary for stochastic stability of an Internet Aydilek, Harun, Optimal decisions under time series congestion control model in terms of recursive utility fluid model stability Huang, Wenying, Spatial processes with Glatt-Holtz, Nathan, Stochastic equations stochastic heteroscadesticity Niedermaier, Andrew, Statistics on wreath in geophysical fluid dynamics products Iverson, Todd, Kernel methods for a Guest, Simon, A solvable version of the discrete state/continuous time process Raleigh, Sean, Ruled Legendrian knots with auxiliary information and Lagrangian surfaces Baer-Suzuki theorem and generaliza- tions Wang, Ke, Spatial models with applica- Szypowski, Ryan, Least-squares finite el- tions in computer experiment ements and constrained evolution sys- Jedwab, Andrea, Representations of finite Yang, Yu, Estimation for L´evy-driven tems dimensional Hopf algebras CARMA processes Yacobi, Oded, An analysis of multiplicity Oleksandr, Lytvak, Fully discrete ap- spaces in classical symplectic branching proach for estimating local volatility in University of Colorado, University of California, the generalized Black-Scholes setting Boulder (13) Piterbarg, Yuliya, Population modeling Department of Applied Mathematics Santa Barbara (12) and Bayesian estimation for the decon- Department of Mathematics volution of blood alcohol concentration Barker, Andrew, Parallel monolithic fluid- from transdermal biosensor data structure interaction algorithms with Dai, Dongyan, Convergence of the Yam- application to blood flow simulation abe flow on manifolds with boundary Psiloyenis, Yiannis, Mixing conditions and long return times on Markov towers Haut, Terry, Analysis and applications of Hendrata, Melisa,Adynamicalsystem nonlocal spectral formulations of fluids model for simulating myxobacteria life Seliger, Philip, Optimal device design with free interfaces and surfaces cycle Song, Qian, Optimal and exact control Sanders, Geoffrey, Two extensions to Kessenich, Paul,Globalexistencewith of deterministic, stochastic or parabolic adaptive smooth aggregation (aSA) small initial data for three-dimensional differential equations multigrid: Eigensolver initialization and incompressible isotropic viscoelastic nonsymmetric problems materials Yang, Yan, Mathematical modeling and analysis of representation error and Simpson, David, Bifurcations in piecewise- Levitt, John, Embeddings of Mori dream smooth, continuous systems spaces system uncertainties for discretized Mohler, George, Efficient, non-stiff, and distributed parameters systems Department of Mathematics multiscale methods for complex fluids Zeng, Yu, Large deviations approach to D’Andrea, Jonas, Wavelet frames on frac- Qi, Jiawei, Effective dynamics for fer- the bi-stable systems tal spaces of Dutkay-Jorgensen type rofluids Zhou, Yuegang, Credit risk of a leveraged Denoncourt, Hugh, Some combinatorial Taylor, Scott, Boring split links and firm in a controlled optimal stopping models for reduced expressions in unknots framework Coxeter group FEBRUARY 2010 NOTICES OF THE AMS 279 Doctoral Degrees Conferred Ernst, Dana, A diagrammatic represen- Levin, Oscar, Computability theory, re- Wei, Ang, Random-harmonic functions tation of an affine C Temperly-Lieb verse mathematics and ordered fields and multivariate Gaussian estimates algebra Markkanen, Tyler, Separating the degree Wu, Junhua, Geometric structures and Formichella, Marc, Functional equations spectra of structures linear codes related to conics in classi- among Barnes’ integrals and hypergeo- Prime, Russell, Averaging quadratic L- cal projective planes of odd orders metric series functions over function fields Xu, Liwei, Computational methods for a Kang, Sooran, The Yang-Mills functional Rong, Zhou, Numerical simulation of cell class of problems in acoustic, elastic and Laplace’s equation on quantum movement and water waves Heisenberg manifolds Sang, Hailin, Asymptotic properties of Zeev, Noam, Direct and inverse scatter- Mishev, Ilia, Coxeter group actions on generalized kernel density estimators ing problems for thin dielectric and supplementary pairs of Saalschutzian¨ Termine, Lisa, Existence of solutions partially coated objects 4F3(1) hypergeometric series to semilinear elliptic differential equa- Pohlmann, Brent, Structural properties tions using computer verification of acyclic heaps with applications to Wooster, Robert, Evolution systems of DISTRICT OF Kazhdan-Lusztig theory measure for stochastic differential Schumacher, Timothy, Removable bound- equation with Levy noise COLUMBIA ary singularities for the equation ∆u = Department of Statistics uα in non-smooth domains George Washington Shaw, Jason, Commutator relations and Xie, Wangang, Bayesian phylogenetic University (4) the clones of finite groups model selection and applications Department of Mathematics Zhao, Yifang, Contributions to microar- University of Colorado, ray data analysis Luse, Terry, Invariants of knots, graphs, Denver (5) and Feynman diagrams Wesleyan University (4) Department of Biostatistics and Pingrey, Sarah, Strong degree spectra of Department of Mathematics and Informatics relations Computer Science Boyd, Adam, Censoring-robust treatment Department of Statistics Gould, Franklin, On certain classes of effect estimation in clinical trials with Hikawa, Hiroyuki, Local linear Peters- time-to-event outcomes minimally almost periodic groups Hall, Becky, An improved method for Belson regression and its applications Department of Mathematical and computing group cohomology of con- to employment discrimination cases Statistical Sciences gruence subgroups of SL3(Z) Yan, Lihan, Group sequential robust Pan, WeiWei, Group actions on categori- designs in genetic studies Beezley, Jonathan, High-dimensional data fied bundles assimilation and morphing ensemble Kalman filters with applications in Shea, Steve, Finitary isomorphisms of Howard University (2) r -processes wildfire modeling Department of Mathematics Flink, Stephen, Truncated quadrics and Yale University (4) elliptic curves Al-Islam, Najja,Pseudoalmostperiodic solutions to some systems of non- Harris, Angela, Cycle structures in graphs Department of Mathematics linear hyperbolic second-order partial Zakharyan, Armen, Stochastic diffusion Aryapoor, Masood, The Penrose trans- differential equations model of heterogeneous populations form and self-dual Yang-Mills equations in split signature Salaam, Lifoma, Combinatorial statistics on phylogenetic trees University of Northern Le Donne, Enrico, Bi-Lipschitz homoge- Colorado (2) neous geodesic manifolds Mohammadi, Amir, Unipotent flows in School of Mathematical Sciences FLORIDA positive characteristics and applica- Phipps, Marnie, A phenomenological in- tions Florida Atlantic vestigation on eighth graders’ number Warshall, Andrew, Depth and related sense of fractions properties of infinite finitely generated University (4) Toney, Allison, Women with advanced groups Department of Mathematical degrees in mathematics in doctoral Sciences programs in mathematics education DELAWARE Bozovic, Vladimir,Algebraicandcombi- CONNECTICUT University of Delaware (8) natorial aspects of group factorizations Department of Mathematical Science Hopkins, Mary, Weakly integrally closed University of Connecticut, monoids and forbidden patterns Gutekunst, Todd, Subsets of finite groups Kasprikova, Eva, Higher-order commuta- Storrs (13) exhibiting additive regularity tors in the method of orbits Department of Mathematics Heryudono, Alfa, Adaptive radial basis Wess, Mark, Computing topological dy- function methods for the numerical so- namics from time series Alagic, Gorjan, Uncertainty principles for lution of partial differential equations compact groups with application to the simulation of Bella, Thomas, Topics in numerical linear the human tear film Florida Institute of algebra related to quasiseparable and Maki, Kara, Computational solutions of Technology (1) other structured matrices linear systems and models of the Department of Mathematical Corluy, Marc, Rates of conveyance in the human tear film Sciences central limit theorem for Markov chains Wang, Zeying, Skew Hadamard difference Jura, Matthew, Reverse mathematics and sets, strongly regular graphs and p-ary Huang, Weijun, Sequential stochastic the coloring number of graphs bent functions games 280 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Florida State Department of Statistics Georgia Institute of University (12) Li, Zhen, Bayesian methodologies for Technology (13) Department of Mathematics genomic data with missing covariates School of Mathematics Roy, Vivekananda, Theoretical and meth- Lebedev, Yuri,Openmathlibraryfor odological developments for Markov Bilinski, Mark, Approximating the cir- computing on Riemann surfaces chain Monte Carlo algorithms for cumference of 3-connected claw-free Lin, Haomin, An optimal control problem Bayesian regression graphs for a time dependent Ginzburg-Landau Bird, Csaba, Problems and results in Wu, Song, A robust approach for genetic model of superconductivity partially ordered sets, graphs and ge- mapping of complex traits Mavroudis, Konstantinos,Constantpro- ometry portions portfolio strategies in an evo- Yap, John, Nonparametric covariance es- Greenberg, Samuel, Random sampling lutionary context under a dividend timation in functional mapping of com- of lattice configurations using local factor model plex dynamic traits Markov chains Salta, Emmanuel, Variance reduction Jim´enez, David,Analysisoftwoprob- (3) techniques in pricing financial deriva- University of Miami lems in signal quantizations and A/D tives Department of Mathematics conversion Shah, Manan,Randomizedquasi-Monte Li, Yongfeng, Nonlinear oscillation and Carlo methods and the computation of Coburn, Brian, Multi-species influenza control in the BZ chemical reaction models with recombination endogenous mortgage rates Litherland, Trevis, On the limiting shape Tzigantcheva, Milena, Stochastic volatil- Donzelli, Fabrizio, Algebraic density prop- of random young tableaux for Marko- ity extensions of the swap market erty of homogeneous spaces vian words model Zivanovic, Sanja, Attractors in dynamics Marcus, Adam, New combinatorial tech- Yoo, Eunjoo, Variance gamma pricing of with choice niques for non-linear orders American futures options Pearson, John, The noncommutative ge- Department of Statistics University of South ometry of ultrametric Cantor sets Florida (1) Pugliese, Alessandro, Theoretical and nu- Balov, Nikolay, Covariance on manifolds merical aspects of coalescing of eigen- Department of Mathematics and Ivanescu, Andrada, Revealing sparse sig- values and singular values of parameter nals in functional data Statistics dependent matrices Lin, Lanjia, Association models for clus- Miladinovic, Branko, Kernel density esti- Savinien, Jean, Cohomology and K-theory tered data with binary and continuous mation of reliability with applications of aperiodic tilings responses to extreme value distributions Xu, Hua, Aspects of random matrix con- Liu, Yang, Transformation models for centration and subsequence problems survival data analysis and applications Young, Stephen, Random dot product Simino, Jeanette, Discrimination and cali- GEORGIA graphs: A flexible model for complex bration of prognostic survival models networks University of Central Emory University (11) Yurchenko, Aleksey,Someproblemsin Department of Biostatistics the theory of open dynamical systems Florida (8) and deterministic walks in random Department of Mathematics Kwee, Lydia, Multilocus methods for environments association mapping of complex traits Boustique, Hatim, Lattice valued conver- University of Georgia (13) gence: Quotient maps Lu, Chengxing, Statistical methods to Bryant, Donald,AnalysisofKolmogorov’s adjust for misclassified repeated ex- Department of Mathematics superposition theorem and its implica- posures in modeling disease-exposure associations Bagci, Irfan, Cohomology and support tions with low and high dimensional varieties in Lie superalgebras Price, Megan, Issues in causal inference data Hower, Jeremiah, On elliptic curves and and applications to public health Galiffa, Daniel J., The Sheffer B-type 1 arithmetical graphs orthogonal polynomial sequences Wiener, Jeffrey, Evaluating agreement Zhuang, Chao, Approximation methods Khosravi, Mehrdad, Pseudoquotients: Con- among observers or methods of mea- and applications in financial optimiza- struction, applications, and their Fourier surement for quantitative data tion problems transform Department of Mathematics and Konate, Souleymane, Efficient cone beam Department of Statistics Computer Science reconstruction for the distorted circle Breazel, Ellen Hepfer, Effects of common and line trajectory Avart, Christian, Colorful flowers errors in microsatellite data on esti- Landon, Benjamin, Degree of approxima- Castle, Mariangely,Polya’stheoremwith mates of population differentiation and tion of Holder continuous functions zeros inferring genotypic structure of com- plex loci using genome-wide expression Lopez, Jerry, Optimal dual frames for Chung, Julianne, Numerical approaches data erasures and discrete Gabor frames for large-scale ill-posed inverse prob- Macon, Lisa Fischer,Almostregular lems Chen, Jien, Applications of empirical graphs and edge face colorings of likelihood to quantile estimation and Dudek, Andrzej, Problems in extremal plane graphs longitudinal data combinatorics Le-Rademacher, Jennifer,Principalcom- University of Florida (6) Malagon, Audrey, Killing forms of Lie ponent analysis for interval-valued and algebras Department of Mathematics histogram-valued data and likelihood Poerschke, Annika, On algorithmic hyper- functions and estimators for symbolic Dashti, Seyyed, Effective symbolic dy- graph regularity data namics Razouk, Nader, Localization phenomena Liu, Shangbin, Statistical issues on mass Nguyen, Hung Ngoc, Representations of in matrix functions: Theory and algo- spectrometry-based protein identifica- finite groups of Lie type rithms tion and quantitation FEBRUARY 2010 NOTICES OF THE AMS 281 Doctoral Degrees Conferred Qiu, Junshan, Variable selection methods Meshes, Jonathan, Functions in class F in Bennett, Hanna, Volume distortion in with applications the unit disc which have integrals not groups Sun, Guoying,Ageneticalgenomicsto in F Biringer, Ian, Algebra versus geometry in genome scan for complex traits Racovitan, Michael, Generalizations for hyperbolic 3-manifolds Tewari, Susanta, Construction of high- the blocks of the symmetric group and Broshi, Michael, Moduli of finite flat group resolution likelihood-based integrated Nakayama conjecture schemes with G-structure genetic and physical map of neurospora Sokolov, Vadim, Quadratic inverse eigen- Conidis, Chris, Applications of com- crassa value problems: Theory, methods and putability theory applications Wang, Qin, Sufficient dimension reduc- Cudney, Richard, A category of represen- tion and sufficient variable selection Northwestern tations at the critical level Ye, Jun, Geostatistical methods for University (11) Fowler, James,Poincar´edualitygroups spatio-temporal analysis of fMRI data and homology manifolds Zhen, Xiaoa, Categorical time series Department of Engineering Sciences Ginensky, Adam, A generalization of and Applied Mathematics the Clifford index and determinantal IDAHO Conway, Jessica, Complex patterns in equations for curves and their secant oscillatory systems varieties University of Idaho (3) Hansen, David,AxisymmetricStokesflow Malestein, Justin, Topological properties solutions and bubble microstreaming of a surface and the lower central series Department of Mathematics McCallum, Matthew, Eigenfunctions of Patel, Deepam, De-Rham epsilon factors Bauer, Ryan, Disjoint cycles and tree cyclic integral transforms Patterson, Eric, Singularities of configura- packings in graphs Menon, Vilas, Building better models of tion and graph hypersurfaces Buzbas, Erkan Ozge, Likelihood methods the CA1 pyramidal neuron: A com- Shulman, Michael, Double categories and for balancing selection under the k- bined computational and experimental base change in homotopy theory approach allele models Smith Zbarsky, Emma, Families of A- Tran, Luong Dinh, Nonlinear extensions Merkey, Brian, Biofilm modeling for infinity algebras and homotopy group of the Erdos-Ginzburg-Ziv˝ theorem wastewater treatment: Multiple species actions and multiple components Sparling, Jonathan,Onthelocalrelative Sample, Christine, Nonlinear dynamics of trace formula ILLINOIS interacting interfaces in electrochemi- cal and biological systems Townsend Beazley, Elizabeth,Codimen- Illinois Institute of sions of Newton strata for SL(3) in the Smith, Bryan, The extended finite ele- Iwahori case Technology (5) ment method for special problems with moving interfaces Department of Applied Mathematics University of Illinois at Department of Mathematics Aijun, Du, Deviation, ergodicity and re- Chicago (15) duction for stochastic dynamical sys- Nanes, Kalman,Ontheergodicityof Division of Epidemiology and tems partially hyperbolic flows Biostatistics Chen, Bao Hua, Stochastic modeling of Ouyang, Cheng, Asymptotics of implied water vapor in the climate system volatility in local volatility models Aryal, Subhash, Small sample tests and Vidozzi, Frederico, Two essays on multi- Qian, Randy, Diffractive theorems for sample size determination for hierar- variate stochastic processes and appli- the wave equation with inverse square chical mixed-effects models cations to credit risk modeling potential Joslin, Charlotte, The association between Vidozzi, Sebastiano, Selected applications Wang, Fang, Minimal measures for La- Acanthamoeba keratitis and domestic of the theory of Markov processes to grangian systems water exposure in the Chicago area mathematical finance Southern Illinois Lee, Jungwha, Variance estimation of Zeng, Xiaoyan, Integration and approxi- partial population attributable fraction mation using lattice designs University, Carbondale (5) and its applications Department of Mathematics Rankin, Kristin, Applying multifactorial Illinois State University (2) population attributable fractions to the A, Min, Solvability of diffusion mod- problem of childhood overweight Department of Mathematics els of insect movement in complex Ricks, Phillip, Tuberculosis control among Casey, Stephanie, Subject matter knowl- landscapes substance users: The indigenous lead- edge for teaching statistical association Ghebremicael, Aman, Latin squares and ership outreach model vs. standard Williams, Nicole, Interpreting scatter- applications care plots: The role of statistical knowledge Schwartz, Andrew, Decompositions of Department of Mathematics, and personal knowledge graphs and trees Teweldemedhin, Amanuel,Nonparamet- Statistics, and Computer Science Northern Illinois ric estimators of mean residual life de los Angeles Ortiz Rodriguez, Rosario, University (6) function A statistical look at day trading Ye, Ping, Tests of symmetry with or- Fazioli, Christopher Carlo, On analyticity Department of Mathematical dered alternatives in three-dimensional Sciences of variations of the Dirichlet-Neumann contingency tables operator, and computational concerns Bilgili, Devrim, Quantitative trait loci University of Chicago (15) Jordan, Richard, Asymptotic methods detection using an accelerated failure applied to finance: Equity and volatility time cure model Department of Mathematics derivatives Corley, Todd, Classification methods for Beceanu, Marius, A critical centre-stable Kakleas, Maria, Numerical simulation of primitive table algebras of dimension 4 manifold for the semilinear Schrodinger¨ a weakly nonlinear model for water Lund, Mark, Nevanlinna theory in annuli equation waves with viscosity 282 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Lu, Chaoxiao, Pricing stock options in Stodolsky, Burak, Domination in sparse Dean, Noah, Vertex reinforced jump mergers and acquisitions with jump- graphs processes on one dimensional weighted diffusion model environments Department of Statistics Piret, Katherine, Computing critical points Fang, Qirong, Efficient and stable spectral of polynomial systems using PHCpack Hong, Hyokyoung, Prediction of condi- methods for acoustic scattering over and Python tional quantiles in ordinal regression bounded obstacles Pitcher, Nicole, Efficient point-counting with applications to aging research Hassapis, George, Bergman coordinates on genus-2 hyperelliptic curves Huebner, Alan, Modeling correlated or- on finite Riemann surfaces Wang, Liqing, The constructibility theo- dinal data: Marginal and conditional Jang, Junmyeong, Generic p-ranks for rem for differential modules approaches semi-stable fibrations Zhao, Shi, Crossover designs under sub- Jien, Yu-Juan, Stochastic analysis and ject dropout differential equation with respect to INDIANA fractional Brownian motion University of Illinois, Indiana University, Joyner, Sheldon, Zeta functions as iter- ated integrals Urbana-Champaign (25) Bloomington (16) Kim, Jaehong, Holomorphic Banach bun- Department of Mathematics Department of Mathematics dles over compact manifolds Kjelgaard Mikkelsen, Carl Christian,Nu- Barrus, Michael, On induced subgraphs, Alan, Muhammed Ali,Weightedpluripo- degree sequences, and graph structure merical methods for large Lyapunov tential theory equations Choi, Jeong Ok, Extremal problems in Altok, Serdar, Reversibility of simple Kwan, Yuen Yick, Fast spectral methods coloring and labeling of graphs and random walk on Galton-Watson trees partial orders for incompressible flows Bateman, Michael, Maximal averages over Marazzi Lopez, Leonardo,Directand Dong, Zhou, The injective envelope as the rectangles in the plane inverse scattering on conformally com- space of extremal functions Chen, Qingshan, Analysis and computa- pact manifolds and asymptotically hy- Duong, Han, Minimal volume lattice d- tion of the inviscid primitive equations perbolic manifolds with polyhomoge- simplices and the corner singularities for some neous metric Gunaydin, Ayhan, Model theory of fields nonlinear diffusive equations Orfanos, Stefanos, Crossed products and with multiplicative group Crawford, Jesse, Interpretation of eigen- their finite dimensional approximation Johnson, Mathew, On the stability of pe- values in multivariate statistical analy- properties riodic solutions of nonlinear dispersive sis Pham, David, G-Frobenius algebras, twist- equations Gash, Justin, On efficient Grobner basis ing, and the Drinfeld double Jung, Jaebum, Pareto optimization in computation Price, James, Weighted homogeneous robotics with acceleration constraints Hackman, Michelle,Anewfamilyof polynomials and the Jacobian Koh, Ngin-Tee, Approximable quasidisks screw-motion invariant minimal sur- Shang, Ning, Low genus algebraic curves Konwerska, Malgorzata, The law of the faces in cryptography iterated logarithm in noncommutative Hagge, Tobias, Graphical calculus for fu- Taylor, Courtney, Computation of fixed probability sion categories and quantum invariants point data from equivariant cohomol- Landquist, Eric, Infrastructure, arith- for 3-manifolds ogy metic, and class number computations Hong, Seung-Moon, Classification and Wang, Jingyue, Error bounds for nu- in purely cubic function fields of char- applications of tensor categories merical methods for the ROF image acteristics at least 5 Kuo, Jung-Miao, The Clifford algebra of a smoothing model Law, Ka Lung, Laurent polynomial in- cubic form Wang, Yi, Variational solutions to SPDE verse matrices and multidimensional Mazur, Justin, Motivic zeta functions and perturbed by a general Gaussian noise multivariate systems the Grothendieck ring for varieties with Wu, Qingyu, Image of transfer from Liu, Qi, Some extremal problems on group actions GL(2) × GL(3) to GL(b) graphs and partial orders Nguyen, Toan, Asymptotic stability of Xie, Zhenqiu, Minimizers of the Lawrence- Lopes, Vincius Cifu, Grothendieck semi- noncharacteristic viscous boundary lay- Poniach model under weak coupling rings and definable endofunctions ers and a parallel or slightly tilted field Marikova, Jana, O-minimal fields with Van Cott, Cornelia, Obstructions to slic- Yang, Kai,Thelogistic,two-sex,age- standard past map ing generalized doubles structured population model with births outside marriages Masri, Nadia, Fourier coefficients of mod- Yarahmadian, Shantia, Pointwise Green ular forms and their applications function bounds and stability of non- Yao, Song, Quadratic nonlinear expecta- tions and risk measures Mizrak, Ozgur, Asymptotic stability of characteristic boundary layers Yuan, Chunyan, Gamma-convergence of the ground states of the nonlinear Yari, Masoud, Formation and long-time the Lawrence-Doniach energy for lay- Schrodinger¨ equation persistence of patterns outside of equi- librium ered superconductors to a three-dimen- Popa, Ana-Maria, On completely bounded sional anisotropic Ginzburg-Landau Yasamin, Ahamed, Maximal invariants multipliers of the Fourier algebra A(G) energy over symmetric cones Rojas, Ricardo, Loops and finite affine Zhang, Ji, A solution method for zero planes Purdue University (37) dimensional equation systems and a Sahin, Mehmet, Moduli space of sheaves pose-free framework for the robust on P 2 with Hilbert polynomial 4n +1 Department of Mathematics solution of the structure from motion Santhanam, Rekha, Units of equivariant Bakhtary, Parsa, On the cohomology of problem ring spectra a simple normal crossings divisor and Zhang, Songfu,Smoluchowski-Kramers Sheikh, Naeem N., Extremal problems for its dual complex, with applications to approximation for stochastic equations partitions of edge-sets of graphs isolated singularities with Levy-noise Stan, Florin,Traceproblemsinalge- Crosby, Charles, A monoidal structure Zhao, Ruijun, Some mathematical models braic number fields and applications to in the category of cosimplicial chain and scientific computations in epidemi- characters of finite groups complexes ology and immunology FEBRUARY 2010 NOTICES OF THE AMS 283 Doctoral Degrees Conferred Department of Statistics Bhattacharyya, Gargi, Terwilliger alge- Jiang, Yu, Inference and prediction in bras of wreath products of association a multiple structural break model of An, Lingling, Dynamic clustering of time schemes economics time series series gene expression Dagli, Mehmet,Liealgebradecompo- Kazmi, Kamran, Mathematical theory and Anderson, David, Multifractal fractional sitions with applications to quantum numerical analysis of integrated biolu- sum-difference models for internet traf- dynamics minescence tomography and diffuse fic Drignei, Mihaela Cristina, Inverse Sturm- optimal tomography Daye, Zhongyin John,Somenewap- Liouville problems using multiple spec- Kim, Sujin, The role of bases and wavelets proaches in high-dimensional variable tra in stochastic processes selection and regression Lee, Jangwoon (Leo), Analysis and finite Ortega, Omayra, Evaluation of rotavirus Jeng, Xinge Jessie, Covariance adaptation element approximations of stochastic vaccination programs and regularization in large-scale hy- optimal control problems constrained pothesis testing and high-dimensional Department of Biostatistics by stochastic elliptic partial differential variable selection equations Bayman, Emine O., Bayesian hierarchical Li, Jinguang Tony, An additive-interactive Mikkelson, Rana, Minimum rank of models for multi-center clinical trials: nonlinear volatility model, estimating graphs that allow loops Power and subgroup analyses and testing Pryporova, Olga, Types of convergence Chen, Yiyi, Optimal adaptive group se- Qi, Xiaoli, Nonparametric calibration of of matrices quential design for Phase II clinical two common susceptibility tests using trials: A Bayesian decision theoretic interval-censored data with measure- Shin, Jaemin, Inverse scattering problems approach ment error for first-order systems Department of Mathematics Shu, Lei, Stepwise model building for Xie, Xiaoliang, Large time-stepping meth- mixed-effect models with random scale ods for higher order time-dependent Chun, Sangmin, The annihilator semi- effects evolution equations group ring Tuzov, Nikita, Mutual fund performance Xu, Ying, Modeling and direct numerical Froelich, Jennifer, Universal deformation evaluation methodology and local false simulation of particle-laden turbulent rings related to the symmetric group S5 flows discovery rate approach Ljumanovich, Leonida, Horn theories Yoo, Suk-Young, Statistical methods for Department of Statistics Stehnova, Jitka, Theta correspondence integrating epigenomic results for unitary groups Zeng, Lingmin, Group variable selection Bingham, Melissa, Likelihood and Bayes methodsinanalysesofgenomicdata inference for a class of distributions on orientations in 3 dimensions KANSAS University of Notre Chapman, Jessica, Computationally effi- cient resource allocation for complex Kansas State University (6) Dame (7) system reliability studies Department of Mathematics Department of Mathematics Dancik, Garrett, Exploring host-pathogen Cipra, James, Waring’s number in a finite relationships through computer simu- field Broad, Steven, Index formulas for higher lations of intracellular infection order Loewner vector fields Shklyarov, Dmytro, Hirzebruch-Riemann- Diao, Lixia, Robust small area estimation Cibotaru, Daniel, Localization formulae Roch theorem for differential graded in odd K-theory Lu, Lu, Statistical methods in a high algebras school transcript survey Hauenstein, Jonathan, Regeneration, local Department of Statistics dimension, and applications in numeri- Ma, Haiming, New developments in plan- cal algebraic geometry ning accelerated life tests Jung, Yoonsung, Effects of unequal treat- ment variances on the analysis of Hoehn, Stacy,Themodulispaceofcom- Ramler, Ivan, Improved statistical meth- crossover designs pact fiber bundle structures on a fibra- ods for k-means clustering of noisy and tion directional data Ndum, Edwin, Individual treatment effect Reising, Monica,Modelinganddiscrimi- heterogeneity in multiple time points Jensen, Raymond, Integral boundary trials invariants for conformally compact nation for spectral-temporal data manifolds of constant curvature and Tang, Chengyong, Parameter estimation Tang, Zhongwen,Lackoffitoflogistic measure-theoretic Einstein condition and bias correction for diffusion pro- GEE models and cost efficient Bayesian optimal designs for nonlinear com- Maher, Christina, On embeddings of com- cesses and a nonparametric approach to census population size estimation binations of parameters in nonlinear putable structures, classes of structures regression models and computable isomorphism Villanueva-Morales, Antonio,Modified pseudo-likelihood estimation for Markov Zhang, Ke (Kurt), Inference of non- Rissler, Matthew, Stochastic models of parametric hypothesis testing on high collective motion in Myxococcus Xan- random fields with Winsorized Poisson conditional distributions dimensional longitudinal data and its thus application in DNA copy number varia- Wang, Jianqiang, Estimating the distance tion and microarray data analysis distributions of subpopulations and IOWA testing observation outlyingness for a University of Kansas (5) large-scale complex survey Department of Mathematics Iowa State University (23) You, Lifeng, Cross-validation in model- assisted estimation Department of Mathematics Kastens, Jude, Some new developments on two separate topics: Statistical cross Ahmed, Haseena, High-resolution alter- University of Iowa (11) validation and floodplain mapping nating evolution schemes for hyper- Department of Applied Mathematics Kummini, Manoj, Homological invariants bolic conservation laws and Hamilton- of monomial and binomial ideals and Computational Sciences Jacobi equations Moen, Kabe, Linear and multilinear frac- Allen, Aaron, Stability results for damped Gillispie, Brian, Instabilities of elastic tional operators: Weighted inequalities, multilayer composite beams and plates objects in motion sharp bounds, and other properties 284 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Niknejad, Jila, Some properties of real- LOUISIANA Chan, Kwun Chuen (Gary), Recurrent compact subspaces and coarser normal marker process before failure event: A spaces Louisiana State backward process approach Yengulalp, Lynne, Coarser connected University, Baton Cheng, Yu-Jen, Statistical methods for topologies and non-normality points failure time data with biased sampling Rouge (6) and measurement errors Wichita State University (1) Department of Mathematics Di, Chongzhi, Likelihood ratio testing un- der nonidentifiability with applications Department of Mathematics and Alali, Bacim, Multiscale analysis of het- to biomedical studies erogeneous media for local and nonlo- Statistics Eckel, Sandrah, Quantifying individual cal continuum theories and city level modification of the health Vargas, German, Spectral methods so- Jara, Patricio, Rational approximation effects of air pollution in older adults lution of Navier-Stokes equations for schemes for solutions of abstract Ho, Yen-Yi, Statistical methods for failure steady viscous flows Cauchy problems and evolution equa- time data with biased sampling and tions measurement errors Meng, Chao, Stochastic and Copula mod- Li, Qing, Genetic association tests: Trio KENTUCKY els for credit derivatives logic regression and score test Morris, Karli, Trace forms of Abelian Shum, Kim Fai (Kenny),Estimatingthe University of Kentucky (13) extensions of number fields mean medical cost in small samples Ozer, Koray, Laplace transform inversion Department of Mathematics and time-discretization methods for Department of Mathematics Bouchat, Rachelle, Algebraic properties evolution equations Agarwala, Susama, The geometry under- of edge ideals Tugurlan, Maria Cristina, Fast marching lying renormalization Brown, Tricia, Rees products of posets methods—Parallel implementation and Banerjee, Abhishek,Nearbycycles,Archi- and inequalities analysis medean complex and periodicity in cyclic homology Busse, Philip, Multivariate list decoding of University of Louisiana at evaluation codes with a Grobner basis Breiner, Christine, Embedded minimal perspective Lafayette (4) surfaces of finite topology and one end: Department of Mathematics Conformal structure and asymptotic Butcher, Gene, A study of projective properties complexes Marinov, Tchavdar, An inverse problem Chen, Yifei, Strong rational connected- Chifman, Julia,Directproductsandthe for the Euler-Bernoulli equation and a ness of toric varieties intersection map of certain classes of new scheme for solving a hierarchical Hezari, Hamid, Eigenvalues and eigen- finite groups size-structured model with nonlinear functions of Schrodinger¨ operators: Frayer, Christopher, Scattering with sin- growth, mortality, and reproduction Inverse spectral problems and zeros of gular Miura potentials on the line rates eigenfunctions Jiang, Teng, Minimizing L-variational Peng, Jie, Sample size calculation and Jiang, Jin-Cheng, Bilinear Strichartz esti- problems with the Dirichlet energy tests for binomial and Poisson distribu- mates in 2 dimensional compact man- constraint tions ifolds and cubic nonlinear Schrodinger¨ Kahn, Eric, The generalized Burnside and Stanford, Alex, Topological rings and equations representation rings associated topological groups Kramer, Joel, Examples of embedded Tragoonsirisak, Patcharin, Quenching and stable minimal surfaces of large area Nicolas, Carlos, Structural and enumer- blow-up phenomena due to concen- ative properties of k-triangulations of Seyyedali, Reza, Balanced metrics in trated nonlinear sources in R − n the n-gon Kahler geometry Wright, Thomas, On convergence of sin- Schmidt, Jack, Rewriting systems for gularseriesforapairofquadratic finite groups MARYLAND forms Stokes, Erik,Theh-vectors of matroids Johns Hopkins Zulkowski, Patrick, The plateau problem and the arithmetic degree of squarefree of Alexandrov spaces satisfying the strongly stable ideals University (22) Perelman conjecture Department of Statistics Department of Applied Mathematics and Statistics University of Maryland, Chirilla, Costel, Empirical processes and Baltimore County (5) Beer, Elizabeth, Random latent position ROC curves with an application to linear Department of Mathematics and combinations of diagnostic tests graphs: Theory, inference, and applica- tions Statistics Yu, Lei, Extended polychotomous logistic Markov model for longitudinal categor- Hutchison, David, Stopping times and Gurtuna, Filiz, Geometric optimization ical responses confidence bounds for small-sample problems with applications to el- stochastic approximation algorithms lipsoidal approximations and quasi- University of Louisville (3) Lin, Xue,Rank-basedmethodsforsta- Newton methods tistical analysis of gene expression Klein, Martin, Statistical analysis based on Department of Mathematics microarray data physiologically-based pharmacokinetic Ye, Xugang, Random disambiguation models Brauch, Timothy, Applications of the paths: Models, algorithms, and anal- Newcomer, Justin, Estimation procedures combinatorial nullstellensatz on bipar- ysis for multinomial models with overdis- tite graphs persion Department of Biostatistics Cochran, John, Model extensions and Wang, Dan, Partial differential equation applications in image processing Carvalho, Benilton, Statistical methods constrained optimization and its ap- Zahedi, Hamid, Data mining to examine and software for high density oligonu- plications to parameter estimation in the treatment of osteomyelitis cleotide arrays models of nerve dendrites FEBRUARY 2010 NOTICES OF THE AMS 285 Doctoral Degrees Conferred Warfield, Joseph, Sequential optimal de- Spiliopoulos, Konstantinos, Asymptotic Wang, Junbo, Diophantine approximation sign with unknown link function problems for stochastic processes with of linear forms reflection and related partial differen- Zhang, Yuqing, Homogeneous flows and University of Maryland, tial equations Diophantine exponents of affine sub- College Park (28) Sun, Weiran, Mathematical problems aris- spaces ing when connecting kinetic to fluid Department of Mathematics regions Harvard University (34) Belding, Juliana, Number theoretic algo- Tang, Guojing, Discrete time stochastic Department of Biostatistics, School volatility model rithms for elliptic curves of Public Health Brody, Justin, On the model theory of Wang, Xia, Applications of genetic al- random graphs gorithms, dynamic programming, and Hoffmann, Thomas, Contributions to linear programming to combinatorial Chen, Qiwen, Dependence structure for multivariate association models for nu- optimization problems Levy processes and its applications in clear families finance Maghsoudi, Kaveh, Transcript mapping Delgado, Robert, Density properties of MASSACHUSETTS with genome tiling microarrays subsets of the Euler characteristic- Manjourides, Justin, Distance based meth- 2surfacegroup,PSL(2,R)character Boston University (7) ods for space time modelling of the variety Department of Mathematics and health of populations Draper, Thomas, Non-linear complexity Statistics Naylor, Melissa, Novel multivariate and of Boolean permutations Bayesian approaches to genetic associ- Emmerling, Thomas, Essays on American ation testing and integrated genomics Furnival, Darran, Iterative methods for and game-type options the stochastic diffusion problem Othus, Megan, Survival analysis with Hamrick, David, Topics in stationarity, complex censoring mechanisms with Galaz-Garcia, Fernando, Non-negatively volatility, and contagion curved fixed-point homogeneous mani- applications in population-based stud- Jiang, Xiaoyu, Network-based informa- ies and clinical trials folds in low dimensions tion integration for protein function Torgovitsky, Roman, Resampling meth- Groer, Christopher, Parallel and serial prediction ods for high-dimensional data and algorithms for the vehicle routing prob- Robertson, Scott, Applications of large lem and several variations nonparametric regression with applica- deviations principles to options pricing tions to brain imaging, bioinformatics, Hoban, Ryan, Local rigidity of triangle and portfolio choice and sleep/circadian medicine groups in Sp(4,R) Russell, Elizabeth,Thecomplexdynamics Wang, Lu, Nonparametric and semipara- Johnson, Gregory, Abstract elementary of rational maps metric regression with missing data classes with Lowenheim-Skolem¨ num- Spurgin, Greg, Dynamics of the four body Wu, Michael, Statistical methods for high- ber cofinal with ω problem with large and small masses dimensional genomic data Kim, Hyejin, Asymptotic problems for Uminsky, David,TheviscousN-vortex stochastic processes and related differ- problem: A Helmholtz/Kirchhoff ap- Zhang, Jing Jenny, Novel methodologies ential equations proach for the analysis of complex failure time data and alternative progression-free Long, Jane, The cohomology of the affine survival estimators group over Fp2andofPSL(3,Fp) Boston University School Lott, Paul Aaron, Fast solvers for models of Public Health (6) Department of Mathematics of fluid flow with spectral elements Department of Biostatistics Barnet-Lamb, Thomas James,Onpoten- Manon, Christopher, Presentations of tial automorphy, and other topics in semigroup algebras of weighted trees Blood, Emily, Performance of mixed ef- fects models in the analysis of medi- number theory Nicholls, Stacey, Column generation in ated longitudinal data generated from Basu, Samik, A∞ structures on Thom predictor-corrector methods for solving a structural equation model spectra linear programs Chibnik, Lori, Development and eval- Bullock, Evan Merrill, Subcanonical points Olsson, James Scott, Combining evidence uation of risk prediction models in on algebraic curves from unconstrained spoken term fre- the presence of correlated markers Chi, Chen Yu, Pseudonorms and the- quency estimation for improved speech and non-linear associations between retrieval orems of Torelli type for birational markers and outcome using logistic equivalence Pauletti, Miguel, Parametric AFEM for geo- regression and net-benefit analysis metric evolution equations and coupled Choi, Suh Hyun, Local deformation lifting Nguyen, Anh-Hoa, Family-based associa- spaces of mod1 Galois representations fluid-membrane interaction tion tests and genetic risk scores for Hornet, Stefan Laurentiu,Ageneraliza- Prakash, Samvit, Pricing volatility deriva- survival data tives using space scaled Levy processes tion of a theorem of Ravenel and Stedman, Margaret,Acomparisonof Wilson Pressley, Joanna,Responsedynamicsof methods for physician randomized tri- integrate-and-fire neuron models als with binary and survival outcomes Isaacson, Samuel Baruch, Cubical ho- motopy theory and monoidal model Sukpraprut, Suporn, The estimation of Price, Carter, Applications of operations categories research models to problems in health the probability of treatment greater care than control in clinical trials Kass, Jesse, Good completions of Neron models Rosario, Dalton, Algorithm development Wu, Hong Sheng, Methods for genetic for hyperspectral anomaly detection association studies using longitudinal Rasmussen, Sarah Dean, A number theo- retic result for Berge’s conjecture Sam, Chon (Denise), Variable selection and multivariate phenotypes in families properties of L1 penalized regression Smith, Camillia, Enumeration of the dis- (3) in generalized linear models Brandeis University tinct shuffles of permutations Shaw, Christopher, Weakly o-minimal Department of Mathematics Tarnita, Corina Elena,Evolutionarydy- namics in structured populations structures and Skolem functions Drake, Brian, An inversion theorem for Song, Xin, A new scheme for monitoring labeled trees and some limits of areas Tosatti, Valentino, Geometry of complex multivariate process dispersion under lattice paths Monge-Amp`ere equations 286 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred School of Engineering and Applied Liu, Ruochuan, On the slope filtration of Department of Mathematics and Science φ-modules over the Robba ring Statistics Maydanskiy, Maksim, Exotic symplectic Abbot, Dorian, A high-latitude convective Grigoryan, Viktor, Stability of geodesic manifolds from Lefschetz fibrations cloud feedback wave maps Ritter, Alexander, The Novikov theory Constantin, Gabriel, Expressiveness and Hall-Seelig, Laura, Asymptotically good optimization under incentive compati- for symplectic cohomology and exact Lagrangian embeddings towers of global function fields and bility constraints in dynamic auctions bounds for the Ihara function Dechadilok, Panadda,Electrokineticef- Sabatini, Silvia,ThetopologyofGKM spaces and GKM fibrations Kucuksakalli, Omer, Class numbers of fects in hindered transport ray class fields of imaginary quadratic Spiridonov, Alexey, Pattern-avoidance in Dias, Joao, Automatically generating the fields back end of a computer using declara- binary fillings of grid shapes Milburn, Brett, Dirac maps and gener- tive machine descriptions Tievsky, Aaron,AnaloguesofK¨ahler alized complex geometry on homoge- geometry on Sasakian manifolds Hendrix, Philip,Learninganddecision neous spaces making with reputation information Xiao, Liang, Nonarchimedean differential modules and ramification theory Shieh, Meng-Shiou, Correction methods, Kang, Laura Suh Young,Openmech- approximate biases, and inference for anism design: Ensuring and verifying Yick, King Yeung, A sphere settling in misclassified data the strategyproofness of mechanisms a stratified fluid at small Reynolds in open environments number Worcester Polytechnic Kordab, Imad, Bilevel optimization meth- Yin, Jingbin,Aq-analogue of spanning ods to evaluate intermodel freight trees: Nilpotent transformations over Institute (1) transportation investments in Colombia finite fields Department of Mathematical McClean, Megan, Principles of signal Sciences processing and fidelity in biological Northeastern networks University (5) Huang, Xueying (Shirley),InvivoMRI- Nesson, Rebecca, Synchronous and mul- based three-dimensional fluid-structure ticomponent tree-adjoining grammars: Department of Mathematics interaction models and mechanical Complexity, algorithms and linguistic Brown, Justin, Some geometric properties image analysis for human carotid applications of certain toric varieties and Schubert atherosclerotic plaques Patel, Ankit, Modeling and inferring cleav- varieties age patterns in proliferating epithelia Cutler, Anthony, Regular polyhedra of Redwine, Kevin, Inference rules plus index 2 MICHIGAN proof-search strategies equals pro- Donovan, Elizabeth, Various parameters grams of subgraphs and supergraphs of the Michigan State Shneidman, Jeffrey, Rationally motivated hypercube University (13) failure in distributed systems Koldan, Nilufer, Semiclassical asymp- Wapinski, Iian, Genome-wide reconstruc- totics on manifolds with boundary Department of Mathematics tions of the evolution of genes and reg- Mukherjee, Himadri,Singularitiesofa ulatory elements in Ascomycota fungi Arikan, Mehmet, Topological invariants certain class of toric varieties of contact structures and planar open Massachusetts Institute books Tufts University (4) of Technology (19) Condori, Alberto, Thematic indices and Department of Mathematics superoptimal singular values of matrix Department of Mathematics functions Espanol, Malena Inez, Multilevel meth- Durusoy, Daniel, Heegaard Floer homol- Abolfathe Beikidezfuli, Salman, Quantum ods for discrete ill-posed problems: ogy of certain 3-manifolds and cobor- proof systems and entanglement theory Application to deblurring dism invariants Bernstein, Jacob, Conformal and asymp- Gonzalez Martinez, Cristian,Someas- totic properties of genus g minimal pects of the geometry and the coho- Dyer, Kimberly, Almost dual FF-modules surfaces with one end mology of the moduli spaces of vector Enders, Joerg, Generalizations of the Chen, Victor, The Gowers norm in the bundles and coherent systems over reduced distance in the Ricci flow— testing of Boolean functions algebraic curves Monotonicity and applications Davis, Christopher, The overconvergent O’Brien, Mark Richard, Right-angled Cox- Geng, Weihua, Interface method and de Rham-Witt complex eter groups and CAT(0) spaces Green’s function based Poisson Boltz- Eu, Ching-Hwa, Hochschild homology/ Zhang, Wei, Haar based multi-resolution mann equations solver and interface cohomology of preprojective algebras stochastic processes technique based molecular dynamics of ADET quivers Kim, Dong-Joong, Comparison of native- Iacovino, Vito, Open strings in the cotan- University of English and native-Korean speaking uni- gent bundle and Morse homotopy Massachusetts, versity students’ discourses on infinity Jung, Kyomin, Approximate inference: and limit Amherst (7) Decomposition methods with applica- Knapp, Adam, Computations of La- tions to networks Department of Biostatistics, Public grangian Floer homology and gauge the- Kilic, Mustafa Sabri, Induced-charge elec- Health oretic invariants for Montesinos twins trokinetics at large voltages Li, Mingfei, Strategies in repeated games Lekili, Yanki, Broken Lefschetz fibrations, Li, Xiaohong, Approaches to estimation of Tsai, Chih-Hsiung, Algorithms for solving Lagrangian matching invariants and haplotype frequencies and haplotype- polynomial systems by homotopy con- Ozsv´ath-Szabo´ invariants trait associations tinuation method and its parallelization Lim, Joungkeun, On the capacity of the Xu, Bo, Predictors of treatment means erasure channel and the construction foraonefactorcompletelyrandomized Zhang, Ying, Mixed volume and total of an ε-randomizing map design degree FEBRUARY 2010 NOTICES OF THE AMS 287 Doctoral Degrees Conferred Department of Statistics and Department of Statistics Tena, Jezania Kira, Test procedures for Probability equality of two variances in delta Hamidieh, Kamal, Topics in statistical distributions Aggarwal, Deepa, On some inference modeling of extremes and their depen- problems for current status data dence Zhu, Yongfang, Statistical models for as- Linn, Matthew, Nonlinear filtering of MINNESOTA sessing the individuality of fingerprints random fields in the presence of long- memory noise and related problems in University of Oakland University (4) stochastic analysis Minnesota-Twin Cities (10) Department of Mathematics and Sen, Bodhisattva, A study of bootstrap Division of Biostatistics, School of and likelihood based methods in non- Statistics Public Health standard problems Coffield, Daniel,Amodelforsingle Singhal, Harsh, Statistical inverse prob- Li, Meijuan, Statistical models for a phase flow in horizontally fractured lems on graphs with applications to censored quantitative trait and can- porous media using homogenization flow volume estimation in computer didate genes association mapping in techniques networks structured populations with multilevel Gan Hewage, Jayantha Lanel,Complex Sun, HoKeun, Hybrid bootstrap for map- genetic relatedness root isolation ping quantitative trait loci and change Wei, Peng, Network-based mixture mod- Jia, Feiyi, Contributions to Bayesian rank- point problems els for genomic data ing and selection rules and to random- Wagaman, Amy, Topics in high-dimen- ized response survey analysis School of Statistics sional inference with applications to Rathugamage, Sriyani Renuka Menike, Raman spectroscopy Adragni, Kofi Placid, Principal fitted com- Models and analysis for adhesive con- ponents on small samples tact, material damage, and cardiac Yang, Lili, Sample based estimation of network traffic flow characteristics Deng, Qiqi, Developing a distribution- waves free change-point model for unknown- Zhao, Ou, Isotonic regression and sta- baseline data tionary random walks University of Michigan (25) Dong, Hua, Statistical properties of Gaus- Department of Mathematics sian reproducing kernel in univariate (5) Wayne State University density estimation Felgueiras, Oscar, The ample cone for a Department of Mathematics Flegal, James, Monte Carlo standard morphism errors for Markov chain Monte Carlo Gentry, Sara,Mathematicalmodelingof Holdai, Veera, Multichannel change point Joo, Jonghoon, Curve/surface estimation mutation acquisition in hierarchical problems with possible jumps/roofs/valleys pre- tissues: Quantification of the cancer Kan, Shaobai, System identification in served stem cell hypothesis stochastic hybrid systems Rendahl, Aaron,Graphicalmethodsof Gomez-Guerra, Jose, Models of twisted Li, Weiyuan, Fundamental solution and determining predictor importance and K-theory p L estimates for higher order subellip- effect Gong, Jasun, Derivations on metric mea- ¨ tic Schrodinger operators on stratified Wang, Huixin, On large margin hierarchi- sure spaces groups cal classification Iwen, Mark, Combinatorial compressive Truong, Bao, Variational analysis and ap- Wu, Seongho, Homotopy, nonconvexity sampling with applications plications to multiobjective optimiza- and regularization Jain, Harsh, Quantitative modeling of tion molecular pathways associated with Zhu, Huiqing, Discontinuous Galerkin intratumoral angiogenesis methods for singularly perturbed prob- MISSISSIPPI Kutch, Jason, Signal in human motor lems unsteadiness: Detecting the action and Mississippi State activity of muscles Western Michigan University (1) Lee, KyungYong, On realization of line University (7) Department of Mathematics and arrangements as multiplier ideals Department of Mathematics Statistics Mackay, John, Conformal dimension and Shahul-Hameed, Jaffar Ali,Multiplepos- the quasisymmetric geometry of metric Madden, Sandra, High school mathemat- itive solutions for classes of elliptic spaces ics teachers’ evolving study of compar- systems with combined nonlinear ef- More, Yogesh, Arc valuations on smooth ing distributions varieties fects Robbins, Hannah, Finiteness of asso- Department of Statistics University of ciated primes of local cohomology Awosoga, Oluwagbohunmi,Metaanalyses Mississippi (4) modules of multiple baseline time series design Sierra, Susan, The geometry of bira- intervention models for dependent and Department of Mathematics tionally commutative graded domains independent series Chism, Lyrial, On independence polyno- Spencer, Craig, Analytic methods for Nantz, Eric, Testing equality of competing mials and independence equivalence in Diophantine problems risks of patient discontinuation across graphs Stubbs, Joe, Potent elements and tight multiple disease states Craddock, Michelle, Reflexivity and the closure in Artinian modules Nyirenda, Themba, Heteroscedastic lin- Grothendieck property for positive ten- Urzua, Giancarlo, Arrangements of curves ear model estimation based on ranks: sor products of Banach lattices and algebraic surfaces An iterated reweighted least squares Lai, Wei-Kai (Brian), The Radon-Nikodym Vavrichek, Diane, Accessibility and JSJ Ponnathapura, Srinand, Statistical proce- property for positive tensor products decompositions of groups dures for bioequivalence analysis of Banach lattices Yang, Bo, Application of perturbation Rosales, Mathew,Robustnessofconfi- Song, Lanzhen, Independence polynomi- methods to pricing credit and equity dence intervals for the mean of delta als of k-tree related and well-covered derivatives distribution graphs 288 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred MISSOURI Brieler, Robert, Symmetric and alternat- Hummel (Miller), Livia, A theory of non- ing groups as monodromy groups of Noetherian Gorenstein rings Missouri University of compact Riemann surfaces: The case of Moore, W. Frank, Cohomology of prod- Science and four branch points ucts of local rings Browder, Jonathan, Proper group actions Technology (3) and the face structure of simplicial Department of Mathematics and complexes NEW HAMPSHIRE Statistics Fan, Chunlin, Contributions to the theory of copula University of New Marciniak, Malgorzata, Holomorphic ex- Gill, James, Functions of finite distortion Hampshire (6) tensions in toric varieties and a lower bound for the weak- Department of Mathematics and Mukhopadhyay, Purna, A modified ap- type constant of the Beurling-Ahlfors Statistics proach for obtaining sieve bootstrap transform prediction intervals for time series Lin, Yonhow, The interplay between har- Das, Paramita, Planar algebras of families Supapakorn, Thidaporn, Estimating monic analysis, functions theory and of group-type subfactors bounds for nonidentifiable parameters operators Li, Weihua, Free entropy dimensions and using potential outcomes Lott, Tim, Dehn fillings of hyperbolic approximate liftings punctured-torus bundles Paddack, Megan,Theprocessofmaking St. Louis University (1) Ronshausen, Emily, The Liouville prop- meaning: The interplay between teach- Department of Mathematics and erty in the discrete group-action setting ers’ knowledge of mathematical proofs Computer Science Standeven, Bennett, The role of first order and their classroom practice Ravichandran, Mohan, Kadison-Singer al- Manche, Jaime, Best representations and logic in analysis of several complex gebras with applications to von Neu- completions in weakened topologies for variables mann algebras the integers MONTANA Shubov, Mikhail, Improved estimation of University of Fourier coefficients for ill-posed inverse Montana State problems Missouri-Columbia (9) Yuan, Wei, Kadison-Singer algebras Department of Mathematics University (3) Department of Mathematical Bihun, Oksana, Approximate isometries NEW JERSEY and distortion energy functionals Sciences Cowell, Simon, Asymptotic uncondition- Alonso, Nicomedes, III, An alternating- New Jersey Institute of ality in Banach spaces direction Sinc-Galerkin method for el- Technology (3) Felts, Kenneth, Harmonics and excitations liptic problems on finite and infinite of oscillators domains Department of Mathematical Sciences Koh, Doowon, Extension theorems in Segal, Sara, Action research in mathe- vector spaces over finite fields matics education: A study of a master’s Espin, Leonardo, Self-similar flows in Shane, Christopher, Uniqueness theo- program for teachers finite or infinite two-dimensional ge- rems for non-symmetric convex bodies Vallines Mira, Raquel, Teachers’ beliefs ometries regarding effective teaching strategies Joshi, Yogesh, Discrete dynamical pop- Department of Statistics for American Indian students in mathe- ulation models: Higher dimensional Lin, Xiaoyan, Bayesian hierarchical mod- matics pioneer-climax models els for the recognition-memory experi- University of Montana, Matakuti, Kamyar, Numerical detection ment of complex singularities in two and Yue, Yu, Spatially adaptive priors for Missoula (2) three dimensions regression and spatial modeling Department of Mathematical Zhang, Jing, Bayesian spatial analysis Sciences Princeton University (17) with application to the Missouri Ozark Department of Mathematics Forest Ecosystem Project Elias, Joran, Randomness of tree ensem- ble methods Brooks, Shimon, Entropy bounds for Zhu, Liang, Semiparametric analysis of quantum limits multivariate longitudinal data Yates, Rebekah, Norm-preserving criteria for uniform algebra isomorphisms Bukh, Boris, Extremal problems in combi- Washington University in natorial geometry and additive combi- natorics (11) NEBRASKA St. Louis Greene, Joshua, Donaldson’s theorem, Department of Electricial and System University of Heegaard Floer homology, and results Engineering on knots Nebraska-Lincoln (6) Guttikar, Chaitanya, Recovering a variety Yang, Ruoting, Data-driven two degree of Department of Mathematics from its derived category freedom control for robust intelligent life science automation Bockelman, Brian, Solving partial differ- Matchett Wood, Melanie,Modulispaces ential equations with Sinc methods for rings and ideals Department of Mathematics Crabbe, Andrew, Hilbert polynomials and Michalogiorgaki, Maria,Onrationalblow- Abel, Haley, The role of positive selec- building large indecomposable modules downs of smooth 4-manifolds tion in molecular evolution: Alternative Einfeld, Duane, Combinatorial and com- Naber, Aaron, Ricci solitons and col- models for within-locus selective ef- mutative manipulations in Feynman’s lapsed spaces fects operational calculi for noncommuting Pierce, Lillian, Discrete analogues in har- Bohanon, Joseph, Groups in which the operators monic analysis normalizer of every non-normal sugroup Ford, Pari, Polynomial LYM inequalities Sarkar, Sucharit, Topics in Heegaard is maximal and association schemes on lattices Floer homology FEBRUARY 2010 NOTICES OF THE AMS 289 Doctoral Degrees Conferred Snowden, Andrew, The Jacquet-Langlands University of New Mosinca, Natalia, Probability on graphs correspondence for GL(2) Mexico (13) and groups: Theory and applications Volpato, Michael, Some counting prob- Pedersen, Helge, Splice diagrams, sin- lems in arithmetic Department of Mathematics and gularity links and universal Abelian Statistics Wong, Willie, On the uniqueness of Kerr- covers Newman black holes Collins, Dave, Nonparametric solution to Ross, Joseph, The Hilbert-Chow morphism Young, Andrew, Complex analytic N´eron flowgraph models and the incidence divisor models for degenerating Abelian vari- Cuadros, Jaime, On null Sasakian struc- Simring, Eric, Representations and com- eties over higher dimensional parame- tures binatorics of Ocneanu’s essential paths ter spaces Deshler, Jessica, Predicting student suc- Welji, Shaffiq, On the Heegaard Floer Yun, Zhiwei, Towards a Springer theory cess in undergraduate mathematics homology of singular knots and their for global function fields courses unoriented resolutions Program in Applied and Grandini, Daniele, Quaternionic K¨ahler Zhang, Wei, Modularity of generating reductions of Wolf spaces functions of special cycles on Shimura Computational Mathematics Guba, Oksana,Thespectraofordinary varieties Eckhoff, Philip, Neuromodulation of de- differential equations Zhou, Yun, A stochastic representation cision making: Incorporating single-cell Meng, Chen, Phylogenetic models for hy- problem with applications in optimal physiology in low-dimensional models bridization in a coalescent framework singular control, Dynkin games and Gevertz, Jana, Multi-scale mathemati- Panek, Danusz, On sharp extrapolation obstacle problems cal modeling of heterogenous tumor theorems Department of Statistics growth Peterson, Kara, Modeling artic sea ice Ward, Rachel, Extracting information using the material-point method and an Ichiba, Tomoyuki, Topics in multi-dimen- from systems with a redundant rep- elastic-decohesive rheology sional diffusion theory: Attainability, resentation: Three problems in signal Ringler, Adam, Positivity problems in reflection, ergodicity and rankings and image processing higher dimensional geometry Jin, Man, Variable selection in canonical Srinivasan, Gowri, Uncertainty quantifi- discriminant analysis Rutgers University-New cation in stochastic processes Paik, Jane, Contributions to the analysis Brunswick (6) Villagren-Hernandez, Alejandro,Monte of survival data subject nonstandard sampling schemes Department of Statistics and Carlo strategies for calibration in cli- Biostatistics mate models Pospisil, Libor,Riskmeasuresandmaxi- Ying, Wenxia, Bayesian modeling of non- mum drawdown Chen, Chuanwen, Admissible, consistent stationary extremal events Cornell University (12) multiple testing with applications Zhu, Min, Functional transformed mod- Li, Jixin, Saddle point approximations els: Frequentist on Bayesian approaches Center for Applied Mathematics Mukherjee, Somnath,Astatisticaltest spectrum—from robust to powerful Diener, Nicolas, Mathematical models for NEW YORK swing options and subprime mortgage Sun, Qiankun, Statistical modeling and derivatives inference for multiple temporal or (2) spatial cluster detection Clarkson University Huerta-Sanchez, Emilia,Ananalysisof Department of Mathematics and sequence polymorphism under alterna- Wang, Jue, Some properties of robust tive population genetic models statistics under asymmetric models Computer Science Owen, Megan, Distance computation in Zhang, Juan, Higher order conditional Tang, Yuefeng, Interpolation algorithms the space of phylogenetic trees inference using parallels with approxi- in program verification mate Bayesian techniques Restrepo, Mateo, Computational meth- Zhou, Guohua, Fourier analysis and trun- ods for static allocation and real-time Rutgers University- cation error estimates of multigrid redeployment of ambulances methods and conservative Jacobians on Verdugo, Angel, Dynamics of gene net- Newark (1) hexagonal grids works with time delays Department of Mathematics and Columbia University (18) Department of Mathematics Computer Science Department of Mathematics Adams, Bryant, Representing processes Puri, Karan, Factorization of isometries as update automata and transducers of hyperbolic 4-space and a discrete- Cohen, Dvora, The existence of asymp- ness condition totically parallel spinors on mani- Horwitz, Noam, Free resolutions of mono- folds asymptotic to hypersurfaces in mial ideals Minkowski space Koch, Sarah, A new link between Te- NEW MEXICO DeLand, Matthew,Geometryofrational ichmuller¨ theory and complex dynam- curves on algebraic varieties ics New Mexico State Goia, Irina, Bessel and volatility-stabilized Matucci, Francesco, Algorithms and clas- University, Las Cruces (3) processes sification in groups of piecewise-linear Λ homeomorphisms Department of Mathematical Li, Zhi,On -adic Saito-Kurokawa lifting and its application Minnes, Mor Mia, Computability and com- Sciences Maharaj, Yogishwar, Line bundles, curves, plexity properties of automatic struc- Eydenberg, Michael, Kernel methods on and quasi-projective surfaces tures and their applications Ω-ultradistributions Mapes, Sonja, Finite atomic lattices and Murgescu, Radu,Onthep-class groups Noussi, Hubert, Stabilization of compe- their relationship to resolutions of of the pure number field Q(N1/p)and 1/p tition models in the chemostat via monomial ideals its Galois closure Q(N ,ζp) feedback linearization McFeron, Donovan,Remarksonsome Workman, John, End-point estimates Tran, Hien, Statistics of course data with non-linear heat flows in Kahler geome- and multi-parameter paraproducts on applications to financial risk analysis try higher dimensional tori 290 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Graduate Center, City Yarmola, Tatiana, Invariant measures for Pallekonda, Seshendra, Bounded category University of New York (7) degenerate random perturbations of of an exact category discrete-time dynamical systems Sullivan, Margaret, DNA computing with Ph.D. Program in Mathematics Yilmaz, Atilla, Large deviations for ran- cutting, pasting, filtering and washing dom walk in a random environment Dean, Margaret, Finitely generated meta- Tuncer, Nigar, Globalization theorems in belian and solvable groups Yin, Yong, Static Born-Infeld theory topology Garbin, Daniel, Asymptotics for the para- Rensselaer Polytechnic bolic, hyperbolic, and elliptic Eisenstein State University of New series through hyperbolic and elliptic Institute (9) York at Stony Brook (15) degeneration Department of Mathematical Kim, Youngju, Rigidity and stability for Sciences Department of Applied Mathematics isometry groups in hyperbolic 4-space and Statistics Kogan, Hana, On critical points for Gaus- Cristobal La Torre, Juan, Effective trans- sian vectors with infinitely divisible plant properties of stochastics biophys- Chang, Su-Wei, Growth mixture modeling squares ical models as an exploratory analysis tool to Ding, Xueru, Statistical equilibria of the identify longitudinal quantitative trait Lovell, Jonathan, Total variation of Gaus- coupled barotropic flow and shallow loci in genome-wide association studies sian processes and local times of water flow on a rotating sphere associated L´evy processes Chu, Adrienne, Classification of gastroin- Fang, Yi, Imaging from sparse measure- testinal bleeding data Orosz, Brooke, Problems in additive num- ments ber theory Kifle, Hagos, Lattice Boltzmann simula- Frenkel, Yevgeniy, Propagation of ultra- Tepper, Michael, Endomorphisms of - tion of immiscible two-phase flow in n short electromagnetic pulses in a dimensional projective space over func- three dimensional porous media Maxwell-Duffing medium tion fields Kim, Daesang, Network flow model for Lin, Kui, Error estimation for the di- the simulation of CO2 sequestration New York University, rect inversion model and numerical process schemes for the full inversion model in Courant Institute (17) elastography Kim, Dongyung, Conservative data col- lections and comparison study for Morales, Jose, Synthetic-aperture radar Courant Institute of Mathematical front-tracking Sciences imaging and waveform design for dis- persive media Lee, Jungyeon, Extending commingling Albuts, Thomas, Dimension and measure Rasmussen, Kathryn, Optical pulse dy- analysis to exponential distributions of SLE on the boundary namics in active media with negative and its genetic applications Bae, Hantaek, On the Navier-Stokes equa- refractive index Yongzhi, Chen, Task mapping on super- tions Saintval, Wendy, Influence of modal at- computers with cellular networks Bringley, Thomas, Analysis of the im- tenuation on shallow water propagation Department of Mathematics mersed boundary method for Stokes Wang, Chaoxiong, Power control for flow multiuser communication systems and Chandramouli, Vasu V. M. Sarma,Renor- Freire, Juliana, Analysis techniques for computation of generalized Nash equi- malization and non-rigidity nudged random processes libria Clarke, Andrew, Rigidity of minimal Goldberg, Daniel,NumericalK theoretical submanifolds treatment of grounding line movement State University of New and ice shelf buttressing in marine ice York at Buffalo (5) Hao, Ning, D-bar spark theory and sheets Deligne cohomology Department of Biostatistics Hand, Paul, Homogenization in cardiac Hazard, Peter,H´enon-like maps renor- electrophysiology and blow-up in bac- Cai, Xueya, Generalized linear model malisation terial chemotaxis approach for estimating and testing Ostrovsky, Stanislov,WeightedL2 inter- Kerzhner, Yakov, Eisenstein series, unique- equality of control correlations polation on non-uniformly separated ness principle, and finite order bounds Miller, Austin, Statistical methods for sequences Le, Nam Quang, Analysis of several biodosimetry in the presence of both Savelyev, Yakov, Quantum characteristic sharp-interface limits in variational Berkson and classical measurement classes error problems Thind, Jaimal, Categorical combinatorial Lee, Darren, Two constructions of com- Department of Mathematics constructions of A, D, E root systems plete minimal surfaces Buettgens, Matthew, Property Tao and Xu, Dezhen, Configurations of graphs and Mintchev, Stanislav, Self-organization phe- von Neumman algebras the master equation nomena in networks of pulse-coupled Iovanov, Miodrag C., The representa- phase oscillators tion theory of profinite algebras— Syracuse University (5) Momin, Al Saeed, Intersection theory in interactions with category theory, al- Department of Mathematics 3-dimensional linearized contact ho- gebraic topology and compact groups mology Li, Bo, Regularity estimates for Berezin’s Graves, Christina, On the structure of Render, Paris, Harmonic functions for operator calculus reliability polynomials random walks in random environments Graves, Stephen, Growth of tessellations Rodriguez, Ignacio, A mathematical model State University of New Lu, Yao, Fast multiscale integral equation of telomere length regulation and cellu- York at Binghamton (4) lar senescence methods for image restoration Department of Mathematics and Schutzer, Emmanuel,Markingthe(1, 2) McCaffery, Adam, Structure of infinitely- points of the Brownian web and appli- Science ended, edge-transitive planar maps and cations Junes, Leandro, Duality of higher order their Petrie walks Tulloch, Ross, Geostrophic dynamics at non-Euclidean property for oriented Zhang, Haizhang,Samplingwithrepro- surfaces in the atmosphere and ocean matroids ducing kernels FEBRUARY 2010 NOTICES OF THE AMS 291 Doctoral Degrees Conferred The University of Albany, North Carolina State Wan, Wei, Dynamic game theoretic mod- SUNY (3) University (40) els in marketing and finance Yang, Yipeng, Path dependent stochastic Department of Mathematics and Department of Mathematics models and their applications in finance Statistics Andjelkovic, Ivan, Auxiliary signal de- and communications Clark, Timothy, Poset resolutions of sign for fault detection for nonlinear Yuhasz, George, Berlekamp/Massey al- monomial ideals systems: Direct approach gorithms for linearly generated matrix sequences Savvopoulou, Anna, Almost everywhere Benedict, Brandy,Axisymmetricporoe- convergence of convolution measures lastic boundary element methods for Department of Statistics biphasic mechanics of articular carti- Wedrychowicz, Christopher, Almost ev- lage Anand, Suraj, Novel statistical approaches erywhere convergence of weighted av- to assessing the risk of QT prolonga- erages Davis, Jimena, Uncertainty quantification in the estimation of probability distribu- tion and sample size calculation in ‘thorough QT/QTc studies’ University of Rochester (7) tions on parameters in size-structured population models Brinkley, Jason, Generalized estimators Department of Biostatistics and Dickson, Kelly, Condition estimates for of the attributable benefit of an optimal Computational Biology numerical continuation with applica- treatment regime tions to atomic and molecular fluids Cho, Yoonjin, Comparing predictive val- Chen, Linlin, The correlation structure ues of two diagnostic tests of microarray data and its statistical Ellwein, Laura, Cardiovascular and respi- implications ratory regulation, modeling and param- Choi, Jungsoon, Multivariate spatial-tem- eter estimation poral modeling of environment-health Ma, Yan, Inference for instrument reli- processes ability and rater agreement within a Ernstberger, Jon, High speed parameter multi-rater and longitudinal data set- estimation for a homogenized energy Curtis, Steven, Variable selection meth- ting model ods with applications to shape re- Ernstberger, Stacey, Sensitivity methods stricted regression Pearson, Alexander, Subset selection for for dynamical systems Demirhan, Eren, Variable selection for high-dimensional data, with applica- multivariate smoothing splines with tions to gene array data Govan, Angela, Ranking theory with application to popular sports correlated random errors Department of Mathematics Janovitz-Freireich, Itnuit, Computation of Krishna, Arun, Shrinkage-based variable the exact and approximate radicals selection for linear regression and Huang, Haijun, Heat equation with noise of ideals: Techniques based on matri- mixed effects models restricted to fractal set ces of traces, moment matrices and Liu, Song, Variable selection in general- Li, Huibin, Invariant measure of Burger’s Bezoutians ized additive models and additive Cox system of equations Jiang, Qunlei, Analysis and computation models with nonnegative garrote Palsu-Andriescu, Emanuel,Fourierinte- for a fluid mixture model of tissue Ogorek, Benjamin, Orthology-based mul- gral operators on Colombeau spaces deformations tilevel modeling of mouse and human Salch, Andrew, Moduli and cohomology Kalhorn, Rebecca, On the solvability of gene pairs nonlinear boundary value problems on Saha, Paramita, Robust inference with time scales quantile regression in stochastic volatil- NORTH CAROLINA Kim, Eun Jung, Computational biome- ity models with application to value at chanical models for the pericellular risk calculation Duke University (9) matrix of articular cartilage Schumann, David, Robust variable selec- Department of Statistical Science Levy, Louis, Multipliers for the lower cen- tion tral series of strictly upper triangular Wu, Yuefeng, Asymptotic behavior of Bullard, Floyd, Exoplanet detection: A matrices some Bayesian nonparametric and semi- comparison of three statistics or how Okay, Irfan, The additional dynamics of parametric procedures long should it take to find a small the least squares completions of linear Zhang, Min, Asymptotic joint distribu- planet differential algebraic equations tions of time to default with application Ghosh, Joyee, Efficient Bayesian com- Olson, Sarah,Mathematicalmodelsfor to the pricing of credit derivatives putation and model search in linear analysis of tissue regeneration in artic- Zhang, Qianyi, Seasonal unit root test: A hierarchical models ular cartilage comparison Mitra, Robin, Bayesian methods to impute Pasteur, Roger, A multiple inhibin model Zhang, Ying, Testing for unit roots in missing covariates for causal inference of the human menstrual cycle seasonal time series with long period and model selection Pope, Scott, Parameter identification in Niemi, Jarad, Bayesian analysis and com- lumped compartment cardiorespiratory University of North putational methods for dynamic mod- models Carolina at Chapel Hill (19) eling Porter, Gloria, Multiobjective optimal Department of Biostatistics Ouyang, Zhi, Bayesian additive regression control methodology for the analysis of kernels certain sociodynamic problems Abraham, Anita, Model selection meth- Puggioni, Gavino, Using data augmenta- Qasimov, Heydar, Lacunae based stabi- ods for the linear mixed model for tion and stochastic differential equa- lization of PMLs longitudinal data tions in spatio temporal modeling Samuels, John, Inverse problems and Andraca Carrera, Eugenio, Approaches Scott, James, Bayesian adjustment for post analysis techniques for a stenosis- to parameter and variance estimation multiplicity driven acoustic wave propagation model in generalized linear models Vance, Eric Alan, Statistical methods for Selee, Teresa, Stochastic matrices: Ergod- Chung, Yeonseung,NonparametricBayes- dynamic network data icity coefficients and applications to ian inferences on predictor-dependent Zhang, Liang, Statistical computation ranking response distributions for model space exploration in high- Siskind, Ryan, Model development for Clement, Meagan, Analysis techniques dimensional problems shape memory polymers for diffusion tensor imaging data 292 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Lu, Tsui-Shan, Statistical inferences for Pile, Angela, The space of regular poly- Department of Mathematics outcome dependent sampling with con- gons with a small number of edges Kuceyeski, Amy, Efficient computational tinuous multivariate outcomes and statistical models of hepactic Park, Ju-Hyun, Bayesian density regres- metabolism sion and predictor-dependent cluster- OHIO ing Ye, Deping, Topics in convex geometry and phenomena in high dimension Saville, Benjamin, Bayesian multilevel Bowling Green State models and medical applications University (3) Department of Statistics Shi, Xiaoyan, Diagnostic measures for missing covariate data and semipara- Department of Mathematics and Liu, Peng, Adaptive mixture estimation metric models for neuroimaging Statistics and subsampling PCA Simpson, Sean, Linear models with gen- Attanayake, Champike, Finite elements Kent State University (6) eralized AR(1) covariance structure for and practical error—analysis of Huxley longitudinal and spatial data and EFK equations Department of Mathematical Sciences Department of Mathematics Dong, Fanglong, Bayesian model check- ing in multivariate discrete regression Bouzarth, Elizabeth, Regularized singu- Dekock, Mienie, Absolute continuity and problems larities and spectral deferred correction on the range of a vector measure methods: A mathematical study of nu- Rashid, Mamunur, Inference on logistic Dugan, Carrie, Solvable groups whose merically modeling Stokes fluid flow regression models character degree graphs have diameter Dodson, Benjamin, Caustics and the in- three definite signature Schrodinger¨ equa- Case Western Reserve Fenta, Aderaw, Lucunary power se- tion, linear and nonlinear University (15) quences and extremal vectors Newport, Jason, Solutions to the non- Ignatyev, Oleksiy, The compact support Department of Epidemiology and linear Schrodinger¨ equation with Dirac property for hyperbolic SPDE: Two mass initial data Biostatistics contrasting equations Olli, Jeanette, Tiling systems, division Chang, Shelley, Predicting methicillin- Smith, Tabrina, Orbits and ranges of point measures, and endomorphisms resistant S. aureus carriage and dissem- operators on Banach spaces Department of Statistics and ination in a veterans affairs medical Wakefield, Thomas, Verifying Huppert’s Operation Research center Conjecture for the nonabelian simple Conroy, Britt, Racial differences in sur- groups of rank two Lim, Changwon, Robust statistical theory gical procedure utilization and cardiac and methodology for nonlinear models procedure mortality among American Ohio State University, with applications to toxicology Indian and Alaska Native (AI/AN), Black, Columbus (32) Samarov, Daniel, Analysis and advanced and White Medicare beneficiaries 65 extensions of canonical correlation years of age and older in 15 states, Department of Mathematics analysis 1997 Arms, Scott, Minimal heights in number Sen, Suman,Classificationonmanifolds Hawthorn, Rachael,Findingarole:Health fields Wang, Fangfang, Statistical analysis of care professional’s perspectives on and Ault, Shaun, On the symmetric homology some financial time series responses to role uncertainty in end-of- of algebras Wang, Zhaohui, Capacity investment life care planning Griesmer, John, Ergodic averages, corre- strategies under operational flexibility Heaphy, Emily, Evaluation of HIV-risk be- lation sequences, and sumsets Yoon, Jungyeon, Contributions to sto- haviors of Puerto Rican women with se- Kar, Aditi, Discrete groups and CAT(0) chastic volatility models and option vere mental illness in Cuyahoga County, asymptotic cones pricing Ohio Khoury, Jr., Michael,Multiplicity-onere- Hixson, Eric, Ambulatory heart failure sults and explicit formulas for quasi- University of North treatment: Process and outcomes ef- split p-adic unitary groups Carolina at Charlotte (4) fects of practice and patient adherance Li, Lingfei, Functoriality of harmonic Department of Mathematics and Lu, Qing, Designing and forming pre- foliations Statistics dictive genetic test using optimal ROC curve Liu, Sheng-Chi, Mass equidistribution Huang, Hongwei, Statistical properties of Hecke eigenforms on the Hilbert Onwuzulike, Kaine, The genetics and def- and applications of Turing’s formula modular varieties inition of salt-sensitivity hypertension Lee, Jimin, Some statistical methods for in African Americans McSweeney, John,Timetocoalescence failure time data for a class of nonuniform allocation Ou, Ju-Chi, Evaluation of exposure/treat- Li, Jing, Risk minimizing portfolio opti- models ment effect via spatial propensity score mization and hedging with conditional Niu, Liang, The vertex primitive and in observation studies value-at-risk vertex bi-primitive s-arc regular graphs Parrado, Tony, Improved individual an- Wang, Xian, Selection of mixed copu- Pikula, Rafal, Enveloping semigroups of cestry estimates for proper adjust- las and finite mixture models with affine skew products and Sturmian-like ments of ancestral confounding in applications in finance systems association analysis Schnell, Christian, The boundary behavior Wang, Yang, Gene discovery for age- of cohomology classes and singularities NORTH DAKOTA related macular degeneration of normal function North Dakota State Won, Sungho, Improving genetic analysis Shi, Ronggang, Equidistribution of ex- of case-control studies University, Fargo (2) panding measures with local maximal Zullo, Melissa, Cardiovascular disease dimension and Diophantine approxima- Department of Mathematics management and functional capacity tion Lambert, Joshua, Determining the bipla- improvement in patients with metabolic Swartz, Eric, 2-arc transitive polygonal syndrome nar crossing number of Ck ×Cl×Cm×Cn graphs of large girth and valency FEBRUARY 2010 NOTICES OF THE AMS 293 Doctoral Degrees Conferred Xiong, Wei, Brownian motion of a particle Clough, Crystal, Square: A new family of OREGON immersed in a viscous, incompressible, multivariate encryption schemes thermally fluctuating solvent Jin, Yan, Bayesian solution to the analysis Oregon State University (6) Yang, Keyan, On orbit equivalent permu- of data with values below the limit of Department of Mathematics tation groups detection (LOD) Cook, Sam, Killing spinors and affine Yeum, Ji-A, Probability of solvability of Sauer, Johnothon, Semigroups and their symmetry tensors in Godel’s¨ universe random systems of 2-linear equations zero-divisor graphs Pierce, Larry, Computing entropy for over GF(2) Teymuroglu, Zeynep, Continuum models Z2-actions Young, Justin, The twisted tensor L- for the spread of alcohol abuse funtion of GSp(4) Schmurr, Jason, Triangular billiards sur- faces and translation covers Zhao, Peng, Quantum variance of Maass- University of Cincinnati, Hecke cusp forms Ultman, Shari, The cohomology rings of Medical College (3) seven dimensional primitive cohomo- Department of Statistics Department of Environmental Health geneity one manifolds Bautista, Diannecarrol,Asequentialde- Wills, Dean, Connections between com- sign for approximating the Pareto front Liu, Xiaolei, Sample size calculations binatorics of permutations and algo- using the expected Pareto improvement in matched case-control studies and rithms and geometry unmatched case-control studies with function Department of Statistics Ding, Jie, Monte Carlo pedigree dis- controls contaminated equilibrium test with missing data and Xi, Huifeng, Allele frequency distribution Taylor, Aimee, Statistical enhancement population structure and its implication in association study of support vector machines Han, Gang, Modeling the output from Yu, Tao, A self-building system for Portland State computer experiments having quantita- ranked set sampling tive and qualitative input variables and University (4) its applications University of Toledo (2) Department of Mathematics and Kohlschmidt, Jessica,Rankedsetsam- Department of Mathematics Statistics pling: A look at allocation issues and Ahmed, Aden Omar, Quaternions, octo- missing data complications Liu, Nanshan, Theory and applications of nions, and quantizations of games Kumar, Arun, Sequential calibration of Legendre polynomials and wavelets Godinez Vazquez, Humberto,Dataas- computer models Yousef, Abdel,Twoproblemsinthe similation, adaptive observations and theory of Toeplitz operators on the Li, Dongmei, Resampling-based multiple applications testing with applications to microarray Bergman space Khan, Faisal Shah, Quantum multiplex- data analysis ers, Parrando games, and proper quan- Markiewicz, Shannon, Nonparametric in- tization ference using order restricted random- OKLAHOMA Swinyard, Craig,Students’reasoning ized designs Oklahoma State about the concept of limit in the context Paul, Rajib, Theoretical and algorithmic of reinventing the formal definition developments in Markov chain Monte University (4) Carlo Department of Mathematics University of Oregon (7) Roy, Soma, Sequential-adaptive design of Department of Mathematics computer experiments for the estima- Dollarhide, John,OntheL-function of tion of percentiles multiplicative character sums Brown, Jonathan Scott, Finite w-algebras Sroka, Christopher, Extending ranked set Fisher, Brian, Students’ conceptualiza- of classical type sampling to survey methodology tions of multivariable limits Collins, John Patrick,GluingBridgeland Walters, Kimberly, The use of post- stability conditions and Z2-equivariant Department of Statistics intervention data from waitlist controls sheaves on curves to improve estimation of treatment Bryan, Jennifer, Rank transforms and Leeman, Aaron Christopher, Stabilization effect in longitudinal randomized con- tests of interaction for repeated mea- of chromatic functors trolled trials sures experiments with various covari- Phan, Christopher Lee, Koszul and gen- Wang, Zhen, Semi-parametric Bayesian ance structures eralized Koszul properties for noncom- models extending weighted least squares Shi, Lei, Homozygosity mapping with mutative graded algebras Yao, Yonggang, Statistical applications unknown common ancestors Vanderpool, Ruth, Non-existence of a sta- of linear programming for feature se- ble homotopy category for P-complete lection via regularization methods University of Oklahoma (7) abelian groups Zhang, Xiuyun, Efficient algorithms for Wade, Jeremy, Summability of Fourier or- Department of Mathematics fitting Bayesian mixture models thogonal expansions and a discretized Cho, Sangbum, A finite presentation of Fourier orthogonal expansion involving Ohio University, Athens (2) the Goeritz group radon projections for functions on the Department of Mathematics Frye, Timothy, Graph reflection groups cylinder Walsh, Mark G., Metrics of positive Blackwood, Brian, A study of partial Martin, Jason, Expert conceptualizations scalar curvature and generalised Morse orders on nonnegative matrices and of the convergence of Taylor series, functions von Neumann regular rings yesterday, today and tomorrow Stover, Derrick, Continuous mappings Mecham, TaraLee, Largeness of graphs and some new classes of spaces of Abelian groups PENNSYLVANIA Mukherjee, Antara, Isoperimetric in- Carnegie Mellon University of Cincinnati (5) equalities using Varopoulos transport University (14) Department of Mathematical Srinivasan, Varadharaj Ravi,Oncertain Sciences towers of extensions by antiderivatives Department of Mathematical Science Baena, John, Fast signature schemes over Tonic, Vera, Bockstein basis and resolu- Arama, Danut, On a variational approach odd characteristic tion theorems in extension theory for Stokes conjectures in water waves 294 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Brunick, Gerard, A weak existence re- Zhang, Lei, Phase field model for the Shonkwiler, Clayton,Poincar´edualityan- sult with applications to the financial nucleation in solid state phase trans- gles on Riemannian manifolds with engineer’s calibration problem formations: Theories, algorithms and boundary Chebolu, Prasad,Topicsinrandom applications Tsay, Joe-Kai, Formal analysis of the graphs Zheng, Bin, Finite element approxima- Kerberos authentication protocol Melsted, Pall,Algorithmsonrandom tions of high order partial differential Vela-Vick, David Shea, Applications of graphs equations Ozsvath-Szabo invariants to contact Obermeyer, Fritz, Automated equational Zhu, Yunrong, Robust preconditioners geometry reasoning in nondeterministic λ-calculi for H(grad), H(curl) and H(div) sys- Yoo, Meesue, Combinatorial formulas ∗ modulo theoreis H tems with strongly discontinuous coef- connected to diagonal harmonics and Offner, David, Extremal problems on the ficients Macdonald polynomials hypercube Department of Statistics Zhu, Tong, Nonlinear Polya urn models Picollelli, Michael, Extremal problems and and self-organizing processes Benaglia, Tatiana, Nonparametric esti- random processes on graphs Zong, Ying, On hyper-symmetric abelian mation in multivariate finite mixture Rand, Alexander, Delannay refinement varieties algorithms for numerical methods models Department of Statistics Sousa, Bernardo, Variational methods for Dong, Yuexiao, Dimension reduction for non-illiptically distributed predictors phase transitions Gupta, Abhishek, Statistical methods for Wimmer, Karl, Fourier methods and Gaugler, Trent, Nonparametric estima- high-dimensional data analysis combinatorics in learning theory tion in multivariate finite mixture mod- els Lewin, David, Causal inference methods for randomized controlled trials with Department of Statistics Kim, Daeyoung, Mixture inference at the noncompliance edge of identifiability Bertolet, Marianne,Toweightornotto Nagaraja, Chaitra, Statistical methods Recta, Virginia, A model-based analysis weight: Incorporating sampling designs for modeling house prices and indices into model-based analyses of semicontinuous spatial data Wang, Lie, Variance function estimation Tyekucheva, Svitlana, Augmenting the Damouras, Sotirios, Nonparametric time in nonparametric regression series analysis using Gaussian pro- bootstrap to analyze high-dimensional cesses genomic data University of Luca, Diana, Genetic matching by ances- Wang, Yu, Nonlinear dimension reduction (22) try in genome-wide association studies in feature space Pittsburgh Xie, Libo, Long-memory stochastic volatil- Zhang, Wei, A general class of agree- Department of Biostatistics ity models: A new method for inference ment coefficients for categorical and Blakesley-Ball, Richard, Parametric con- and applications in options pricing continuous responses trol familywise error rates with depen- Zhu, Junjia, Essays on monitoring profile dent P-values Lehigh University (2) data Cheng, Chunrong, Enhanced inter-study Department of Mathematics Temple University (2) prediction and biomarker detection in microarray with application to cancer Mohler, Joel, Residues of Weyl group Department of Mathematics multiple Dirichlet series studies Qian, Huiyu, On data-driven chi square Du, Xiuhong, Additive Schwarz precondi- Haile, Sarah, Inference on competing statistics tioned GMRES, inexact Krylov subspace risks in breast cancer data methods and applications of inexact CG Jakobsdottir, Johanna, Genetics of age- Pennsylvania State Wang, Linhong, Noetherian skew power related maculopathy and score statis- University (21) series rings tics for X-linked quantitative trait loci Li, Jia, Statistical issues in meta-analysis Department of Mathematics University of for identifying signature genes in the Endres, Erik, Shocks and wave interac- Pennsylvania (18) integration of multiple genomic studies tions for compressible flow Department of Mathematics Mi, Zhibao, Robust cross-platform dis- Ethier, John, Strong forms of orthogonal- ease prediction using gene expression ity for sets of hypercubes Auel, Asher N., Cohomological invari- microarrays George, Chris, The Mackey analogy for ants of line bundle-valued symmetric Oh, Sunghee, Effects of missing value SL(n, R) bilinear forms imputation on down-stream analyses in Kursungoz, Kagan, Parity considerations Bressler, Andrew E., Quantum random microarray data in Andrews-Gordon identities and the walks on the integer lattice via generat- Scott, John, Spatio-temporal mixed mod- k-marked Durfee symbols ing functions els for diffusion tensor magnetic reso- Li, Hengguang, Elliptic equations with Favero, David, A study of the geometry nance imaging singularities: A-priori analysis and nu- of the derived category Tudorascu, Dana, Partial least squares merical approaches Herreros, Pilar, Rigidity of magnetic and on data with missing covariates: A Otgonbayar, Uuye, The local index theo- geodesic flows comparison approach rem in noncommutative geometry Holschbach, Armin, A Chebotarev-like Zhu, Fang, An index of local sensitivity Pan, Jia, Optimal intermediated invest- density theorem in algebraic geometry to nonignorability and a penalized ment in a liquidity-driven business Jankowski, Christopher,OntypeII0 E0- pseudolikelihood method for data with cycle semigroups induced by q-pure maps on nonignorable nonresponse M (C) Weaver, Bryce,Growthrateofperiodic n Department of Mathematics orbits for geodesic flows on surfaces Obus, Andrew, Ramification of primes in with regions of positive curvature fields of moduli of three-point covers Balwe, Chetan, Geometric motivic inte- Xue, Guangri,Mathematicalmodeling Rowe, Paul, Policy compliance, confiden- grationonArtinn-stacks and computation of multiphysics fuel tiality and complexity in collaborative Berry, Robert, Lipschitz estimates for cells systems geodesics in the Heisenberg group FEBRUARY 2010 NOTICES OF THE AMS 295 Doctoral Degrees Conferred Labovschii, Alexandr,Mathematicalar- University of Rhode Parrish, Andrew, Pointwise convergence chitecture for models of fluid flow Island (2) of ergodic averages in Orlicz spaces phenomena Manica, Evandro, Effects of coupling Department of Mathematics University of Tennessee, and heterogeneity in the pre-Botzinger¨ Basu, Sukanya,Globalbehaviorofsolu- Knoxville (4) complex cells using first return maps tionstoaclassofsecond-orderrational Department of Mathematics Poerio, Thomas, Topological algebraic difference equations structure in the density topology and Kostrov, Yevgeniy,Globalbehaviorin Clayton, Timothy,Optimalcontrolof on Souslin lines rational difference equations epidemic models involving rabies and Reynolds, Angela,Mathematicalmodels West Nile viruses of acute inflammation and a full lung Khader, Maisa, The dissipative nonlin- model of gas exchange under inflam- SOUTH CAROLINA ear wave equations with space-time matory stress dependent potentials Stanculescu, Iuliana, Turbulence model- University of South Labuz, Brendon, Generalized uniform ing as an ill-posed problem Carolina (12) covering maps characterized as inverse limits of uniform covering maps Department of Statistics Department of Epidemiology and Biostatistics Neilan, Michael, Numerical methods for Ghebregiorgis, Ghideon,Modelingand fully nonlinear second order partial analyzing multivariate longitudinal left- Ogbuanu, Ikechukwu,Doprenatalim- differential equations censored biomarker data munological exposures modify the ef- fect of Interleukin-13 (IL13) gene on (2) Ha, Wonho, Applications of the reflected Vanderbilt University childhood atopy? Ornstein-Uhlenbeck process Department of Mathematics Parker, David, HIV risk behaviors and Han, Sangdae, Comparing spectral densi- knowledge: Associations with HIV preva- Acosta Reyes, Ernesto, Optimal non-linear ties in replicated time series by smooth- lence in Estonia signal models and stability of sampling- ing spline ANOVA Rasathurai, Sumithran, Bayesian identifi- reconstruction Nickleach, Scott, Numerical algorithms cation methods Kozakova, Iva,PercolationandIsing for stock option valuation Wagner, Sara, Spatial assessment of model on tree-like graphs Wang, Jie, Bathtub failure rates in re- groundwater uranium and cancer in liability and dependence in multiple South Carolina testing TEXAS Zhen, Huiling, Multiple imputation of missing data based on prediction of Baylor University (7) conditional quantiles RHODE ISLAND Department of Statistical Sciences Zhou, Li, Quantile regression with ordinal Brown University (11) and discrete data Crixell, Jo Anna, Logistic regression with covariate measurement error in an Center for Statistical Science Department of Mathematics adaptive design: A Bayesian approach Dunsiger, Shira, Analysis of longitudinal Sircar, Sarthok,Dynamicsandrheology Greer, Brandi, Bayesian and likelihood binary data from behavioral medicine of biaxial liquid crystal polymers interval estimation for comparing two Li, Hong, Statistical methods for moni- Department of Statistics Poisson rate parameters using under- toring disease recurrence reported data Chen, Peng, Topics in binary regression Hetzer, Joel, Statistical considerations for Division of Applied Mathematics models with group testing data the analysis of multivariate Phase II Carvalho, Luis, Bayesian centroid estima- McLain, Alex, Accelerated testing with re- testing tion current events and fundamental issues Miyamoto, Kazutoshi,Bayesianandmax- in marginal models Grinberg, Leopold,Topicsinultrascale imum likelihood methods for some scientific computing with application in Pritchard, Nick, Geometric group testing two-segment generalized linear models biomedical modeling Taylor, Laura,Competingrisksina Powers, Stephanie, Bayesian approaches Luo, Xian, A spectral element/smoothed recurrent event setting to inference and variable selection profile method for complex-geometry Wu, Meng, Based hypothesis test for for misclassified or under-reported re- flows unidimensionality sponse models Narayan, Akil, A generalization of the Wang, Jie, Sample size determination Wiener rational basis functions on TENNESSEE for EMAX model, equivalence/non- infinite intervals inferiority test and drug combination in fixed dose trials Papamichael, Giorgios, Event-by-event University of Memphis (5) Zhang, Lin, Semiparametric AUC regres- Monte Carlo simulation of radiation sion for testing treatment effect in transport in vapor and liquid water Department of Mathematical Sciences clinical trials Pollard, Thomas, Comprehensive 3-d change detection Clarke, Teddy J., Asymptotic analysis of Rice University (16) Schiemenz, Alan, Advances in the dis- the abstract telegraph equation Department of Computational and continuous Galerkin method: Hybrid Garland, Sarah Elizabeth, Factorial cross- Applied Mathematics schemes and applications to the re- over designs with randomization re- active infiltration instability in an up- strictions and balanced cross-over de- Gonzalez del Cueto, Fernando,Filtering welling compacting mantle sign from terraces random layering effects for imaging Srinivasan, Ravi,Closureandcomplete Naheed, Naima, Mathematical contribu- and velocity estimation integrability in Burgers turbulence tions to spin-polarized Thomas-Fermi Gonzalez, Edward, Efficient alternating Zhang, Wei, Statistical inference and theory gradient-type algorithms for the ap- probabilistic modeling in compositional Parrish, Anca, On the geometric structure proximate non-negative matrix factor- vision of Lorentz and Marcinkiewicz spaces ization problem 296 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Papakonstantinou, Joanna,Historicalde- Oeding, Luke, G-varieties and the princi- Snyder, Jason, The global structure of velopment of the BFGS secant method pal minors of symmetric matrices iterated function systems and its characterization properties Sendova, Tsvetanka, A new approach to Sun, Kai, Domain decomposition and the modeling and analysis of fracture University of Texas at model reduction for parabolic optimal through an extension of continuum Arlington (1) control problems mechanics to the nanoscale Department of Mathematics Department of Mathematics Sethuraman, Swaminathan, Volumes of certain cones of multi homogeneous Saha, Snehanshu, A study on the b family Douglas, Casey, Perturbed genus one polynomials of shallow water wave equations Scherk surfaces and their limits Tucci, Gabriel, Some results in the hyper- Dunning, Ryan, Asymptotics under self- invariant subspace problem and free University of Texas at intersection for minimizers of self- probability Austin (20) avoiding energies Vega, Maria, Hypergeometric functions Hardway, Heather, Pattern formation in over finite fields and their relations to Department of Mathematics systems of reaction diffusion equations algebraic curves modeling genetic networks Alonso, Ricardo, The Boltzmann equation: Yang, Haibo,RO(G)-graded equivariant Sharp Povzner inequalities applied to Horn, Peter, Higher-order analogues of Bredon cohomology and sheaves regularity theory genus and slice genus of classical knots Anthropelos, Michail, Agents’ agreement McLelland, Matthew, Deformation of sym- Department of Statistics and partial equilibrium pricing in in- metric Scherk type minimal surfaces Chen, Min, Modeling covariance structure complete markets Department of Statistics in unbalanced longitudinal data Callahan, Jason, The arithmetic and George, Nysia,Mixturemodelingand Ahern, Charlotte, Statistical modeling geometry of two-generator Kleinian in the optimization of breast cancer outlier detection in microarray data groups analysis screening schedules Calleja, Renato, Existence and persis- Chatman, Jamie, Computing diversity in Jeong, Jae Sik, Some applications of tence of invariant objects in dynamical undergraduate admissions wavelets to time series data systems and mathematical physics Christian, James, Incorporating anno- Lee, Seokho, Principal components analy- Carneiro, Emanuel, Extremality, symme- tation data in quantitative trait loci sis for binary data try and regularity issues in harmonic mapping with mRNA transcripts Maity, Arnab, Efficient inference in gen- analysis Fofanov, Viacheslav, Statistical models in eral semiparametric regression models Charters, Philippa, Generalizing binary protein structural alignments quadratic residue codes to higher power Mathaes, Matthias, Statistical tests of University of Houston (11) residues over larger fields neutrality based on SNP and Alu repeat Department of Mathematics DiTanna, Anthony, The optimal control data of a L´evy process Neeley, Eleanor Shannon, Models for the Branton, Sheena, Sub-actions for Young Kahle, Alexander, Superconnections and preprocessing of reverse phase protein towers index theory arrays Dulock, Matthew, Uniqueness of holo- Mallmann, Katja, The discriminant alge- Zhu, Hongxiao, Functional data classifi- morphic curves encountering divisors bra in cohomology cation and covariance estimation Filipliski, Natasha, Periodic solutions in Ortiz, Michael, Differential equivariant systems with finite abelian symmetries K-theory Southern Methodist Kashyap, Upasana, Morita equivalence of Salerno, Adriana, Hypergeometric func- University (2) dual operator algebras tions in arithmetic geometry Department of Mathematics Kouan, Cedvic, Pricing spread options Schwab, Russell, Random and periodic under stochastic volatility homogenization for some nonlinear Haque, Mohammed Ziaul, An adaptive Larson, Craig, Graph theoretic indepen- partial differential equations finite element method for systems of dence and critical independent sets Stirling, Spencer, Abelian Chern-Simons second-order hyperbolic partial differ- Pons, Victoria, Numerical methods for theory with toral gauge group, modular ential equations in one space dimension non-smooth problems from calculus of tensor categories, and group categories Taylor, Michael, Epidemic models for variations: Applications Swenson, Michelle, Phylogenetic supertree partial-temporary immunity with delay Raghupathi, Mrinal, Constrained Nevan- methods linna-Pick interpolation Texas A&M University (18) Zhou, Ti, Essays on pricing and portfolio Wang, Tong, Numerical methods for free choice in incomplete markets Department of Mathematics boundary and particulate flow prob- Institute for Computational Chen, Xianjin, Analysis and computa- lems: Applications to biomechanics tion of multiple unstable solutions to Xie, Weiwei, Valuation of spread options Engineering and Sciences nonlinear elliptic systems by bivariate Edgeworth expansion De Basabe, Jonas, High-order finite ele- Fulkerson, Michael, Radial limits of holo- Yavich, Nikolay, Multilevel precondition- ment methods for seismic wave propa- morphic functions on the ball ers for strongly anisotropic problems gation Jiang, Lijian, Multiscale finite element Iglesias-Hernandez, Marco Antonio,An methods using limited global informa- University of North iterative representer-based scheme for tion and applications Texas (3) data inversion in reservoir modeling Kimball, James, Bounds on codes from Department of Mathematics Santillana, Mauricio, Analysis and numer- smooth toric threefolds with Pic(X)=2 ical simulation of the diffusive wave Muntyan, Yevgen, Automata groups Kazemi, Parimah, A constructive method approximation of the shallow water Nam, Dukjin, Multiscale numerical meth- for finding critical points of the equations ods for some types of parabolic equa- Ginzburg-Landau energy functional Stogner, Roy, Parallel adaptive C1 macro- tions Shao, Chuang, Urysohn ultrametric spaces elements for nonlinear thin film and Nguyen, Nga,Surgeryonframes and isometry groups non-Newtonian flow problems FEBRUARY 2010 NOTICES OF THE AMS 297 Doctoral Degrees Conferred Tharkabhushanam, Sri Harsha,Acon- Touron, Charles, An adaptive method for White, Toby, Extensions of latent class servative deterministic spectral method calculating blow-up solution transition models with application to for rarefied gas flows chronic disability survey data University of Virginia (7) Youn, Ahrim, Learning transcriptional Department of Mathematics regulatory networks from the integra- UTAH ton of heterogeneous high-throughout Brigham Young Drupieski, Christopher, Cohomology of data Frobenius-Lusztig kernels of quantized Zhang, Shengyu, Statistical analysis of University (2) enveloping algebras portfolio risk and performance mea- Department of Mathematics Kapp, Brian, On the structure of certain sures: The influence function approach groups associated in division algebras Lian, Zeng, Lyapunov exponents and in- over fields of power series Washington State variant manifold for random dynamical Khongsap, Totrakool, Spin Hecke algebras systems in a Banach space University (2) Pitsillides, Achilleas, Structure of unitary Department of Mathematics Rimmasch, Gretchen, Complete tropical groups over number fields Bezout’s theorem and its consequences Schmitz, Rebecca, Toeplitz-composition Morris, De Anne, Combinatorial prop- ∗ University of Utah (6) C -algebras with piecewise continuous erties of nonnegative and eventually symbols nonnegative matrices Department of Mathematics Tysse, Jill, The centers of spin sym- Yielding, Amy, Spectrally arbitrary zero- Centeleghe, Tommaso,Aconjectural metric and spin hyperoctahedral group nonzero patterns mass formula for Galois representa- algebras tions modp Wang, Yao, On a stochastic wave equation WEST VIRGINIA Copene, Elizabeth, Ephaptic coupling of modeling heat flow cardiac cells West Virginia Crofts, Scott, Duality of Spin(m +1,n) WASHINGTON University (1) Preszler, Jason, Nilpotent orbits of sym- plectic p-adic Lie algebras and quadratic University of Department of Mathematics forms Washington (17) Zhou, Ju, Cycles in graph theory and Simeonova, Lyubima,Wavepropagation matroids in composite materials: Effective prop- Department of Mathematics erties and optimization Ballard, Matthew, Derived categories of Thompson, Joshua, Grafting real complex WISCONSIN quasi-projective schemes projective structures Koo, Tak-Lun, Change of Selmer group Marquette University (1) Utah State University (3) under isogeny, Iwasawa invariants of Λ-adic representation, and coefficients Department of Mathematics, Department of Mathematics and of newforms Statistics and Computer Science Statistics Kunkel, Christopher, Quaternionic con- Ratnakumar, Shivani, Protocol for esti- Maslova, Inga, Testing and estimation tact pseudohermitian normal coordi- mation of target registration error in for functional data with applications to nates registration of acquired CT to live x-ray magnetometer records Moody, Dustin, The Diffie-Hellman prob- left atrial images Morphet, William, Simulation, kringing, lem and generalization of Verheul’s and visualization of circular-spatial theorem Medical College of data Sekar, Anusha, Earthquake source inver- Wisconsin (2) Yurk, Brian, Modeling the evolution of sion for tsunami runup prediction Division of Biostatistics insect phenology with particular refer- Vance, Stephanie, Lattices and sphere ence to mountain pine beetle packings in Euclidean space Fan, Xiaolin, Bayesian nonparametric in- Whitcher, Ursula, Families of polarized ference for competing risks data VIRGINIA K3 hypersurfaces Pajewski, Nicholas, Bayesian semipara- metric hierarchical models for genetic Department of Statistics George Mason association studies in the presence of Alkema, Leontine, Uncertainty assess- population structure and multiplicities (1) University ments of demographic estimates and Department of Statistics projections University of Wisconsin, Markaryan, Tigran,Exactdistributional Di, Yanming, Conditional tests for local- Madison (21) properties of Efron’s biased coin design izing trait genes Department of Mathematics with application to clinical trials Fu, Quiyan, Models and inference of transmission of DNA methylation pat- Akers, Benjamin, On model equations or Old Dominion terns in mammalian somatic cells gravity-capillary waves Andrejko, Erik, Between O and Os- University (3) Gile, Krista, Inference from partially- observed network data taszewski Department of Mathematics and Li, Qunhua, Statistical methods for pep- Darnall, Matthew, On the discrepancy of Statistics tide and protein identification using random (t,m,s)-nets Fernando, Anne, DGM—A finite differ- mass spectometry Galindo, Diego,Selfsimilarsolutionsof ence scheme based on the discontinu- Sloughter, James McLean, Probabilistic cold ion plasma equations ous Galerkin method weather forecasting using Bayesian Hua, Zheng, Derived categories of toric Grant, Terri, Improved constrained global model averaging stacks and Calabi-Yau varieties optimization for estimating molecular Stanberry, Larissa, Statistical solutions Kang, Hye-Won, Multiple scaling methods structure from atomic distance to some problems in medical imaging in chemical reaction networks 298 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Doctoral Degrees Conferred Kim, Yeon Hyang, Representations of WYOMING almost periodic functions using L2- frames University of Wyoming (2) McGinn, Daniel, Quantifier elimination in Department of Mathematics intuitionistic JRS theories Rahunanthan, Arunasalam,Astudyof Milovich, David, Order-theoretic invari- spatial and time discretizations for ants in set-theoretic topology discontinuous Galerkin methods Nam, Sangnam, Construction and anal- Department of Statistics ysis of local wavelet-like pyramidal representations in several dimensions Li, Yumei, Hidden Markov modeling of Otto, Benjamin, Supercharacters of alge- earthquake swarms bra groups Owen, Robert, Outer model theory and the definability of forcing Petro, Matthew,ModulispacesofRie- mann surfaces Rault, Patrick, On uniform bounds for rational points on rational curves and thin sets Rhoades, Robert, The interplay between harmonic weak Maass forms and classi- cal modular forms Sengun, Mehmet Haluk, Serre’s conjec- ture over imaginary quadratic fields Shi, Yingzhe, Numerical methods for coupling of multispecies kinetic and hydrodynamic equations Tang, Yudong, Geodesic rays and test configurations Thorne, Frank, Extensions of results on the distribution of primes Tonejc, Jernej, Formal normal forms for almost complex structures Yang, Xu, Numerical methods for mul- tiscale kinetic transport and high fre- quency waves University of Wisconsin, Milwaukee (8) Department of Mathematical Sciences Chiappetta, Shawn, Non-overlapping do- main decomposition parallel algorithms for convection-diffusion equations Fuhrman, Kseniya,Mathematicalmodel- ing and analysis of virus-host interac- tion in aquatic systems He, Hao, Utility maximization of a port- folio that includes an illiquid asset Michael, Martin, Local and global solu- tions to quasilinear wave equations via Nash-Moser algorithm Rus, George, Finite element methods for control of singular stochastic processes Sebert, Florian, Algebraic and geomet- ric properties of generalized wavelet matrices Shomberg, Joseph, Explicit construction of a robust family of compact inertial manifolds Zaglauer, Katharina, Fair pricing of par- ticipating life insurance contracts in a regime-switching market environment FEBRUARY 2010 NOTICES OF THE AMS 299 2009 Election Results In the elections of 2009 the Society elected a president Nominating Committee elect, a vice president, a trustee, five members at large of Elected as new members of the Nominating Committee are the Council, three members of the Nominating Committee, William Beckner from University of Texas at Austin and two members of the Editorial Boards Committee. Richard T. Durrett from Cornell University Carla D. Savage from North Carolina University President Elect Terms are three years (1 January 2010–31 December 2012). Elected as the president elect is Eric M. Friedlander from the University of Southern California. Term is one Editorial Boards Committee year as president elect (1 February 2010–31 January 2011), Elected as new members of the Editorial Boards Commit- two years as president (1 February 2011–31 January 2013), tee are and one year as immediate past president (1 February Anatoly Libgober from the University of Illinois at 2013–31 January 2014). Chicago Simon Tavener from Colorado State University Vice President Terms are three years (1 February 2010–31 January 2013). Elected as the new vice president is Sylvain Cappell from Courant Institute of Mathematical Sciences, New York University. Term is three years (1 February 2010–31 January 2013). Trustee Elected as new trustee is Mark L. Green from the University of California, Los Angeles. Term is five years (1 February 2010– 31 January 2015). Members at Large of the Council Elected as new members at large of the Council are Alejandro Adem from the University of British Co- lumbia Richard Hain from Duke University Jennifer Schultens from the University of California, Davis Janet Talvacchia from Swarthmore College Christoph Thiele from the University of California, Los Angeles Terms are three years (1 February 2010–31 January 2013). 300 NOTICES OF THE AMS VOLUME 57, NUMBER 2 2010 AMS Election Nominations by Petition Vice President or Rules and Procedures Use separate copies of the form for each candidate for vice Member at Large president, member at large, or member of the Nominating One position of vice president and member of the Council and Editorial Boards Committees. ex officio for a term of three years is to be filled in the elec- tion of 2010. The Council intends to nominate at least two 1. To be considered, petitions must be addressed to candidates, among whom may be candidates nominated Robert J. Daverman, Secretary, American Mathemati- by petition as described in the rules and procedures. cal Society, Department of Mathematics, 302C Aconda Five positions of member at large of the Council for a Court, University of Tennessee, 1534 Cumberland Ave- term of three years are to be filled in the same election. nue, Knoxville, TN 37996-0614, USA, and must arrive by The Council intends to nominate at least ten candidates, 24 February 2010. among whom may be candidates nominated by petition in 2. The name of the candidate must be given as it appears the manner described in the rules and procedures. in the Combined Membership List (www.ams.org/cml). Petitions are presented to the Council, which, according If the name does not appear in the list, as in the case to Section 2 of Article VII of the bylaws, makes the nomi- of a new member or by error, it must be as it appears nations. The Council of 23 January 1979 stated the intent in the mailing lists, for example on the mailing label of of the Council of nominating all persons on whose behalf the Notices. If the name does not identify the candidate there were valid petitions. uniquely, append the member code, which may be ob- Prior to presentation to the Council, petitions in sup- tained from the candidate’s mailing label or by the can- port of a candidate for the position of vice president or didate contacting the AMS headquarters in Providence of member at large of the Council must have at least fifty ([email protected]). valid signatures and must conform to several rules and 3. The petition for a single candidate may consist of sev- operational considerations, which are described below. eral sheets each bearing the statement of the petition, including the name of the position, and signatures. Editorial Boards Committee The name of the candidate must be exactly the same Two places on the Editorial Boards Committee will be filled on all sheets. by election. There will be four continuing members of the 4. On the next page is a sample form for petitions. Peti- Editorial Boards Committee. tioners may make and use photocopies or reasonable The President will name at least four candidates for facsimiles. these two places, among whom may be candidates nomi- 5. A signature is valid when it is clearly that of the mem- nated by petition in the manner described in the rules and ber whose name and address is given in the left-hand procedures. column. The candidate’s assent and petitions bearing at least 100 6. The signature may be in the style chosen by the signer. valid signatures are required for a name to be placed on However, the printed name and address will be checked the ballot. In addition, several other rules and operational against the Combined Membership List and the mail- considerations, described below, should be followed. ing lists. No attempt will be made to match variants of names with the form of name in the CML. A name Nominating Committee neither in the CML nor on the mailing lists is not that of a member. (Example: The name Robert J. Daverman Three places on the Nominating Committee will be filled by election. There will be six continuing members of the is that of a member. The name R. Daverman appears Nominating Committee. not to be.) The President will name at least six candidates for these 7. When a petition meeting these various requirements three places, among whom may be candidates nominated appears, the secretary will ask the candidate to indicate by petition in the manner described in the rules and willingness to be included on the ballot. Petitioners can procedures. facilitate the procedure by accompanying the petitions The candidate’s assent and petitions bearing at least 100 with a signed statement from the candidate giving valid signatures are required for a name to be placed on consent. the ballot. In addition, several other rules and operational considerations, described below, should be followed. FEBRUARY 2010 NOTICES OF THE AMS 301 From the AMS Secretary Nomination Petition for 2010 Election The undersigned members of the American Mathematical Society propose the name of as a candidate for the position of (check one): Vice President Member at Large of the Council Member of the Nominating Committee Member of the Editorial Boards Committee of the American Mathematical Society for a term beginning 1 February, 2011 Return petitions by 24 February 2010 to: Secretary, AMS, 302C Aconda Court, University of Tennessee, Knoxville, TN 37996-0614 USA Name and address (printed or typed) Signature Signature Signature Signature Signature Signature 302 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Mathematics Calendar Please submit conference information for the Mathematics Calendar through the Mathematics Calendar submission form at http://www.ams.org/cgi-bin/mathcal-submit.pl. The most comprehensive and up-to-date Mathematics Calendar information is available on the AMS website at http://www.ams.org/mathcal/. February 2010 post-doctoral researchers in a recently emerged area of Mathematics and Theoretical Physics: SECONDARY CALCULUS. A “diffiety” is a new * 1–March 6 Math-Info 2010: Towards new interactions between geometrical object that properly formalizes the concept of the solu- mathematics and computer science, International Center for Math- tion space of a given system of (nonlinear) PDEs, much as an algebraic ematical Meetings (C.I.R.M.), Marseille, France. variety does with respect to solutions of a given system of algebraic Description: This thematic month will be split into five weeks: * equations. SECONDARY CALCULUS is a natural diffiety analogue of from Feb. 1–5: Lattice Reduction. * from Feb. 8–12: Dynamics and Computation. * from Feb. 15–19: Multi-dimensional Subshifts and the standard Calculus on smooth manifolds, and as such leads to a Tilings. * from Feb. 22–26: Sage Days. * from March 3–5: Topologi- very rich general theory of nonlinear PDEs. It appears that it is this cal Methods for the Study of Discrete Structures. The mornings are the only natural language of quantum physics, just as the standard devoted to lectures (around three lectures of four hours each for Calculus is for classical physics. each week), given by renowned researchers. The specificity of this Information: http://www.levi-civita.org/Activities/ conference is that participants can propose topics and names for DiffietySchools/XIWDS. the afternoon working sessions in order to speak about their results * 14–19 Young Set Theory Workshop 2010, Seminarzentrum Raach, or discuss some problems with other participants. *You can propose near Vienna, Austria. such working sessions* by sending us an email or directly on the Description: The third annual Young Set Theory Workshop will webpage: http://www.lirmm.fr/arith/wiki/MathInfo2010/ take place between 15-19 February 2010 at Seminarzentrum Raach CallForWorkshopProposals. If you intend to participate, *you (http://www.szr.at) located one hour south of Vienna in Raach need to register* as soon as possible on the website: http://www. am Hochgebirge. The aim of this conference is to bring together Ph.D. lirmm.fr/arith/wiki/MathInfo2010/Pre-registration. students and postdocs in Set Theory in order to learn from leading Information: http://www.lirmm.fr/MathInfo2010; http:// researchers in the field, hear about the latest research, and to discuss www.lirmm.fr/arith/wiki/MathInfo2010/Location. research issues in a co-operative environment. The conference format * 1–12 X Winter Diffiety School, Academic Gymnasium, St. Petersburg will be similar to previous years, including tutorials, postdoc research State University, St. Petersburg, Russia. talks, and discussion sessions. Description: The aim of this permanent School is to introduce under- Information: http://www.math.uni-bonn.de/people/ graduate and Ph.D. students in Mathematics and Physics as well as logic/events/young-set-theory-2010/. This section contains announcements of meetings and conferences in the mathematical sciences should be sent to the Editor of the Notices in of interest to some segment of the mathematical public, including ad care of the American Mathematical Society in Providence or electronically hoc, local, or regional meetings, and meetings and symposia devoted to [email protected] or [email protected]. to specialized topics, as well as announcements of regularly scheduled In order to allow participants to arrange their travel plans, organizers of meetings of national or international mathematical organizations. A meetings are urged to submit information for these listings early enough complete list of meetings of the Society can be found on the last page to allow them to appear in more than one issue of the Notices prior to of each issue. the meeting in question. To achieve this, listings should be received in An announcement will be published in the Notices if it contains a call Providence eight months prior to the scheduled date of the meeting. for papers and specifies the place, date, subject (when applicable), and The complete listing of the Mathematics Calendar will be published the speakers; a second announcement will be published only if there are only in the September issue of the Notices. The March, June/July, and changes or necessary additional information. Once an announcement December issues will include, along with new announcements, references has appeared, the event will be briefly noted in every third issue until to any previously announced meetings and conferences occurring it has been held and a reference will be given in parentheses to the month, year, and page of the issue in which the complete information within the twelve-month period following the month of those issues. appeared. Asterisks (*) mark those announcements containing new or New information about meetings and conferences that will occur later revised information. than the twelve-month period will be announced once in full and will In general, announcements of meetings and conferences carry only not be repeated until the date of the conference or meeting falls within the date, title of meeting, place of meeting, names of speakers (or the twelve-month period. sometimes a general statement on the program), deadlines for abstracts The Mathematics Calendar, as well as Meetings and Conferences of or contributed papers, and source of further information. If there is any the AMS, is now available electronically through the AMS website on application deadline with respect to participation in the meeting, this the World Wide Web. To access the AMS website, use the URL: http:// fact should be noted. All communications on meetings and conferences www.ams.org/. FEBRUARY 2010 NOTICES OF THE AMS 303 Mathematics Calendar * 27–28 2nd International Conference on Wireless Information Net- * 14–15 A Celebration of Mathematics and the 40th Anniversary of works & Business Information System (WINBIS2010), Kathmandu, Jeffery Hall, Queen’s University, Kingston, Ontario, Canada. Nepal. Description: In the post-Sputnik era, Canada and the United States Description: The aims of WINBIS’10 is bring together researchers, faced an immense challenge to bring the scientific research levels of engineers, and practitioners interested on wireless information Net- both nations to a higher standard. In the mathematical field, this chal- works & Business information systems to make international forum lenge was met in Canada by several mathematicians, the most notable WINBIS’10 will not only present the papers, also give the ideas on how being Professor R. L. Jeffery of Queen’s University. By strengthening to analyze & approach problems by combining wireless information the Canadian Mathematical Society and by organizing the summer system & Business information system and it’s applications. research seminars held at Queen’s University during the 1950s and Information: http://www.win-bis.com. 1960s, Professor Jeffery was instrumental in raising the level of math- ematical research in Canada. Jeffery Hall, which presently houses the March 2010 Department of Mathematics and Statistics at Queen’s University, is * 14–17 2010 Interpore Conference and Annual Meeting, Texas A&M, named after him. This building was 40 years old in October, 2009. Mar 2010, Texas A&M University, College Station, Texas. The world now faces a similar challenge to the one Professor Jeffery Description: The goal of the conference is to provide a forum for faced in the 1950s. To keep the importance of mathematical research discussing the academic and industrial challenges of porous media, in the forefront of our scientific consciousness, as well as to com- as well as to foster collaboration among theoreticians, modelers and memorate the 40th anniversary of Jeffery Hall and honour Professor experimentalists working in porous media research. The *Interna- Jeffery, the Department of Mathematics and Statistics at Queen’s tional Society for Porous Media* (InterPore) is a non-profit-making University is planning a two-day celebration from May 14-15, 2010. Several presentations are planned. Professor Sir David Cox of the independent scientific organization established in 2008. The general University of Oxford, Professor Hale Trotter of Princeton University aim of the Society is to advance and disseminate knowledge for the and Professor Gerhard Frey of the University of Duisburg, Essen, have understanding, description, and modeling of natural and industrial already agreed to speak. A lecture surveying the development of re- porous media systems (see http://www.interpore.org/ for more search in mathematics in Canada, as well as R. L. Jeffery’s role in the information). The conference will include keynote lectures, invited oral Canadian Mathematical Society, is also planned. It is anticipated that presentations, and poster sessions. No parallel sessions are planned, this celebration will galvanize and reinforce the mathematical talent and oral presentations are by invitation only. in Canada to meet the present scientific and technological challenges. Deadline: The abstracts for poster session presentations are to be It is expected that many alumni and others will attend. sent to: [email protected]. Information: If you would like to be put on the mailing list to receive Information: See http://isc.tamu.edu/news-and-events/ more information, please write by mail to: Dr. A. M. Herzberg FRSC, 2010-interpore-conference-and-annual-meeting.html. Department of Mathematics, Queen’s University, Kingston, Ontario * 15–19 Arizona School of Analysis with Applications, University of K7L 3N6; by email: Miss A. Burns; [email protected]. Arizona, Tucson, Arizona. * 16–22 ESF Mathematics Conference in partnership with EMS and Description: This school is primarily organized for graduate students ERCOM Algebraic Methods in Dynamical Systems, The Institute of and postdocs, but everybody is welcome. The core program consists Mathematics Conference Centre, Bedlewo, Poland. of four minilectures by Rafael Benguria (Isoperimetric Inequalities for Description: The conference is intended to give an account of the new Eigenvalues of the Laplacian), Laszlo Erdos (Universality of Wigner results concerning algebraic methods in dynamical systems. Morales- Random Matrices), Michael Loss (Kinetic Theory and Kac’s Master Ramis theorem establishing the quasi-abelianity of the differential Equation), and Gunter Stolz (Localization in Disordered Media). Thanks Galois group of the variational equation along a particular solution to support by the University of Arizona and the NSF, we can contrib- of a meromorphically completely integrable Hamiltonian system have ute to the travel costs of many participants. led during the recent years to a big number of new results on non- Information: http://www.mathphys.org/AZschool/. integrability of Hamiltonian systems. More recently, the application * 22–26 Conference “Recent Advances in Function Related Operator of differential Galois theory to the study of Hamiltonian systems is Theory”, Hotel “Rincon of the Seas”, Rincon, Puerto Rico. no longer limited to Picard-Vessiot theory but is extended to the more Topics: Operator theory, Function theory in spaces of analytic and general setting based on recent results of Malgrange and Umemura. harmonic functions, Approximation theory, Applications of opera- Information: http://www.esf.org/conferences/10320. tor theory. * 17–20 The Seventh International Conference on Computational Information: More information through the conference web page: Physics, Fragrant Hill Hotel, Beijing, China. http://www.albany.edu/rafrot; or email: rafrot@albany. Focus: The conference will focus on key aspects of computational edu. physics, including theory, numerical methods, and applications. Twenty-two invited speakers have confirmed their participation in May 2010 ICCP7. Papers submitted to ICCP7 will have an opportunity to be pub- * 13–15 International Conference Devoted to the Memory of Acade- lished in one of two journals (Communications in Computational Phys- mician M. Kravchuk (1892–1942), National Technical University of ics, Chinese Journal of Computational Physics). The Scientific Commit- Ukraine, Kyiv, Ukraine. tee will give recommendation for publication of quality papers. For any Description: The Programme of the Conference includes the follow- enquiry you are welcome to send an email to: [email protected]. ing 4 sections: 1. Differential and integral equations, its applications. Organizer: ICCP7 is organized by the Institute of Applied Physics and 2. Algebra, geometry. Mathematical and numerical analysis. 3. Theory Computational Mathematics. of probability and mathematical statistics. 4. History, methods of Information: For further information on the conference venue, please teaching of mathematics. visit the website: http://www.xsfd.com/; http://www.iapcm. Opening ceremony: May 13, at 2:30 pm. ac.cn/iccp7/. Registration fee: US $50.00 (you can pay, when you arrive to Kyiv). * 23–28 Conformal maps: from probability to physics, Monte Verita, All registration fees include a book of abstracts, daily coffee breaks Ascona, Ticino, Switzerland. and excursion. Description: The conference centers on random structures in the Information: [email protected]. context of complex analysis, inspired by interactions between math- 304 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Mathematics Calendar ematics and physics. The central example is perhaps the Stochastic * 7–11 7th Annual Conference on Theory and Applications of Mod- Loewner Evolution, introduced by Oded Schramm. SLE arises as the els of Computation- TAMC 2010, Charles University, Prague, Czech scaling limit of interfaces in 2D lattice models at criticality (percola- Republic. tion, Ising model, self avoiding polymers), and its elegant combination Description: TAMC is an international conference series with an inter- of probability and complex analysis has led to the proofs of many con- disciplinary character, bringing together researchers working in com- jectures originating in physics. Other topics include Diffusion Limited puter science, mathematics and the physical sciences. The crossdisci- Aggregation (a model for electrodeposition and other phenomena), plinary character, together with its focus on algorithms, complexity Hele-Shaw flow (describing interfaces between fluids of different vis- and computability theory, gives the conference a special flavor and distinction. The TAMC conference series arose naturally in response cosities, e.g. oil and water), 2D Quantum Gravity and Random Maps. to important scientific developments affecting how we compute in Organizers: K. Astala, S. Rohde, S. Smirnov. the twenty-first century. Originally aimed to become a significant Information: http://www.unige.ch/~hongler/ascona/. contributor to the scientific resurgence seen in East Asia, TAMC is * 26–29 Workshop on Combinatorial and Additive Number Theory now playing an important international role in contemporary applica- (CANT 2010), CUNY Graduate Center, New York, New York. tions of theoretical computer science. After six successful conferences Description: This is the eighth in a series of annual spring work- held in China (Bejing 2004 and 2006, Kunming 2005, Shanghai 2007, shops sponsored by the New York Number Theory Seminar on prob- Xi’an 2008, ChangSha 2009) TAMC is leaving Asia for the first time. lems in combinatorial and additive number theory and related parts Information: http://www.tamc2010.cz. of mathematics. In addition to long and short talks, each day of the * 15–19 The Thirteenth International Conference on “Hyperbolic conference will include a problem and discussion session. Previous Problems: Theory, Numerics and Applications”, Beijing, People’s CANT conferences have attracted many graduate students as well as Republic of China. research mathematicians, and students are encouraged to attend. A Description: The objective of this conference is to bring together list of lecturers and other information will be posted on the confer- researchers, practitioners, and students with interest in theoretical ence website. Mathematicians who wish to speak at the meeting are analysis, numerical simulations, and applications of hyperbolic PDEs encouraged to submit a title and abstract to: melvyn.nathanson@ and related mathematical models appearing in the area of applied lehman.cuny.edu. sciences. The conference will keep the traditional balance of the HYP Information: http://www.theoryofnumbers.com. series, blending theory, numerics and applications. In addition, addi- tional new attracting points are also taken into granted. June 2010 Information: For more information about the conference list of the * 7–11 International Conference on Applied Mathematics (with the Scientific Committee and Invited Speakers, registration fee and the first William Benter Prize in Applied Mathematics), City University related things including possible support for young mathematicians from developing countries, please visit the HYP2010 conference web- of Hong Kong, Hong Kong. site at: http://www.amt.ac.cn/hyp2010/. Description: In recent years, huge advances have been achieved through the application of mathematical ideas and techniques to * 16–21 XI International Conference “Current Geometry”, Vietri sul a wide variety of fields. The International Conference on Applied Mare Salerno, Italy. Mathematics will cover a wide range of topics including all aspects of Description: The power of synthesis of Geometry, which led in the applied mathematics. The objective of this conference is to provide past to the formulation of “grand unification theories”, has got an a forum for the exchange of expertise, experience, and insights by essential role nowadays, especially because of the growing fragmenta- tion of knowledge due to scientific progress. In order to avoid too big mathematical scientists, physicists, and young researchers who are a dispersion, geometers need a constant dialogue. Therefore, a stable active in the area of applied mathematics, and to encourage leading experience of personal meetings, apart from telematic interchanges, scientists from abroad to further strengthen their cooperation with cannot be renounced. Current Geometry was born to allow a periodic local scientists. The first William Benter Prize in Applied Mathemat- update about actual progresses in Geometry (and its applications) on ics will be awarded and the winner will be presented with the prize the international scene. (http://www6.cityu.edu.hk/rcms/WBP/the_prize.html) Information: http://www.levi-civita.org/Activities/ during the conference. The aim of the prize is to recognize outstand- Conferences/xicurrentgeometry. ing mathematical contributions that have had a direct and fundamen- * 17–19 [FG60] Computational and Geometric Topology, Bertinoro, tal impact on scientific, business, finance, or engineering applications. Forli, Italy. Information: http://www6.cityu.edu.hk/rcms/WBP/ Description: The research groups of Geometric Topology and Compu- int_conf.html. tational Topology of the Universities of Bologna and Modena-Reggio * 7–11 International Functional Analysis Meeting in Valencia on the Emilia, Departments of Mathematics, will be organizing a conference Occasion of the 80th Birthday of Professor Manuel Valdivia, Uni- in order to celebrate the 60th birthdays of Massimo Ferri and Carlo versity of Valencia, Valencia, Spain. Gagliardi. Description: This conference will emphasize different areas of func- Organizers: P. Bandieri, M. R. Casali, A. Cattabriga (secretary), P. Cris- tional analysis. It will provide a venue for established scholars to in- tofori, P. Frosini, L. Grasselli, C. Landi, M. Mulazzani (chair). teract with each other and with junior scholars. Partial support for Information: All the information on the conference and the form to a small number of participants is expected to be available. Recent register and/or submit a talk can be found on the conference’s web recipients of doctoral degrees and pre-doc students are encouraged page http://fg60.dm.unibo.it/. to apply. The meeting will take place in Valencia as a joint venture * 19–24 International Algebraic Conference dedicated to the 70th of the University of Valencia and the University Politécnica of Valen- birthday of A. V. Yakovlev, St. Petersburg, Russia. cia. It will take place in the Mathematics building (and surrounding Organizing committee: A. Generalov (chair), N. Gordeev, I. Fesenko, large lecture theaters) in Burjassot (Valencia) from June 7–11, 2010 G. Leonov, A. Merkuriev, E. Novikova, I. Panin, A. Semenov, N. Vavilov, (both days included). The structure of the conference is of plenary S. Vostokov, I. Zhukov. and long invited lectures in the morning, and twelve parallel sessions Information: http://www.pdmi.ras.ru/EIMI/2010/iac/. in the afternoon. * 20–25 The Twelfth National Conference in Algebra (China), North- Information: http://www.adeit.uv.es/fav2010/. west Normal University, Lanzhou, Gansu, China. FEBRUARY 2010 NOTICES OF THE AMS 305 Mathematics Calendar Description: The National Conference in Algebra is the largest na- 12-hour courses of this year will be given by Franco Flandoli, Giam- tional conference in China. It is held every two years. All areas of alge- battista Giacomin, and Takashi Kumagai. bra and their applications will be covered. At the last conference, there Information: http://math.univ-bpclermont.fr/stflour/. were several hundred presentations and over 500 participants. It will * 12–14 2010 International Conference on Theoretical and Math- be a good opportunity for both the experts and beginning research- ematical Foundations of Computer Science (TMFCS-10), Orlando, ers to learn the most recent developments in algebra in China. The Florida. conference encourages participants from within and outside of China. Description: TMFCS is an important event in the theoretical, math- Information: http://www.nwnu.edu.cn/algebra. ematical and logical areas of computer science. The conference will * 21–25 Harmonic Analysis and Related Topics, Instituto Superior be held at the same time and location where several other major in- Técnico, IST, Lisbon, Portugal. ternational conferences will be taking place. Description: The Summer School and Workshop “Harmonic Analysis Information: For other conferences associated with MULTICONF-10, and Related Topics” is intended both for Ph.D. students and young please visit: http://www.promoteresearch.org. researchers and also experts in various topics related to Harmonic * 28–30 The Mathematics of Klee & Grunbaum: 100 Years in Seattle, Analysis and Function Spaces. Summer School includes three short University of Washington, Seattle, Washington. courses: “Sobolev, capacitary and isocapacitary inequalities” (Vladi- Description: A celebration of the long and illustrious careers of Vic- mir Maz’ya), “Weighted problems for operators of Harmonic Analysis tor Klee and Branko Grunbaum and their fundamental contributions in some Banach Function Spaces” (Vakhtang Kokilashvili), “Variable to discrete mathematics and geometry. There will be 18 invited talks, Lebesgue Spaces” (David Cruz-Uribe). The workshop is supposed to a poster session and an open problems session. Details available at cover a number of topics in Harmonic Analysis, Function spaces, and the above website. related areas. Information: http://sites.google.com/a/alaska.edu/ Information: http://www.math.ist.utl.pt/~hart2010/. kleegrunbaum/. * 21–26 2nd International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences (AMiTaNS’10), August 2010 Sozopol, Bulgaria. * 9–13 Permutation Patterns 2010, Dartmouth College, Hanover, New Description: The conference will be scheduled in plenary and keynote Hampshire. lectures followed by special and contributed sessions. The accents of Topics: The conference topics include (but are not limited to) enu- the conference will be on Difference and Spectral methods; Applied meration questions, forbidden pattern questions, study of the per- Analysis, Biomathematics, Continuum Mathematics, which can be mutation pattern order, algorithms for computing with permutation complemented by new topics. You are welcomed to announce and patterns, applications and generalizations of permutation patterns, organize special sessions that should be within the general topic of and others. the conference. Everybody, who is interested in attending AMiTaNS’10 Plenary Lectures: There will be two invited plenary lectures by Nik please let him/her prepare a short abstract within 300 words clearly Rusˇkuc and Richard P. Stanley, as well as contributed talks. stating the goal, tools and results and submit it at: conference@ Information: http://math.dartmouth.edu/~pp2010/. eac4amitans.org and fill out the online Application form. In case * 27–31 Differential Geometry and its Applications, Masaryk Univer- of problems please send its text version as attachment to e-mail: sity, Faculty of Science, Brno, Czech Republic, Europe. [email protected]. Description: The DGA conferences take place regularly at one of the Deadline: For both submissions is March 31, 2010. Czech universities every three years; the tenth conference in the se- Information: http://2010.eac4amitans.org/. ries took place in Olomouc in 2007. * 22–25 Group Representation Theory and Related Topics, EPFL, Invited plenary speakers: Shun-ichi Amari (Japan), Robert Bryant Centre Interfacultaire Bernoulli, Lausanne, Switzerland. (MSRI, USA), Andreas Cap (Vienna, Austria), Anna Fino (Torino, Italy), Description: This is an international conference that will take place Joseph M. Landsberg (Texas, USA), Franz Pedit (Amsherst & Tuebin- in Lausanne, Switzerland. Group representation theory is a very ac- gen), Lorenz Schwachhoefer (Dortmund, Germany), Zhongmin Shen tive area of mathematics which has connections and overlaps with (Indianapolis, USA), Jian Song (Rutgers, USA), Vladimir Soucek (Prague, areas such as algebraic topology, K-theory, algebraic geometry and Czech Republic), Gudlaudur Thorbergsson (Koeln, Germany). The Edi- commutative ring theory. Research in the area has been driven by torial Board meeting of the journal of the same name, cf. http:// several conjectures and open problems which have pushed the de- www.elsevier.com/locate/difgeo will take place during the velopment of new methodology and broader interactions with other conference. areas of mathematics. The aim of the conference is to stimulate ac- Information: http://dga.math.muni.cz/dga2010. tivity in and enhance interaction between representation theory and other areas of mathematics, like those presented above. In addition September 2010 to the lectures, there will be a conference dinner in honor of Profes- * 7–11 International Conference “Modern Stochastics: Theory and sor Jacques Thevenaz. Applications II”, Kyiv National Taras Shevchenko University, Kyiv, Information: http://grt.epfl.ch. Ukraine. * 29–July 4 23nd International Conference on Operator Theory, West Description: Conference is dedicated to the anniversaries of promi- University of Timisoara, Timisoara, Romania. nent Ukrainian scientists: A. Skorokhod, V. Korolyuk, I. Kovalenko. Description: The conference is devoted to operator theory, operator Sessions: Diffusion Processes, Fractal Analysis, Gaussian and Related algebras and their applications (differential operators, complex func- Processes and Fields, Generalized Renewal Processes, Information tions, mathematical physics, matrix analysis, system theory, etc.). Security, Interacting Particle Systems, Limit Theorems, Markov and Information: http://www.imar.ro/~ot/. Semi-Markov Processes, Mathematics of Finance, Queuing Systems, Risk Processes and Actuarial Mathematics, Statistics of Stochastic July 2010 Processes, Stochastic Analysis, Stochastic Differential Equations, Sto- * 4–17 40th Probability Summer School, Saint-Flour, France. chastic Models of Evolution Systems. Description: Founded in 1971, this school is intended for Ph.D. stu- Keynote speakers: Yu. Belyaev, Sweden; V. Buldygin, Ukraine; A. Bu- dents, teachers and researchers who are interested in probability linski, Russia; J. M. Corcuera, Spain; P. Doukhan, France; M. Dozzi, theory, statistics, and in applications of these techniques. The three France; A. Dudin, Belarus; A. Gushchin, Russia; V. Konakov, Russia; 306 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Mathematics Calendar Yu. Kondratiev, Germany; K. Kubilius, Lithuania; N. Leonenko, UK; and software systems, we welcome the participation of mathemati- N. Limnios, France; E. Merzbach, Israel; I. Molchanov, Switzerland; E. cians and scientists who are interested in using mathematical soft- Orsingher, Italy; G. Peccati, France; D. Silvestrov, Sweden; E. Valkeila, ware for their research. The proceedings of the congress is planned Finland; V. Vatutin, Russia; A. Veretennikov, UK. and all accepted papers and short communications will be published Information: http://probability.univ.kiev.ua/msta2conf. as Springer Lecture Notes in Computer Science (LNCS). Information: http://www.mathsoftware.org/. * 7–11 Logic, Algebra and Truth Degrees 2010, Prague, Czech Repub- lic. * 17–19 S 4 Conference on Symmetry, Separation, Superinteg- Description: Mathematical Fuzzy Logic is a subdiscipline of Math- rability and Special Functions, School of Mathematics, University of ematical Logic which studies the notion of comparative truth. The Minnesota, Minneapolis, Minnesota. assumption that “truth comes in degrees” has proved to be very Description: This conference is in honor of Willard Miller Jr., who useful in many, both theoretical and applied, areas of Mathematics, will be retiring from the University of Minnesota. Topics will include Computer Science, and Philosophy. symmetry methods for differential equations, higher symmetries, Goal: The main goal of this meeting is to foster collaboration between separation and multi-separation of variables, special functions and researchers in the area of Mathematical Fuzzy Logic, and to promote orthogonal polynomials, Hamiltonian systems, both classical and communication and cooperation with members of neighbouring fields. quantum, super-integrability and its connections with separability and Invited speakers: Arnon Avron, Félix Bou, Agata Ciabattoni, Roberto quasi-exact solvability, and applications to physical systems arising Cignoli, Ioana Leustean, Franco Montagna, James G. Raftery, and Hi- in classical and quantum mechanics and relativity. roakira Ono. Information: http://www.math.umn.edu/conferences/s4/. Tutorials by: George Metcalfe and Vilém Novák. November 2010 Program committee: Petr Hájek (Chair), Antonio Di Nola, Christian Fermüller, Siegfried Gottwald, Daniele Mundici, and Carles Noguera. * 8–10 2010 IEEE International Conference on Technologies for Deadline for submissions: March 20, 2010. Homeland Security, Westin Hotel, Waltham, Massachusetts. Information: http://www.mathfuzzlog.org/latd2010/. Description: The tenth annual IEEE Conference on Technologies for Homeland Security (IEEE HST ‘10) will focus on innovative technolo- * 7–12 Geometry, Dynamics, Integrable Systems 2010, Mathematical gies for deterring and preventing attacks, protecting critical infra- Institute SANU, Belgrade, Serbia. structure and individuals, and mitigating damage and expediting Main Topics: Integrable Systems, Classical Mechanics, Nonholonomic recovery. Submissions are desired in the broad areas of critical in- Mechanics, Rigid Body Dynamics, Lie Algebras and Lax Representa- frastructure and key resources protection (CIKR), border protection tion, Separation of Variables. and monitoring, and disaster recovery and response, with application Scientific Committee: Alain Albouy, Sergio Benenti, Alexander Bo- within about five years. benko, Alexey Bolsinov, Alexey Borisov, Victor Buchstaber, Pantelis Information: http://www.ieee-hst.org. Damianou, Vladimir Dragovic, Boris Dubrovin, Valery Kozlov, Igor Krichever, Sergey Novikov (Chairman), Anatol Odzijewicz, Tudor Ratiu, Vered Rom-Kedar, Andrey Tsiganov, Marcelo Viana, Jean-Claude The following new announcements will not be repeated until Zambrini, Rade Zivaljevic. the criteria in the next to the last paragraph at the bottom of Information: Contact: [email protected]; http://www. the first page of this section are met. mi.sanu.ac.rs/~gdis2010/. * 12–17 ESF Mathematics Conference in Partnership with EMS and September 2011 ERCOM/INI: Highly Oscillatory Problems: From Theory to Applica- * 10–16 Turning Dreams into Reality: Transformations and Paradigm tions, The Isaac Newton Institute, Cambridge, United Kingdom. Shifts in Mathematics Education, Rhodes University, Grahamstown, Description: Highly oscillatory phenomena occur in a wide range of South Africa. mathematical applications: from fluid to solid mechanics, electromag- Description: International Conference of The Mathematics Education netics, acoustics, combustion, computerised tomography and imag- into the 21s t Century Project. ing, molecular dynamics, quantum chemistry, plasma transport and Preliminary Announcement and Call for Papers: The Mathematics electrical engineering. Such phenomena have attracted a great deal Education into the 21s t Century Project has just completed its tenth of mathematical attention, mainly in harmonic analysis, asymptotic successful international conference in Dresden, Germany, following analysis, homogenisation, differential geometry, theory of Hamilto- conferences in Egypt, Jordan, Poland, Australia, Sicily, Czech Repub- nian systems and theory of integrable systems. They have an oft- lic, Malaysia and the USA. Our project was founded in 1986 and is undeserved reputation of being hopelessly difficult to analyse and dedicated to the planning, writing and disseminating of innovative to compute: the truth of the matter is that, once they have been un- ideas and materials in Mathematics, Statistics, Science and Computer derstood from the mathematical standpoint, effective computational Education. algorithms are bound to follow. Organizing Committee: The chairman is Professor Marc Schafer of Information: http://www.esf.org/conferences/10340. Rhodes University. * 13–17 Third International Congress on Mathematical Software Information: For further conference details please email Alan Rog- [ICMS’2010—developers meeting], Department of Mathematics, erson, Chairman of the International Programme Committee: alan@ Kobe University, Kobe, Japan. rogerson.pol.pl. Description: This congress is the third in the series, where the first meeting was held in Beijing in 2002 and the previous one in Castro Urdiales, Spain in 2006; see http://www.icms2006.unican.es/. We will welcome developers of mathematical software systems as well as researchers in algorithms and mathematicians who are interested in the development of mathematical software and systems. This is an almost unique chance to meet people in different disciplines in math- ematics and computer science and exchange ideas on developments on mathematical software and systems. While the main audience of this meeting is assumed to be developers of mathematical software FEBRUARY 2010 NOTICES OF THE AMS 307 New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to http://www.ams.org/bookstore-email. Analysis integrability of the Riccati equation; M. E. H. Ismail, Two discrete systems of q-orthogonal polynomials; J. Ławrynowicz, L. F. Reséndis O., and L. M. Tovar S., Like-hyperbolic Bloch-Bergman classes; J. T. Lázaro, Some words about the application of Differential Algebra, Tchebycheff systems to weak Hilbert’s 16th problem; D. Mond, From the index of a differential operator to the Milnor number of Complex Analysis a singularity; J. J. Morales-Ruiz and J.-P. Ramis, Integrability of dynamical systems through differential Galois theory: a practical and Orthogonal guide; M. Paredes and S. Pinzón, Tournaments and parabolic Polynomials almost complex structures on flag manifolds. Primitivo B. Acosta-Humánez, Contemporary Mathematics, Volume 509 Universidad Sergio Arboleda, April 2010, approximately 235 pages, Softcover, ISBN: 978-0-8218- Bogotá, Colombia, and Francisco 4886-9, 2000 Mathematics Subject Classification: 05C20, 12H05, Marcellán, Universidad Carlos 14E20, 14L99, 14M15, 20C20, 30C45, 33C50, 33D45, 34A26, 34C07, III de Madrid, Leganés, Spain, 34C08, 34M15, 35C05, 41A60, 42C05, 46E25, 53C15, 54C40, AMS Editors members US$63, List US$79, Order code CONM/509 This volume represents the 2007–2008 Jairo Charris Seminar in Algebra and Analysis on Differential Algebra, Complex Analysis and Advances in p-adic Orthogonal Polynomials, which was held at the Universidad Sergio and Non-Archimedean Arboleda in Bogotá, Colombia. It provides the state of the art in the theory of Integrable Dynamical Analysis Systems based on such approaches as Differential Galois Theory Martin Berz, Michigan State and Lie Groups as well as some recent developments in the theory of University, East Lansing, MI, and multivariable and q-orthogonal polynomials, weak Hilbert’s 16th Khodr Shamseddine, University Problem, Singularity Theory, Tournaments in flag manifolds, and spaces of bounded analytic functions on the unit circle. of Manitoba, Winnipeg, Manitoba, Canada, Editors The reader will also find survey presentations, an account of recent developments, and the exposition of new trends in the This volume contains the proceedings of areas of Differential Galois Theory, Integrable Dynamical Systems, the Tenth International Conference on p-adic and Non-Archimedean Orthogonal Polynomials and Special Functions, and Bloch–Bergman Analysis, held at Michigan State University in East Lansing, Michigan, classes of analytic functions from a theoretical and an applied on June 30–July 3, 2008. perspective. This volume contains a kaleidoscope of papers based on several of The contributions present new results and methods, as well as the more important talks presented at the meeting. It provides a applications and open problems, to foster interest in research in cutting-edge connection to some of the most important recent these areas. developments in the field. Through a combination of survey papers, This item will also be of interest to those working in algebra and research articles, and extensive references to earlier work, this algebraic geometry. volume allows the reader to quickly gain an overview of current A co-publication of the AMS and Instituto de Matemáticas y sus activity in the field and become acquainted with many of the recent Aplicaciones (IMA). sub-branches of its development. , , and , Strict topologies Contents: D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Contents: J. Aguayo S. Navarro M. Nova on spaces of vector-valued continuous functions over Galois theory of algebraic Lie-Vessiot systems; L. Fernández, non-Archimedean field; , Some subalgebras of the algebra F. Marcellán, T. E. Pérez, and M. A. Piñar, Recent trends on B. Diarra of bounded linear operators of the one variable Tate algebra; two variable orthogonal polynomials; C. A. Gomez S., On the A. Escassut and N. Maïnetti, The ultrametric corona problem; 308 Notices of the AMS Volume 57, Number 2 New Publications Offered by the AMS A. K. Katsaras, Vector-valued p-adic measures; H. A. Keller and representations to actions; Unitary representations of abelian H. Ochsenius, On the Clifford algebra of orthomodular spaces over groups; Induced representations and actions; The space of unitary Krull valued fields; K.-O. Lindahl, Divergence and convergence representations; Semidirect products of groups; Bibliography; of conjugacies in non-Archimedean dynamics; H. M. Moreno, Index. A criterion for the invertibility of Lipschitz operators on type Mathematical Surveys and Monographs, Volume 160 separating spaces; M. Nilsson and R. Nyqvist, On monomial dynamical systems on the p-adic n-torus; H. Ochsenius and February 2010, 237 pages, Hardcover, ISBN: 978-0-8218-4894-4, E. Olivos, On the value group and norms of a form Hilbert space; LC 2009042253, 2000 Mathematics Subject Classification: 37A15, H. Ochsenius and W. H. Schikhof, Compact perturbations of 37A20, 03E15, AMS members US$62, List US$77, Order code Fredholm operators on Norm Hilbert spaces over Krull valued SURV/160 fields; J. Ojeda, Applications of the p-adic Nevanlinna theory to problems of uniqueness; C. Pérez-García and W. M. Schikhof, Tensor products of p-adic locally convex spaces having the strongest locally convex topology; C. G. Petalas and A. K. Katsaras, Differential Equations Tensor products of p-adic measures; A. Rodionov and S. Volkov, p-adic arithmetic coding; K. Shamseddine and M. Berz, Analysis on the Levi-Civita field, a brief overview; P.-A. Svensson, Criteria for non-repelling fixed points; F. Tangara,A p-adic q-deformation of Lectures on Elliptic the Weyl algebra, for q a pN -th root of unity. Boundary Value Contemporary Mathematics, Volume 508 Problems March 2010, 269 pages, Softcover, ISBN: 978-0-8218-4740-4, Shmuel Agmon, Hebrew LC 2009042367, 2000 Mathematics Subject Classification: 46S10, University of Jerusalem, Israel 11S80, 12J25, 16W30, 46G10, 32P05, 11D88, 30G06, 47B37, AMS members US$71, List US$89, Order code CONM/508 This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order Global Aspects of elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, Ergodic Group Actions such as the problem of existence and regularity of solutions of Alexander S. Kechris, California higher-order elliptic boundary value problems. It also contains a Institute of Technology, Pasadena, study of spectral properties of operators associated with elliptic boundary value problems. Weyl’s law on the asymptotic distribution CA of eigenvalues is studied in great generality. 2 The subject of this book is the study of Contents: Notations and conventions; Calculus of L 2 ergodic, measure preserving actions of derivatives—Local properties; Calculus of L derivatives—Global countable discrete groups on standard properties; Some inequalities; Elliptic operators; Local existence probability spaces. It explores a direction theory; Local regularity of solutions of elliptic systems; Gårding’s that emphasizes a global point of view, inequality; Global existence; Global regularity of solutions of concentrating on the structure of the space of measure preserving strongly elliptic equations; Coerciveness; Coerciveness results actions of a given group and its associated cocycle spaces. These are of Aronszajn and Smith; Some results on linear transformations equipped with canonical topological actions that give rise to the on a Hilbert space; Spectral theory of abstract operators; usual concepts of conjugacy of actions and cohomology of cocycles. Eigenvalue problems for elliptic equations; The self-adjoint case; Structural properties of discrete groups such as amenability, Non-self-adjoint eigenvalue problems; Completeness of the Kazhdan’s property (T) and the Haagerup Approximation Property eigenfunctions; Bibliography; Notation index; Index. play a significant role in this theory as they have important AMS Chelsea Publishing, Volume 369 connections to the global structure of these spaces. One of the main topics discussed in this book is the analysis of the complexity of March 2010, 210 pages, Hardcover, ISBN: 978-0-8218-4910-1, the classification problems of conjugacy and orbit equivalence LC 2009047651, 2000 Mathematics Subject Classification: 35J40; of actions, as well as of cohomology of cocycles. This involves 35P10, AMS members US$36, List US$40, Order code CHEL/369.H ideas from topological dynamics, descriptive set theory, harmonic analysis, and the theory of unitary group representations. Also included is a study of properties of the automorphism group of a standard probability space and some of its important subgroups, such as the full and automorphism groups of measure preserving equivalence relations and connections with the theory of costs. The book contains nine appendices that present necessary background material in functional analysis, measure theory, and group representations, thus making the book accessible to a wider audience. Contents: Measure preserving automorphisms; The space of actions; Cocycles and cohomology; Realifications and complexifications; Tensor products of Hilbert spaces; Gaussian probability spaces; Wiener chaos decomposition; Extending February 2010 Notices of the AMS 309 New Publications Offered by the AMS Mathematical Analysis Probability of Partial Differential Equations Modeling Electrostatic MEMS Continuous Time Pierpaolo Esposito, Università Markov Processes degli Studi Roma Tre, Rome, Italy, An Introduction Nassif Ghoussoub, University of British Columbia, Vancouver, Thomas M. Liggett, University of BC, Canada, and Yujin Guo, California, Los Angeles, CA University of Minnesota, Markov processes are among the most Minneapolis, MN important stochastic processes for both theory and applications. This book Micro- and nanoelectromechanical systems (MEMS and NEMS), develops the general theory of these which combine electronics with miniature-size mechanical processes, and applies this theory to various special examples. devices, are essential components of modern technology. It is the The initial chapter is devoted to the most important classical mathematical model describing “electrostatically actuated” MEMS example—one-dimensional Brownian motion. This, together with a that is addressed in this monograph. Even the simplified models chapter on continuous time Markov chains, provides the motivation that the authors deal with still lead to very interesting second- and for the general setup based on semigroups and generators. fourth-order nonlinear elliptic equations (in the stationary case) Chapters on stochastic calculus and probabilistic potential theory and to nonlinear parabolic equations (in the dynamic case). While give an introduction to some of the key areas of application of nonlinear eigenvalue problems—where the stationary MEMS models Brownian motion and its relatives. A chapter on interacting particle fit—are a well-developed field of PDEs, the type of inverse square systems treats a more recently developed class of Markov processes nonlinearity that appears here helps shed a new light on the class of that have as their origin problems in physics and biology. singular supercritical problems and their specific challenges. This is a textbook for a graduate course that can follow one Besides the practical considerations, the model is a rich source of that covers basic probabilistic limit theorems and discrete time interesting mathematical phenomena. Numerics, formal asymptotic processes. analysis, and ODE methods give lots of information and point to Contents: One-dimensional Brownian motion; Continuous time many conjectures. However, even in the simplest idealized versions Markov chains; Feller processes; Interacting particle systems; of electrostatic MEMS, one essentially needs the full available Stochastic integration; Multidimensional Brownian motion and the arsenal of modern PDE techniques to do the required rigorous Dirichlet problem; Appendix; Bibliography; Index. mathematical analysis, which is the main objective of this volume. This monograph could therefore be used as an advanced graduate Graduate Studies in Mathematics, Volume 113 text for a motivational introduction to many recent methods April 2010, approximately 278 pages, Hardcover, ISBN: 978-0-8218- of nonlinear analysis and PDEs through the analysis of a set of 4949-1, 2000 Mathematics Subject Classification: 60J25, 60J27, equations that have enormous practical significance. 60J65; 35J05, 60J35, 60K35, AMS members US$44, List US$55, Titles in this series are co-published with the Courant Institute of Order code GSM/113 Mathematical Sciences at New York University. Contents: Introduction; Part 1. Second-order equations modeling stationary MEMS: Estimates for the pull-in voltage; The branch of stable solutions; Estimates for the pull-in distance; The first branch of unstable solutions; Description of the global set of solutions; Power-law profiles on symmetric domains; Part 2. Parabolic equations modeling MEMS dynamic deflections: Different modes of dynamic deflection; Estimates on quenching times; Refined profile of solutions at quenching time; Part 3. Fourth-order equations modeling nonelastic MEMS: A fourth-order model with a clamped boundary on a ball; A fourth-order model with a pinned boundary on convex domains; Appendix A. Hardy–Rellich inequalities; Bibliography; Index. Courant Lecture Notes, Volume 20 March 2010, 318 pages, Softcover, ISBN: 978-0-8218-4957-6, LC 2009045518, 2000 Mathematics Subject Classification: 35J60, 35B45, 35B35, 35B40, 35P30, 74K15, 74F15, 35J20, 58E07, 74M05, AMS members US$40, List US$50, Order code CLN/20 310 Notices of the AMS Volume 57, Number 2 New AMS-Distributed Publications Advertise in New AMS-Distributed the Publications Notices of the American Mathematical Society Number Theory Formes Quadratiques sur Un Corps Target your Bruno Kahn, Institut de message Mathématiques de Jussieu, to 33,000 Paris, France active readers! This book presents the theory of quadratic forms over a field, focusing on the Pfister-Arason-Knebusch technique of extensions to function fields of quadrics. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Make an impact through Contents: La théorie de Witt; La théorie de Pfister; Corps de fonctions de quadriques; La théorie de Knebusch; Formes devenant Display Advertising isotropes sur le corps des fonctions d’une quadrique; Invariants élémentaires; Le théorème de réduction d’indice et ses applications; Increase your sales and connect with new Formes de basse dimension; Invariants supérieurs; Descente; customers. Rappels sur le groupe de Brauer; Rappels de cohomolgie galoisienne; Courbes algébriques; Un aperçu sur les formes quadratiques en Th e Notices of the AMS has the largest caractéristique 2; Formes quadratiques et cycles algébriques; circulation of any publication in the fi eld. Solutions de certains exercises; Bibliographie; Glossaire; Index. Select premium positions are available. Cours Spécialisés—Collection SMF, Number 15 November 2009, 303 pages, Hardcover, ISBN: 978-2-85629-261-7, Target your audience through 2000 Mathematics Subject Classification: 11E04, 11E81, Individual member US$74, List US$82, Order code COSP/15 Classifi ed Advertising An eff ective and economical way to get the most applicants for your open position. “We have been a long-time advertiser in the Notices and consider it a highly eff ective way to reach our customers in the mathematical community.” —Cambridge University Press Please contact Anne Newcomb to create a program that will address your advertising needs. Anne Newcomb Phone: 401-455-4084 Email: [email protected] Please visit our online media kit: http://www.ams.org/notices/adnot.html February 2010 Notices of the AMS 311 nnoti-adsales-halfpg-ad-bw.inddoti-adsales-halfpg-ad-bw.indd 1 11/4/10/4/10 12:4212:42 PMPM Classified Advertisements Positions available, items for sale, services available, and more KANSAS applications begins December 1, 2009, can contribute to this mission, as well as and continues until positions are closed. to the specific position for which you are KANSAS STATE UNIVERSITY Kansas State University is an Equal Op- applying. Department of Mathematics portunity Employer and actively seeks Review of applications will begin im- diversity among its employees and en- mediately and continue until the position Applications are invited for two Visiting courages applications from women and is filled. Position offers excellent state Assistant Professorships commencing Au- minorities. A background check is re- benefits. To request accommodations, gust 8, 2010. These will be annual appoint- quired. call (201) 684-7718. Additional supportive ments with the possibility of two subse- 000018 materials in nonelectronic format may quent one-year appointments depending be sent to Dr. Maxim Goldberg-Rugalev, on performance, funding, and need of Search Committee chair, Ramapo College services. A Ph.D. in mathematics or a Ph.D. NEW JERSEY of New Jersey, 505 Ramapo Valley Road, dissertation—accepted with only formali- Mahwah, NJ 07430. ties to be completed—is required by the RAMAPO COLLEGE OF NEW JERSEY Ramapo College is a member of the time of appointment. The department Assistant Professor of Mathematics Council of Public Liberal Arts Colleges seeks candidates whose research interests Tenure-Track–Fall 2010 (COPLAC), a national alliance of leading mesh well with current faculty. The de- liberal arts colleges in the public sector. partment has research groups in algebra, JOB DESCRIPTION: Responsibilities en- EEO/AFFIRMATIVE ACTION. analysis, differential equations, geometry/ compass a wide range of undergraduate 000019 topology, and number theory. Successful mathematics courses, and the teaching candidates are expected to participate in and development of General Education the department’s programs integrating mathematics courses. BRAZIL undergraduate and graduate research, REQUIREMENTS: Ph.D. in mathematics including mentoring undergraduate stu- by September 1, 2010, is required. College INSTITUTE FOR PURE AND APPLIED dents during summer programs. The teaching experience preferred. Faculty MATHEMATICS (IMPA) successful candidate should have strong members are expected to maintain active Rio de Janeiro, Brazil research credentials as well as strong ac- participation in research, scholarship, complishments or promise in teaching, college governance, service, academic The Institute for Pure and Applied Math- and should value working with colleagues advisement, and professional develop- ematics (IMPA), invites applications for and students from diverse backgrounds. ment activities. a two-year postdoctoral position in any Applicants must submit the following: All applications must be completed field of mathematics, with a monthly take- a letter of application; curriculum vita; online at http:www.ramapojobs.com. home salary of R$6.000 (about US$3,500 outline of teaching philosophy; a state- Attach resume, cover letter, statement of at the current exchange rate). Candidates ment of research objectives; and four teaching philosophy, research interests, must have obtained their Ph.D. degrees letters of reference, at least one of which and a list of three references to your com- after March 31, 2006. IMPA, located in Rio addresses the applicant’s teaching abil- pleted application. Since its beginning, de Janeiro, Brazil, is widely recognized as ity or potential. All application materi- Ramapo College has had an intercultural/ one of the leading mathematical research als must be submitted electronically via international mission. Please tell us how centers worldwide. Its main goal is the http:wwwmathjobs.org. Screening of your background, interest, and experience generation of high-level mathematical Suggested uses for classified advertising are positions available, books or U.S. laws prohibit discrimination in employment on the basis of color, age, lecture notes for sale, books being sought, exchange or rental of houses, sex, race, religion, or national origin. “Positions Available” advertisements and typing services. from institutions outside the U.S. cannot be published unless they are The 2010 rate is $3.25 per word. No discounts for multiple ads or accompanied by a statement that the institution does not discriminate on the same ad in consecutive issues. For an additional $10 charge, an- these grounds whether or not it is subject to U.S. laws. Details and specific nouncements can be placed anonymously. Correspondence will be wording may be found on page 536 (vol. 56). forwarded. Situations wanted advertisements from involuntarily unemployed math- Advertisements in the “Positions Available” classified section will be set ematicians are accepted under certain conditions for free publication. Call with a minimum one-line headline, consisting of the institution name above toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or 401-455-4084 body copy, unless additional headline copy is specified by the advertiser. worldwide for further information. Headlines will be centered in boldface at no extra charge. Ads will appear Submission: Promotions Department, AMS, P.O. Box 6248, Providence, in the language in which they are submitted. Rhode Island 02940; or via fax: 401-331-3842; or send email to There are no member discounts for classified ads. Dictation over the [email protected]. AMS location for express delivery packages is telephone will not be accepted for classified ads. 201 Charles Street, Providence, Rhode Island 20904. Advertisers will be Upcoming deadlines for classified advertising are as follows: March 2010 billed upon publication. issue–December 28, 2009; April 2010 issue–January 28, 2010; May 2010 issue–February 26, 2010; June/July 2010 issue–April 28, 2010; August 2010 issue–May 28, 2010; September 2010–June 28, 2010. 312 NOTICES OF THE AMS VOLUME 57, NUMBER 2 classified-feb.indd 308 12/28/09 2:16 PM Classified Advertisements research. It offers also graduate level programs at the Ph.D. and MSc level. Cur- POHANG UNIVERSITY OF SCIENCE AND rently, its faculty includes specialists in TECHNOLOGY (POSTECH) Real and Complex Dynamical Systems, Analysis, Algebra, Geometry, Probability, Department of Mathematics Fluid Dynamics, Optimization, Mathemati- Tenure-track Faculty Positions cal Economics, and Computer Graphics. Available Applications should be sent to: opening@ Moving? impa.br until March 31, 2010. Further The Department of Mathematics at Po- inquiries should be addressed to the same hang University of Science and Technol- email address. For information on applica- ogy (POSTECH) invites applications for tion submissions, see: http://www.impa. several tenure-track faculty positions in br/opencms/en/pesquisa/postdoc/ any areas of Pure and Applied Mathemat- opening.html. ics including Statistics. The positions are 000017 primarily at the Assistant Professor level, although senior candidates with strong credentials will also be considered. The KOREA positions are open to mathematicians of Please make any nationality who are fluent in English, POHANG UNIVERSITY OF SCIENCE AND and applications are accepted throughout TECHNOLOGY (POSTECH) the year. To be considered for Spring & sure that the Department of Mathematics Fall, they must be received by early Janu- ary & early July, respectively. New faculty P-Assistant Professorship Positions AMS Notices members are provided with a competitive The department of mathematics at Pohang benefits package and free use of faculty University of Science and Technology apartments for ten years. and Bulletin fi nd (POSTECH) solicits applications for non- POSTECH was founded in 1986 by the tenure-track P-Assistant Professorship. world-leading steel company, POSCO, and their new home. All areas of pure and applied mathematics is an internationally renowned academic are considered. Applicants are expected to institution dedicated to education and have demonstrated exceptional research research in science and engineering. Suc- potential, including major contributions cessful candidates are expected to estab- beyond or through the doctoral disserta- lish strong research programs and to excel tion. The applicants should be within the in teaching at both the undergraduate and first 4 years of their research activities graduate levels. since gaining their Ph.D. degrees. No lin- 1. Qualification: Applicants must hold guistic ability in Korean is required for a Ph.D. in Mathematics or a related disci- successful candidates who are fluent in pline with an outstanding research record English. and should be able to teach in English. The P-Assistant Professorship is a two- 2. Applicants should submit year position and is non-renewable. Excep- a. A signed cover letter tional research performance during the b. A curriculum vitae two years may warrant considerations for c. A description of research possible tenure-track appointment. The d. Three letters of recommendation annual salary is up to 60,000,000 KRW. • Email your new address to An on-campus apartment unit may be sent directly by recommenders provided at a low cost. e. Copies of publications including us: [email protected] This position will remain open until it the Ph.D. dissertation is filled. For more information, please visit our webpage: POSTECH Homepage: http:// • or make the change yourself Applicants should submit; www.postech.ac.kr. online at: a. A signed cover letter Mathematics Department Homepage: www.ams.org/cml-update http://math.postech.ac.kr. b. A curriculum vitae Please have your package sent by email, c. A description of research fax, and regular mail, to • or send the information to: d. Three letters of recommendation sent Hyungju Park, Chair directly by recommenders Department of Mathematics Member and Customer Services e. Copies of publications including the POSTECH American Mathematical Society, Ph.D. dissertation Pohang, 790-784 Korea; 201 Charles Street, Providence, RI Please have your applications sent to: tel:+82-54-279-2302; Chair, Department of Mathematics fax:+82-54-279-2799; 02904-2294 USA Pohang University of Science and email: [email protected]. Technology 000015 Phone: (800) 321-4267 Pohang, 790-784 (US & Canada) The Republic of Korea; fax: +82-54-279-2799; (401) 455-4000 (Worldwide) phone: +82-54-279-2302; Inquiries can be made to: Yong-kyoo Lee (Mr.) Department Administrative Manager; email: [email protected]. 000016 FEBRUARY 2010 NOTICES OF THE AMS 313 classified-feb.indd 309 12/28/09 2:16 PM Meetings & Conferences of the AMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http://www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting. Special Sessions Lexington, Kentucky Advances in Algebraic Coding Theory (Code: SS 6A), University of Kentucky Heide Gluesing-Luerssen, University of Kentucky, and Jon-Lark Kim, University of Louisville. March 27–28, 2010 Advances in Algebraic Statistics (Code: SS 2A), Sonja Saturday – Sunday Petrovic´, University of Illinois, Chicago, and Ruriko Yo- shida, University of Kentucky. Meeting #1057 Advances in Algebraic and Geometric Combinatorics Southeastern Section (Code: SS 14A), Richard Ehrenborg and Margaret A. Re- Associate secretary: Matthew Miller addy, University of Kentucky. Announcement issue of Notices: January 2010 Analysis and Control of Dispersive Partial Differential Program first available on AMS website: February 11, 2010 Equations (Code: SS 25A), Michael J. Goldberg and Bingyu Program issue of electronic Notices: March 2010 Zhang, University of Cincinnati. Issue of Abstracts: Volume 31, Issue 2 Combinatorial Algebra (Code: SS 7A), Juan C. Migliore, University of Notre Dame, and Uwe Nagel, University of Deadlines Kentucky. For organizers: Expired Commutative Algebra (Code: SS 1A), Alberto Corso, For consideration of contributed papers in Special Ses- University of Kentucky, Claudia Polini, University of Notre sions: Expired Dame, and Bernd Ulrich, Purdue University. For abstracts: January 26, 2010 Complex Analysis and Potential Theory (Code: SS 4A), James E. Brennan and Vladimir Eiderman, University of The scientific information listed below may be dated. Kentucky. For the latest information, see www.ams.org/amsmtgs/ Financial Mathematics and Statistics (Code: SS 22A), sectional.html. Kiseop Lee, University of Louisville, and Jose Figueroa- Invited Addresses Lopez, Department of Statistics, Purdue University. Function Theory, Harmonic Analysis, and Partial Dif- Percy A. Deift, Courant Institute–New York University, ferential Equations (Code: SS 5A), , Centre Col- Open problems in integrable systems and random matrix Joel Kilty theory. lege, Irina Mitrea, Worcester Polytechnic Institute, and Irina Mitrea, Worcester Polytechnic Institute, Recent Katharine Ott, University of Kentucky. progress in the area of elliptic boundary value problems Geometric Function Theory and Analysis on Metric on rough domains. Spaces (Code: SS 3A), John L. Lewis, University of Ken- Bruce Reznick, University of Illinois at Urbana Cham- tucky, and Nageswari Shanmugalingam, University of paign, The secret lives of polynomial identities. Cincinnati. Bernd Ulrich, Purdue University, Title to be announced. Homotopy Theory and Geometric Aspects of Algebraic Doron Zeilberger, Rutgers University, 3x+1 (Erdo˝s Topology (Code: SS 16A), Serge Ochanine, University of Memorial Lecture). Kentucky, and Marian F. Anton, Centre College. 314 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences Interactions between Logic, Topology, and Complex Analysis (Code: SS 23A), Matt Insall, Missouri University St. Paul, Minnesota of Science and Technology, and Malgorzata Marciniak, University of Toledo. Macalester College Inverse Problems, Riemann-Hilbert Problems, and Non- linear Dispersive Equations (Code: SS 10A), Peter A. Perry, April 10–11, 2010 University of Kentucky, and Peter Topalov, Northeastern Saturday – Sunday University. Large Scale Matrix Computation (Code: SS 19A), Qiang Meeting #1058 Ye, University of Kentucky, and Lothar Reichel, Kent State Central Section University. Associate secretary: Georgia Benkart Mathematical Economics (Code: SS 21A), Adib Bagh and Announcement issue of Notices: February 2010 Robert E. Molzon, University of Kentucky. Program first available on AMS website: February 25, 2010 Mathematical Problems in Mechanics and Materials Sci- Program issue of electronic Notices: April 2010 ence (Code: SS 20A), Michel E. Jabbour and Chi-Sing Man, Issue of Abstracts: Volume 31, Issue 2 University of Kentucky, and Kazumi Tanuma, Gunma University. Deadlines Mathematical String Theory (Code: SS 18A), Al Sha- For organizers: Expired pere, Department of Physics and Astronomy, University For consideration of contributed papers in Special Ses- of Kentucky, Eric Sharpe, Physics Department, Virginia sions: Expired Polytechnic Institute and State University, and Mark A. For abstracts: February 16, 2010 Stern, Duke University. Mathematics Outreach (Code: SS 26A), Carl W. Lee and The scientific information listed below may be dated. David C. Royster, University of Kentucky. For the latest information, see www.ams.org/amsmtgs/ Matroid Theory (Code: SS 9A), Jakayla Robbins, Uni- sectional.html. versity of Kentucky, and Xiangqian Zhou, Wright State University. Invited Addresses Multivariate and Banach Space Polynomials (Code: SS Charles Doering, University of Michigan, Title to be 15A), Richard A. Aron, Kent State University, and Law- announced. rence A. Harris, University of Kentucky. Matthew James Emerton, Northwestern University, Noncommutative Algebraic Geometry (Code: SS 24A), Title to be announced. Dennis S. Keeler and Kimberly Retert, Miami University. Vladimir Touraev, Indiana University, Title to be an- Partial Differential Equations in Geometry and Varia- nounced. tional Problems (Code: SS 8A), Luca Capogna, University Peter Webb, University of Minnesota, Title to be an- of Arkansas, and Changyou Wang, University of Kentucky. nounced. Recent Progress in Numerical Methods for Partial Differ- ential Equations (Code: SS 12A), Alan Demlow, University Special Sessions of Kentucky, and Xiaobing H. Feng, University of Tennes- Applications of a Geometric Approach to Chaotic Dy- see at Knoxville. namics (Code: SS 16A), Evelyn Sander, George Mason Relative Homological Algebra (Code: SS 11A), Edgar University, Judy Kennedy, Lamar University, and James E. Enochs, University of Kentucky, and Alina C. Iacob, Yorke, University of Maryland. Georgia Southern University. Cohomology and Representation Theory of Algebraic Sharp Spectral Estimates in Analysis, Geometry, and Groups and Related Structures (Code: SS 7A), Christopher Probability (Code: SS 17A), Richard S. Laugesen and Bart- Bendel, University of Wisconsin-Stout, Bobbe Cooper, lomiej Siudeja, University of Illinois. University of Minnesota, and Terrell Hodge, Western Spectral and Transport Properties of Schrödinger Opera- Michigan University. tors (Code: SS 13A), Peter D. Hislop, University of Ken- Combinatorial Representation Theory (Code: SS 3A), tucky, and Jeffrey H. Schenker, Michigan State University. Tom Halverson, Macalester College, and Victor Reiner, University of Minnesota. Commutative Ring Theory (Code: SS 5A), Michael Ax- tell, University of St. Thomas, and Joe Stickles, Millikin University. Differential Equations and Applications (Code: SS 15A), Nicolai Tarfulea, Purdue University Calumet, and Catalin Turc, Case Western Reserve University. Fractals, Convolution Measures, and Frames (Code: SS 13A), Keri Kornelson, University of Oklahoma, and Karen Shuman, Grinnell College. FEBRUARY 2010 NOTICES OF THE AMS 315 Meetings & Conferences Geometric Flows, Moving Frames and Integrable Systems place. The meeting will provide free bus transportation (Code: SS 8A), Gloria Mari-Beffa, University of Wisconsin- to Macalester College on Saturday and Sunday morning Madison, and Peter Olver, University of Minnesota. and back to the hotel on Saturday evening. There are also Hecke Algebras and Deformations in Geometry and public buses between the hotel and Macalester which run Topology (Code: SS 11A), Matthew Douglass and Anne every half hour on Saturday and every hour on Sunday. For Shepler, University of North Texas. those with a car, parking is available at the hotel for US$10 Mathematical Developments in Cell and Systems Biology per day, and it is an easy 12-minute drive to Macalester (Code: SS 14A), Anastasios Matzavinos, Iowa State Uni- along Saint Paul’s Grand Avenue. versity, and Nicoleta Eugenia Tarfulea, Purdue University Calumet. Food Service Matrices and Graphs (Code: SS 9A), Luz M. DeAlba, The Campus Center Dining Hall will be open for lunch Drake University, Adam Berliner, St. Olaf College, Leslie and dinner on Saturday and Sunday. All-you-can-eat-meals Hogben, Iowa State University, and In-Jae Kim, Minnesota cost approximately US$8. Macalester is located on Saint State University. Paul’s well-known Grand Avenue, which offers many var- Partition Theory and the Combinatorics of Symmetric ied dining opportunities. A list of nearby restaurants is Functions (Code: SS 6A), Eric S. Egge, Carleton College, and available at http://www.macalester.edu/admissions/ Kristina Garrett, St. Olaf College. campusvisit/restaurants.html. A list of local restau- Pattern Formation in Biological Systems (Code: SS 12A), rants will be available at the registration desk. Magdalena Skolarska, University of St. Thomas, and Chad Topaz, Macalester College. Local Information and Campus Map Physical Knotting and Linking and its Applications For further information please consult the following sites: (Code: SS 10A), Eric Rawden, University of St. Thomas, Campus Map: http://www.macalester.edu/about/ Yuanan Diao, University of North Carolina at Charlotte, mapbynumber.html and Claus Ernst, Western Kentucky University. Macalester College: http://www.macalester.edu Probabilistic and Extremal Combinatorics (Code: SS 2A), The Macalester College Math Department: http://www. Ryan Martin and Maria Axenovich, Iowa State University. macalester.edu/mathcs Quantum Invariants of 3-manifolds and Modular Cat- Grand Avenue: http://www.grandave.com/ egories (Code: SS 1A), Thang Le, Georgia Institute of Tech- Visit Saint Paul: http://www.visitsaintpaul.com/ nology, Eric Rowell, Texas A&M University, and Vladimir Touraev, Indiana University. Other Activities Universal Algebra and Order (Code: SS 4A), Jeffrey AMS Book Sale: Stop by the on-site AMS Bookstore— Olson, Norwich University, Jeremy Alm, Illinois College, review the newest titles from the AMS, enter the FREE Kristi Meyer, Wisconsin Lutheran College, and Japheth book drawing, enjoy up to 25% off all titles or even take Wood, Bard College. home the new AMS T-shirt! Complimentary coffee will be served courtesy of AMS Membership Services. Accommodations AMS Editorial Activity: An acquisitions editor from the Participants should make their own arrangements directly AMS Book program will be present to speak with prospec- with the hotel. When making a reservation identify your- tive authors. If you have a book project that you would like self as being with the AMS-Macalester College meeting. The to discuss with the AMS, please stop by the book exhibit. AMS is not responsible for rate changes or for the quality of the accommodations. Hotels have varying cancellation Parking or early checkout penalties; be sure to ask for details Parking is readily available on campus and is free. We rec- when making your reservation. ommend that you park in the lot just south of the Leonard A special hotel rate has been negotiated at the Crowne Center accessible via Snelling Avenue or in the lots just Plaza Saint Paul-Riverfront Hotel. The room rate is west of Olin-Rice or west of the Janet Wallace Fine Arts US$109 per night, plus occupancy tax. Please be advised to Center, both of which are accessed via Macalester Drive. make your reservations as soon as possible. This special See the campus map: http://www.macalester.edu/ rate is valid while rooms are available up to and includ- about/mapbynumber.html. ing March 15, 2010. Participants can reserve their room through the direct link below, or by calling 651-292-1900 Social Event or 866-422-3185 and mentioning that they will be attend- The Department of Mathematics will host a reception for ing the AMS-Macalester College meeting. A complimentary all participants. The reception will be held from 6:15 p.m. continental breakfast is included with this room rate. to 7:45 p.m. in the Smail Gallery of Olin-Rice Hall. Hearty Link to hotel reservations at the group rate: https:// appetizers, beer, and wine will be served. The AMS thanks secure.ichotelsgroup.com/h/d/cp/1/en/cwshome/ the department for its hospitality. DPRD-7X8K3X/MSPSP?secure=true&secure=true& justLoggedIn=true&_requestid=638587. Registration and Meeting Information The Crowne Plaza is located in downtown Saint Paul, an Registration will take place in Olin-Rice Hall, from area with many restaurants and museums. The hotel is 4 7:30 a.m. to 4:00 p.m. on Saturday, April 10, and 8:00 a.m. miles from Macalester College, where the meeting will take to noon on Sunday, April 11. The Invited addresses will 316 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences take place in the J. B. Davis Lecture Hall of the Campus Car Rental Center. Most Special Sessions will take place in Olin-Rice Avis is the official car rental company for the sectional Hall and the others will take place in the Humanities Build- meeting in Albuquerque. All rates include unlimited free ing, Carnegie Hall, and Old Main. mileage. Weekend daily rates are available from noon Registration fees: (payable on-site only) US$40/AMS Thursday to Monday at 11:59 p.m. Rates do not include members; US$60/nonmembers; US$5/emeritus mem- any state or local surcharges, tax, optional coverages, bers, students, or unemployed mathematicians. Fees are or gas refueling charges. Renters must meet Avis’s age, payable by cash, check, VISA, Mastercard, Discover, or driver, and credit requirements. For the best available rate American Express. and to make a reservation please call Avis at 800-331-1600 or go online at http://www.avis.com. Please use the AMS Travel meeting Avis Discount Number J098887. Macalester College is located on Saint Paul’s Grand Avenue Weather in a pleasant neighborhood which is 4 miles west of down- town Saint Paul and 6 miles east of downtown Minneapolis. April weather in Saint Paul is unpredictable. The average high for the month is 58 degrees F and the average low is By Air: 37 degrees F. It has been known to snow in mid April and All major airlines service the Minneapolis-Saint Paul In- it has been known to be warm, bright, and sunny. For up- ternational Airport (MSP). The airport is 6.3 miles from to-the-minute weather please visit http://www.weather. Macalester College and 8.5 miles to the Crowne Plaza Hotel com/outlook/driving/interstate/local/55105. in downtown St. Paul. Cab fares from the airport to Macalester cost about Information for International Participants US$20 and from the airport to the Crowne Plaza cost Visa regulations are continually changing for travel to the about US$25. United States. Visa applications may take from three to Super Shuttle (http://www.supershuttle.com) costs four months to process and require a personal interview, about US$14 each way between the airport and the Crowne as well as specific personal information. International Plaza. participants should view the important information Metro Transit (http://metrotransit.org) provides about traveling to the U.S. found at http://www7. public bus options. The 54 bus from the airport to the nationalacademies.org/visas/Traveling_to_ Crowne Plaza takes about 25 minutes. Buses from the US.html and http://travel.state.gov/visa/index. airport to Macalester take 35–45 minutes and require at html. If you need a preliminary conference invitation in least one transfer. order to secure a visa, please send your request to wsd@ If you rent a car, it is easy to drive to the Crowne Plaza: ams.org. Follow the signs for St. Paul/Highway 5. Hwy 5 turns into If you discover you do need a visa, the National Acad- 7th Street in Saint Paul. Travel along Hwy 5 (7th Street) emies website (see above) provides these tips for success- for 6.7 miles until you reach Kellogg Blvd. Turn right on ful visa applications: Kellogg and travel for .5 miles until you reach the hotel * Visa applicants are expected to provide evidence that at 11 E. Kellogg Blvd. they are intending to return to their country of residence. Driving from the airport to Macalester: Follow the signs Therefore, applicants should provide proof of “binding” for St. Paul/Highway 5. After you cross the river, exit right or sufficient ties to their home country or permanent onto Edgecumbe Road (keep left as you exit) and follow for residence abroad. This may include documentation of 1 mile; stay to the left and take Fairview Avenue. Follow the following: Fairview Avenue for 1.5 miles. Turn right (east) on Saint – family ties in home country or country of legal per- Clair Avenue and go .5 miles to Snelling Avenue. Turn left manent residence – property ownership (north) on Snelling, travel .25 miles to the Leonard Center – bank accounts parking lot. – employment contract or statement from employer By Car stating that the position will continue when the employee Driving from the East or the West: Take interstate I94 to returns; Exit 238 at Snelling Avenue. Travel 1 mile south on Snel- * Visa applications are more likely to be successful if ling Avenue to the corner of Snelling and Grand (you are done in a visitor’s home country than in a third country; now at the corner of campus). Continue .25 miles further * Applicants should present their entire trip itinerary, south along Snelling to reach the Leonard Center parking including travel to any countries other than the United lot, just before the football stadium. States, at the time of their visa application; Driving from the South: Take interstate I35 to I35E. * Include a letter of invitation from the meeting orga- Take Exit 104A at Randolph Avenue. Turn left (west) and nizer or the U.S. host, specifying the subject, location, and travel 1 mile along Randolph Avenue until you reach Snel- dates of the activity, and how travel and local expenses ling Avenue. Turn right (north) on Snelling and travel .8 will be covered; miles. The Leonard Center parking lot is accessible from * If travel plans will depend on early approval of the Snelling Avenue just after the football stadium. visa application, specify this at the time of the application; FEBRUARY 2010 NOTICES OF THE AMS 317 Meetings & Conferences * Provide proof of professional scientific and/or Geometric Function Theory (Code: SS 14A), Lukas educational status (students should provide a university Geyer, Montana State University, and Donald Marshall transcript). and Steffen Rohde, University of Washington. This list is not to be considered complete. Please visit Geometric Structures and PDEs (Code: SS 8A), Charles the websites above for the most up-to-date information. Boyer and Dimiter Vassilev, University of New Mexico. Harmonic Analysis and Partial Differential Equations (Code: SS 5A), Matthew Blair, University of New Mexico, Albuquerque, New and Hart Smith, University of Washington. Kleinian Groups and Teichmueller Theory (Code: Mexico SS 15A), Kasra Rafi, University of Oklahoma, Hossein Namaze, University of Texas, and Kenneth Bromberg, University of New Mexico University of Utah. Positivity in Noncommutative Settings (Code: SS 12A), April 17–18, 2010 Roger Roybal, California State University Channel Islands, Saturday – Sunday and Terry Loring, University of New Mexico. Random Matrix Theory and Applications (Code: SS 13A), Meeting #1059 Ioana Dumitriu, University of Washington, and Raj Rao, Western Section University of Michigan. Associate secretary: Michel L. Lapidus Selected Topics in Analysis and Numerics for PDEs Announcement issue of Notices: February 2010 (Code: SS 11A), Thomas Hagstrom, Southern Methodist Program first available on AMS website: March 4, 2010 University, and Stephen Lau and Jens Lorenz, University Program issue of electronic Notices: April 2010 of New Mexico. Issue of Abstracts: Volume 31, Issue 3 Strongly-nonlinear Phenomena: Theory and Applica- tions to Nonlinear Optics, Hydrodynamics, Bose–Einstein Deadlines Condensation and Biology (Code: SS 9A), Alejandro For organizers: Expired Aceves, Southern Methodist University, and Alexander For consideration of contributed papers in Special Ses- Korotkevich and Pavel Lushnikov, University of New sions: Expired Mexico. For abstracts: February 23, 2010 Subjects in between Pure and Applied Mathematics (Code: SS 7A), Hanna Makaruk and Robert Owczarek, Los The scientific information listed below may be dated. Alamos National Laboratory. For the latest information, see www.ams.org/amsmtgs/ Topics in Geometric Group Theory (Code: SS 1A), Mat- sectional.html. thew Day, California Institute of Technology, Daniel Peter Groves, University of Illinois at Chicago, Jason Manning, Invited Addresses SUNY at Buffalo, and Henry Wilton, California Institute Kenneth Bromberg, University of Utah, Title to be an- of Technology. nounced. Trends in Commutative Algebra (Code: SS 3A), Louiza Danny Calegari, California Institute of Technology, Fouli, New Mexico State University, and Janet Vassilev, Title to be announced. University New Mexico. Ioana Dumitriu, University of Washington, Title to be announced. Accommodations Steffen Rohde, University of Washington, Title to be Participants should make their own arrangements directly announced. with the hotel. When making a reservation identify your- self as being with the UNM Math and Stat group attend- Special Sessions ing the AMS Meeting. The AMS is not responsible for rate Dyadic and Non-Dyadic Harmonic Analysis (Code: SS changes or for the quality of the accommodations. Hotels 2A), M. Cristina Pereyra, University of New Mexico, and have varying cancellation or early checkout penalties; Stephanie A. Salomone, University of Portland. be sure to ask for details when making your reservation. Financial Mathematics: The Mathematics of Financial Plaza Inn Downtown Albuquerque, 900 Medical Arts Markets and Structures (Code: SS 4A), Maria Cristina Ave NE, Albuquerque, NM 87102; phone 505-243-5693 Mariani, University of Texas at El Paso, Ionut Florescu, or 866-925-7881. Rates start at US$56.99 + tax. Deadline Stevens Institute of Technology, and Maria P. Beccar- for reservations is March 26, 2010. Be sure to check the Varela, University of Texas at El Paso. cancellation policy. Amenities include: Shuttle to/from Function Spaces, PDEs and Nonlinear Analysis (Code: SS airport and UNM from 6:30 a.m. to 11 p.m., complimentary 10A), Osvaldo Mendez, Behzad Rouhani, and Mohamed continental breakfast, complimentary high-speed wireless Amine Khamsi, University of Texas at El Paso. Internet access in public areas and guest rooms, hot tub, Geometric Combinatorics (Code: SS 6A), Art M. Duval, pool, restaurant. The hotel is approximately one mile to University of Texas at El Paso, and Jeremy Martin, Uni- campus. For more information please visit http://www. versity of Kansas. plazainnabq.com/. 318 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences Albuquerque Airport Fairfield Inn by Marriott, 2300 I40 (East-West). The Martin Luther King exit 1 mile south Centre Ave. SE, Albuquerque, NM 87106, phone: 505-247- of the I25-I40 interchange allows access to the Plaza Inn 1621. Price: US$59 + tax. Deadline for reservations is Albuquerque (take the service road north from Martin March 19, 2010. Be sure to check the cancellation policy. Luther King, on the east side of I25) and UNM (go east to Amenities include: shuttle to/from airport and UNM, University Blvd., enter campus, turn right on Redondo Rd. continental breakfast, coffee/tea in room, outdoor pool & and follow it to visitor parking or residence halls). The spa, complimentary breakfast, wired high-speed Internet Albuquerque Airport Fairfield Inn by Marriott is located in guest rooms. The hotel is approximately 1.5 miles to off of Yale (east of Yale) between the airport and UNM. campus. Transportation from/to the Sunport: Shuttle and taxi service available after all flights outside at baggage claim Food Service level. Albuquerque Cab (505-883-4888, typical charge A list of local restaurants will be available at the registra- US$18.00). Sunport provides a shuttle to Rental Car Cen- tion desk. ter. Call 505-247-1621 for the free shuttle to Albuquerque Airport Fairfield Inn by Marriott, and 505-243-5693 for Local Information and Campus Map the free shuttle to Plaza Inn Albuquerque. For further information please consult the website main- Getting around Albuquerque is easiest by car, but tained by the Department of Mathematics at the Uni- the city has regular bus service along Central Ave. from versity of New Mexico: http://www.math.unm.edu. To downtown to UNM for participants who wish to stay down- view a campus map please visit: http://www.unm.edu/ town, see http://www.cabq.gov/transit/routes-and- campusmap.html. Dane Smith Hall is building 48 at F-G-3. schedules. Taxi service is available but best arranged For travel information please visit: http://www.math. beforehand. unm.edu/about/index.php. Car Rental Other Activities Avis is the official car rental company for the sectional AMS Book Sale: Stop by the on-site AMS Bookstore—re- meeting in Albuquerque. All rates include unlimited free view the newest titles from the AMS, enter the FREE book mileage. Weekend daily rates are available from noon drawing, enjoy up to 25% off all titles or even take home Thursday to Monday at 11:59 p.m. Rates do not include the new AMS T-shirt! Complimentary coffee will be served any state or local surcharges, tax, optional coverages, courtesy of AMS Membership Services. or gas refueling charges. Renters must meet Avis’s age, driver, and credit requirements. For the best available rate AMS Editorial Activity: An acquisitions editor from the and to make a reservation please call Avis at 800-331-1600 AMS Book program will be present to speak with prospec- or go online at http://www.avis.com. Please use the AMS tive authors. If you have a book project that you would like meeting Avis Discount Number J098887. to discuss with the AMS, please stop by the book exhibit. Weather Parking April weather is generally pleasant with daytime tempera- Visitors can park anywhere on campus on weekends (6:00 tures in the 60 degree F. range, and nighttime tempera- p.m. Friday until 8:00 a.m. Monday) without a permit. City tures in the 30–45 degree F. range. For up-to-the-minute parking rules still apply. For further information please weather please visit http://www.weather.com/out- visit http://pats.unm.edu/visitors.cfm. look/driving/local/USNM0004. Registration and Meeting Information Information for International Participants Registration will take place in the atrium of Dane Smith Visa regulations are continually changing for travel to the Hall located on Las Lomas Blvd., across from University United States. Visa applications may take from three to House on Yale Avenue, from 7:30 a.m. to 4:00 p.m. on Sat- four months to process and require a personal interview, urday, April 17, and 8:00 a.m. to noon on Sunday, April 18. as well as specific personal information. International Registration fees: (payable on-site only) US$40/AMS participants should view the important information members; US$60/nonmembers; US$5/emeritus mem- about traveling to the U.S. found at http://www7. bers, students, or unemployed mathematicians. Fees are nationalacademies.org/visas/Traveling_to_ payable by cash, check, VISA, Mastercard, Discover, or US.html and http://travel.state.gov/visa/index. American Express. html. If you need a preliminary conference invitation in order to secure a visa, please send your request to dls@ Travel ams.org. By Air: If you discover you do need a visa, the National Acad- Albuquerque Sunport International Airport: Albuquer- emies website (see above) provides these tips for success- que is served by many of the major commercial carriers ful visa applications: and several commuter airlines. The Sunport is located two * Visa applicants are expected to provide evidence that (2) miles south of UNM. they are intending to return to their country of residence. Traveling to Albuquerque by car: Albuquerque is Therefore, applicants should provide proof of “binding” served by two major interstates, I25 (North-South) and or sufficient ties to their home country or permanent FEBRUARY 2010 NOTICES OF THE AMS 319 Meetings & Conferences residence abroad. This may include documentation of the Richard E. Schwartz, Brown University, Polygonal outer following: billiards. – family ties in home country or country of legal per- manent residence Special Sessions – property ownership Automorphic Forms, L-functions, and Applications – bank accounts (Code: SS 6A), Ameya Pitale, American Institute of Math- – employment contract or statement from employer ematics, and Anantharam Raghuram, Oklahoma State stating that the position will continue when the employee University. returns; Biomembranes: Modeling, Analysis, and Computation * Visa applications are more likely to be successful if (Code: SS 16A), Ricardo H. Nochetto and Dionisios Mar- done in a visitor’s home country than in a third country; getis, University of Maryland. * Applicants should present their entire trip itinerary, Elliptic and Parabolic Problems in Geometry (Code: SS including travel to any countries other than the United 12A), Simon Brendle, Stanford University, and Mu-Tao States, at the time of their visa application; Wang, Columbia University. * Include a letter of invitation from the meeting orga- Expandable Computations, Algorithms, Methodologies nizer or the U.S. host, specifying the subject, location, and and Experiments for Engineering Interpretation (Code: SS dates of the activity, and how travel and local expenses 1A), Mustapha S. Fofana, Worcester Polytechnic Institute, will be covered; Marie D. Dahleh, Harvard School of Engineering and Ap- * If travel plans will depend on early approval of the plied Sciences, Harvard University, and Kenji Kawashima, visa application, specify this at the time of the application; Precision and Intelligence Laboratory, Tokyo Institute of * Provide proof of professional scientific and/or Technology. educational status (students should provide a university Financial Mathematics (Code: SS 9A), Tim S.T. Leung, transcript). Johns Hopkins University. This list is not to be considered complete. Please visit Graph Theory (Code: SS 10A), Nathan W. Kahl, Seton the websites above for the most up- to-date information. Hall University, Michael J. Ferrara, University of Colorado at Denver, and Arthur H. Busch, University of Dayton. Groups, Computations, and Applications (Code: SS 2A), Newark, New Jersey Delaram Kahrobaei, City University of New York. Homology Theories for Knots and Skein Modules (Code: New Jersey Institute of Technology SS 3A), Mikhail Khovanov, Columbia University, and Jozef H. Przytycki and Radmila Sazdanovic, George Washing- May 22–23, 2010 ton University. Saturday – Sunday Invariants of Knots, Links, and 3-Manifolds (Code: SS 4A), Abhijit Champanerkar and Ilya S. Kofman, College Meeting #1060 of Staten Island, CUNY, and Philip J. P. Ording, Medgar Eastern Section Evers College, CUNY. Associate secretary: Steven H. Weintraub Lie Algebras and Representation Theory (Code: SS 8A), Announcement issue of Notices: March 2020 Gautam Chinta, City College, City University of New York, Program first available on AMS website: April 8, 2010 Andrew Douglas, New York City College of Technology, Program issue of electronic Notices: May 2020 City University of New York, and Bart Van Steirteghem, Issue of Abstracts: Volume 31, Issue 3 Medgar Evers College, City University of New York. Logic and Groups (Code: SS 17A), Peggy Dean, Claire Deadlines Wladis, and Marcos Zyman, Borough of Manhattan Com- For organizers: Expired munity College, City University of New York. For consideration of contributed papers in Special Ses- Mathematical Neuroscience: Modeling, Analysis, and sions: February 2, 2010 Simulations (Code: SS 14A), Horacio G. Rotstein, New For abstracts: March 30, 2010 Jersey Institute of Technology. Mathematics and Computations of Fluid Dynamics The scientific information listed below may be dated. (Code: SS 15A), Yuan N. Young, New Jersey Institute of For the latest information, see www.ams.org/amsmtgs/ Technology. sectional.html. Mathematics of Optics and Matter Waves (Code: SS 13A), Roy Goodman, New Jersey Institute of Technology. Invited Addresses Nonlinear Waves (Code: SS 19A), A. David Trubatch, Simon Brendle, Stanford University, Hamilton’s Ricci Montclair State University. flow and the sphere theorem in geometry. Recent Trends in Cayley Graphs to Model Interconnec- Konstantin M. Mischaikow, Rutgers University, Com- tion Networks (Code: SS 18A), Daniela Ferrero, Texas State putational topology applied to the global dynamics of University, and Beth Novick, Clemson University. nonlinear systems. Teichmüller Theory, Hyperbolic Geometry, and Complex Ricardo H. Nochetto, University of Maryland, Curvature Dynamics (Code: SS 5A), Zheng Huang, College of Staten driven flows in deformable domains. Island, CUNY, and Ren Guo, University of Minnesota. 320 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences Topological and Computational Dynamics (Code: SS Complex Analysis and Operator Theory (Code: SS 10A), 7A), Jean-Philippe Lessard, Institute for Advanced Study Maribel Loaiza, Enrique Ramirez de Arellano, and Nikolai and Rutgers University, and Konstantin M. Mischaikow, Vasilevski, CINVESTAV, Ilya M. Spitkovsky, College of Rutgers University. William & Mary, and Kehe Zhu, State University of New Vortex Dynamics: Theory and Applications (Code: SS York at Albany. 11A), Denis Blackmore, New Jersey Institute of Technol- Dynamical Systems (Code: SS 4A), Alberto Verjovsky, ogy, Morten Brøns, Technical University of Denmark, and IM-UNAM, and Rodrigo Perez, Indiana University-Purdue Chjan Lim, Rochester Polytechnic Institute. University, Indianapolis. Graph Theory and Combinatorics with Emphasis on Geometric and Topological Aspects (Code: SS 9A), Gelasio Berkeley, California Salazar, Instituto de Fisica, Universidad Autonoma de San Luis Potosi, and Dan S. Archdeacon, University of University of California Berkeley Vermont. Harmonic Analysis, Microlocal Analysis, and Partial Dif- June 2–5, 2010 ferential Equations (Code: SS 1A), Gunther Uhlmann, Uni- Wednesday – Saturday versity of Washington, and Salvador Perez Esteva, UNAM. Low-Dimensional Topology (Code: SS 8A), Kenneth L. Meeting #1061 Baker, University of Miami, and Enrique Ramirez Losada, Eighth Joint International Meeting of the AMS and the CIMAT. Sociedad Matemática Mexicana. Singularity Theory and Algebraic Geometry (Code: SS Associate secretary: Susan J. Friedlander 6A), David Eisenbud, University of California, Berkeley, Announcement issue of Notices: March 2010 Anatoly S. Libgober, University of Illinois at Chicago, Jose Program first available on AMS website: April 22, 2010 Seade, UNAM, and Xavier Gomez-Mont, CIMAT. Program issue of electronic Notices: June 2010 Toeplitz Operators and Discrete Quantum Models (Code: Issue of Abstracts: Volume 31, Issue 3 SS 5A), Alejandro Uribe, University of Michigan, and Ma- ciej Zworski, University of California, Berkeley. Deadlines For organizers: Expired For consideration of contributed papers in Special Ses- Syracuse, New York sions: February 16, 2010 For abstracts: April 13, 2010 Syracuse University The scientific information listed below may be dated. October 2–3, 2010 For the latest information, see www.ams.org/amsmtgs/ Saturday – Sunday internmtgs.html. Meeting #1062 Invited Addresses Eastern Section Alejandro Adem, University of British Columbia and Associate secretary: Steven H. Weintraub PIMS, Title to be announced. Announcement issue of Notices: To be announced Peter W-K Li, University of California Irvine, Title to be Program first available on AMS website: August 19, 2010 announced. Program issue of electronic Notices: October Ernesto Lupercio, CINVESTAV, Title to be announced. Issue of Abstracts: Volume 31, Issue 4 Victor Perez Abreu, CIMAT, Title to be announced. Alberto Verjovsky, IM-UNAM, Title to be announced. Deadlines Maciej Zworski, University of California Berkeley, Title For organizers: March 2, 2010 to be announced. For consideration of contributed papers in Special Ses- sions: June 15, 2010 Special Sessions For abstracts: August 10, 2010 Algebraic Topology and Related Topics (Code: SS 3A), Alejandro Adem, University of British Columbia, Gunnar The scientific information listed below may be dated. E. Carlsson and Ralph L. Cohen, Stanford University, and For the latest information, see www.ams.org/amsmtgs/ Ernesto Lupercio, CINVESTAV. sectional.html. Analytic Aspects of Differential Geometry (Code: SS 2A), Nelia Charalambous, ITAM, Lizhen Ji, University of Michi- Invited Addresses gan, and Jiaping Wang, University of Minnesota. Alan Frieze, Carnegie-Mellon University, Title to be Commutative Algebra and Representation Theory announced. (Code: SS 7A), David Eisenbud and Daniel M. Erman, Uni- Yan Guo, Brown University, Title to be announced. versity of California, Berkeley, Jose Antonio de la Pena, William Minicozzi, Johns Hopkins University, Title to UNAM, and Rafael Villareal, Cinvestav-IPN. be announced. FEBRUARY 2010 NOTICES OF THE AMS 321 Meetings & Conferences Andrei Zelevinsky, Northeastern University, Title to Special Sessions be announced. Applications of Nonlinear PDE (Code: SS 5A), Susan J. Friedlander and Igor Kukavica, University of Southern Special Sessions California. Difference Equations and Applications (Code: SS 2A), Combinatorics and Probability on Groups (Code: SS Michael Radin, Rochester Institute of Technology. 3A), Jason Fulman and Robert Guralnick, University of Graphs Embedded in Surfaces, and Their Symmetries Southern California, and Igor Pak, University of California (Code: SS 4A), Jack E. Graver and Mark E. Watkins, Syra- Los Angeles. cuse University. Extremal and Probabilistic Combinatorics (Code: SS 4A), Mathematical Image Processing, Lixin Shen and Yuesh- Benny Sudakov, University of California Los Angeles, and eng Xu, Syracuse University. Jacques Verstraete, University of California San Diego. Nonlinear Analysis and Geometry (Code: SS 1A), Ta- Large Cardinals and the Continuum (Code: SS 2A), deusz Iwaniec, Leonid V. Kovalev, and Jani Onninen, Matthew Foreman, University of California Irvine, Alekos Syracuse University. Kechris, California Institute for Technology, Itay Neeman, Several Complex Variables (Code: SS 3A), Dan F. Coman University of California Los Angeles, and Martin Zeman, and Evgeny A. Poletsky, Syracuse University. University of California Irvine. Recent Trends in Probability and Related Fields (Code: SS 6A), Marek Biskup, University of California Los Angeles, Yuval Peres, Microsoft Research, and Sebastien Roch, Los Angeles, University of California Los Angeles. Topology and Symplectic Geometry (Code: SS 1A), Rob- California ert Brown and Ciprian Manolescu, University of California Los Angeles, and Stefano Vidussi, University of California University of California Los Angeles Riverside. October 9–10, 2010 Saturday – Sunday Notre Dame, Indiana Meeting #1063 Notre Dame University Western Section Associate secretary: Michel L. Lapidus October 29–31, 2010 Announcement issue of Notices: August 2010 Friday – Sunday Program first available on AMS website: August 26, 2010 Program issue of electronic Notices: October 2010 Meeting #1064 Issue of Abstracts: Volume 31, Issue 4 Central Section Associate secretary: Georgia Benkart Deadlines Announcement issue of Notices: August 2010 For organizers: March 10, 2010 Program first available on AMS website: September 16, For consideration of contributed papers in Special Ses- 2010 sions: June 22, 2010 Program issue of electronic Notices: October 2010 For abstracts: August 17, 2010 Issue of Abstracts: Volume 31, Issue 4 Deadlines The scientific information listed below may be dated. For organizers: February 19, 2010 For the latest information, see www.ams.org/amsmtgs/ For consideration of contributed papers in Special Ses- sectional.html. sions: July 20, 2010 Invited Addresses For abstracts: September 7, 2010 Greg Kuperberg, University of California Davis, Title The scientific information listed below may be dated. to be announced. For the latest information, see www.ams.org/amsmtgs/ Cris Moore, University of New Mexico, Title to be an- sectional.html. nounced. Stanley Osher, University of California Los Angeles, Invited Addresses Title to be announced. Laura DeMarco, University of Illinois at Chicago, Title Terence Tao, University of California Los Angeles, Title to be announced. to be announced (Einstein Public Lecture in Mathematics). Jordan Ellenberg, University of Wisconsin, Title to be Melanie Wood, Princeton University, Title to be an- announced. nounced. David Fisher, Indiana University, Title to be announced. 322 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences Jared Wunsch, Northwestern University, Title to be Michael J. Field, University of Houston, Title to be an- announced. nounced. Sharon R. Lubkin, North Carolina State University, Title Special Sessions to be announced. Commutative Algebra and Its Interactions with Alge- Stefan Richter, University of Tennessee, Knoxville, Title braic Geometry (Code: SS 2A), Claudia Polini, University to be announced. of Notre Dame, Alberto Corso, University of Kentucky, and Bernd Ulrich, Purdue University. Special Sessions Groups, Representations, and Characters (Code: SS 4A), Operator Theory (Code: SS 2A), Stefan Richter, Uni- James P. Cossey, University of Akron, and Mark Lewis, versity of Tennessee, and William T. Ross, University of Kent State University. Richmond. Hilbert Functions in Commutative Algebra and Alge- braic Combinatorics (Code: SS 3A), Fabrizio Zanello, Michi- gan Technological University, Juan Migliore, University Pucon, Chile of Notre Dame, and Uwe Nagel, University of Kentucky. Interdisciplinary Session on Deterministic and Stochastic December 15–18, 2010 Partial Differential Equations (Code: SS 5A), Nathan Glatt- Wednesday – Saturday Holtz, Indiana University, and Vlad Vicol, University of Southern California. Meeting #1066 Quasigroups, Loops, and Nonassociative Division Al- First Joint International Meeting between the AMS and the gebras (Code: SS 6A), Clifton E. Ealy, Western Michigan Sociedad de Matematica de Chile. University, Stephen Gagola, University of Arizona, Julia Associate secretary: Steven H. Weintraub Knight, University of Notre Dame, J. D. Phillips, Northern Announcement issue of Notices: June 2010 Michigan University, and Petr Vojtechovsky, University Program first available on AMS website: To be announced of Denver. Program issue of electronic Notices: To be announced Singularities in Algebraic Geometry (Code: SS 1A), Nero Issue of Abstracts: To be announced Budur, University of Notre Dame, and Lawrence Ein, Uni- versity of Illinois at Chicago. Deadlines For organizers: April 15, 2010 For consideration of contributed papers in Special Ses- Richmond, Virginia sions: To be announced For abstracts: To be announced University of Richmond The scientific information listed below may be dated. November 6–7, 2010 For the latest information, see www.ams.org/amsmtgs/ Saturday – Sunday internmtgs.html. Meeting #1065 AMS Invited Addresses Southeastern Section Ricardo Baeza, Universidad de Talca, Chile, Title to be Associate secretary: Matthew Miller announced. Announcement issue of Notices: September Igor Dolgachev, University of Michigan, Title to be an- Program first available on AMS website: September 23, nounced. 2010 Andres Navas, Universidad de Santiago de Chile, Title Program issue of electronic Notices: November to be announced. Issue of Abstracts: Volume 31, Issue 4 Rodolfo Rodriguez, Universidad de Concepcion, Title to be announced. Deadlines Gunther Uhlmann, University of Washington, Title to For organizers: March 8, 2010 be announced. For consideration of contributed papers in Special Ses- S. R. Srinivasa Varadhan, New York University, Title sions: July 27, 2010 to be announced. For abstracts: September 14, 2010 AMS Special Sessions The scientific information listed below may be dated. Arithmetic of Quadratic Forms and Integral Lattices. For the latest information, see www.ams.org/amsmtgs/ (Code: SS 6A), Maria Ines Icaza, Universidad de Talca, sectional.html. Chile, Wai Kiu Chan, Wesleyan University, and Ricardo Baeza, Universidad de Talca, Chile. Invited Addresses Automorphic Forms and Dirichlet Series (Code: SS 5A), Matthew H. Baker, Georgia Institute of Technology, Yves Martin, Universidad de Chile, Chile, and Solomon Title to be announced. Friedberg, Boston College. FEBRUARY 2010 NOTICES OF THE AMS 323 Meetings & Conferences Complex Algebraic Geometry (Code: SS 1A), Giancarlo For consideration of contributed papers in Special Ses- Urzua and Eduardo Cattani, University of Massachusetts. sions: To be announced Foliations and Dynamics (Code: SS 4A), Andrés Navas, For abstracts: To be announced Universidad de Santiago de Chile, and Rostislav Grigor- chuk, University of Texas. Group Actions: Probability and Dynamics (Code: SS 3A), Iowa City, Iowa Andrés Navas, Universidad de Santiago de Chile, and Rostislav Grigorchuk, University of Texas. University of Iowa Non-Associative Algebras (Code: SS 2A), Alicia Labra, Universidad de Chile, and Kevin McCrimmon, University March 18–20, 2011 of Virginia. Friday – Sunday Central Section Associate secretary: Georgia Benkart Announcement issue of Notices: To be announced New Orleans, Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Louisiana Issue of Abstracts: To be announced New Orleans Marriott and Sheraton New Deadlines Orleans Hotel For organizers: July 16, 2010 For consideration of contributed papers in Special Ses- January 5–8, 2011 sions: To be announced Wednesday – Saturday For abstracts: To be announced Joint Mathematics Meetings, including the 117th Annual Meeting of the AMS, 94th Annual Meeting of the Math- ematical Association of America, annual meetings of the Worcester, Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), Massachusetts with sessions contributed by the Society for Industrial and College of the Holy Cross Applied Mathematics (SIAM). Associate secretary: Steven H. Weintraub April 9–10, 2011 Announcement issue of Notices: October 2010 Saturday – Sunday Program first available on AMS website: November 1, 2010 Eastern Section Program issue of electronic Notices: January 2011 Associate secretary: Steven H. Weintraub Issue of Abstracts: Volume 32, Issue 1 Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: April 1, 2010 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses- sions: To be announced Deadlines For abstracts: To be announced For organizers: September 9, 2010 For consideration of contributed papers in Special Ses- sions: To be announced Statesboro, Georgia For abstracts: To be announced Georgia Southern University March 12–13, 2011 Las Vegas, Nevada Saturday – Sunday University of Nevada Southeastern Section Associate secretary: Matthew Miller April 30 – May 1, 2011 Announcement issue of Notices: To be announced Saturday – Sunday Program first available on AMS website: To be announced Western Section Program issue of electronic Notices: To be announced Associate secretary: Michel L. Lapidus Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: August 12, 2010 Issue of Abstracts: To be announced 324 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings & Conferences Deadlines For organizers: To be announced Boston, For consideration of contributed papers in Special Ses- sions: To be announced Massachusetts For abstracts: To be announced John B. Hynes Veterans Memorial Conven- The scientific information listed below may be dated. tion Center, Boston Marriott Hotel, and For the latest information, see www.ams.org/amsmtgs/ Boston Sheraton Hotel sectional.html. January 4–7, 2012 Special Sessions Wednesday – Saturday Advances in Modeling, Numerical Analysis and Compu- Joint Mathematics Meetings, including the 118th Annual tations of Fluid Flow Problems (Code: SS 2A), Monika Neda, Meeting of the AMS, 95th Annual Meeting of the Math- University of Nevada Las Vegas. ematical Association of America, annual meetings of the Geometric PDEs (Code: SS 1A), Matthew Gursky, Notre Association for Women in Mathematics (AWM) and the Dame University, and Emmanuel Hebey, Universite de National Association of Mathematicians (NAM), and the Cergy-Pontoise. winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Associate secretary: Michel L. Lapidus Lincoln, Nebraska Announcement issue of Notices: October 2011 University of Nebraska-Lincoln Program first available on AMS website: November 1, 2011 Program issue of electronic Notices: January 2012 October 14–16, 2011 Issue of Abstracts: Volume 33, Issue 1 Friday – Sunday Deadlines Central Section Associate secretary: Georgia Benkart For organizers: April 1, 2011 For consideration of contributed papers in Special Ses- Announcement issue of Notices: August 2011 sions: To be announced Program first available on AMS website: To be announced For abstracts: To be announced Program issue of electronic Notices: October 2011 Issue of Abstracts: To be announced Deadlines San Diego, California For organizers: To be announced San Diego Convention Center and San For consideration of contributed papers in Special Ses- Diego Marriott Hotel and Marina sions: To be announced For abstracts: To be announced January 9–12, 2013 Wednesday – Saturday Joint Mathematics Meetings, including the 119th Annual Salt Lake City, Utah Meeting of the AMS, 96th Annual Meeting of the Math- University of Utah ematical Association of America, annual meetings of the Association for Women in Mathematics (AWM) and the October 22–23, 2011 National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), Saturday – Sunday with sessions contributed by the Society for Industrial and Western Section Applied Mathematics (SIAM). Associate secretary: Michel L. Lapidus Associate secretary: Georgia Benkart Announcement issue of Notices: To be announced Announcement issue of Notices: October 2012 Program first available on AMS website: To be announced Program first available on AMS website: November 1, 2012 Program issue of electronic Notices: To be announced Program issue of electronic Notices: January 2012 Issue of Abstracts: To be announced Issue of Abstracts: Volume 34, Issue 1 Deadlines Deadlines For organizers: To be announced For organizers: April 1, 2012 For consideration of contributed papers in Special Ses- For consideration of contributed papers in Special Ses- sions: To be announced sions: To be announced For abstracts: To be announced For abstracts: To be announced FEBRUARY 2010 NOTICES OF THE AMS 325 Meetings & Conferences Baltimore, Maryland Seattle, Washington Baltimore Convention Center, Baltimore Washington State Convention & Trade Hilton, and Marriott Inner Harbor Center and the Sheraton Seattle Hotel January 15–18, 2014 January 6–9, 2016 Wednesday – Saturday Wednesday – Saturday Joint Mathematics Meetings, including the 120th Annual Joint Mathematics Meetings, including the 122nd Annual Meeting of the AMS, 97th Annual Meeting of the Math- Meeting of the AMS, 99th Annual Meeting of the Math- ematical Association of America, annual meetings of the ematical Association of America, annual meetings of the Association for Women in Mathematics (AWM) and the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic, with winter meeting of the Association of Symbolic Logic, with sessions contributed by the Society for Industrial and Ap- sessions contributed by the Society for Industrial and Ap- plied Mathematics (SIAM). plied Mathematics (SIAM). Associate secretary: Matthew Miller Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2013 Announcement issue of Notices: October 2015 Program first available on AMS website: November 1, 2013 Program first available on AMS website: To be announced Program issue of electronic Notices: January 2013 Program issue of electronic Notices: January 2016 Issue of Abstracts: Volume 35, Issue 1 Issue of Abstracts: Volume 37, Issue 1 Deadlines Deadlines For organizers: April 1, 2013 For organizers: April 1, 2015 For consideration of contributed papers in Special Ses- For consideration of contributed papers in Special Ses- sions: To be announced sions: To be announced For abstracts: To be announced For abstracts: To be announced San Antonio, Texas Atlanta, Georgia Henry B. Gonzalez Convention Center and Hyatt Regency Atlanta and Marriott Grand Hyatt San Antonio Atlanta Marquis January 10–13, 2015 January 4–7, 2017 Saturday – Tuesday Wednesday – Saturday Joint Mathematics Meetings, including the 121st Annual Joint Mathematics Meetings, including the 123rd Annual Meeting of the AMS, 98th Annual Meeting of the Math- Meeting of the AMS, 100th Annual Meeting of the Math- ematical Association of America, annual meetings of the ematical Association of America, annual meetings of the Association for Women in Mathematics (AWM) and the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic, with winter meeting of the Association of Symbolic Logic, with sessions contributed by the Society for Industrial and Ap- sessions contributed by the Society for Industrial and Ap- plied Mathematics (SIAM). plied Mathematics (SIAM). Associate secretary: Steven H. Weintraub Associate secretary: Georgia Benkart Announcement issue of Notices: October 2014 Announcement issue of Notices: October 2016 Program first available on AMS website: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: January 2015 Program issue of electronic Notices: January 2017 Issue of Abstracts: Volume 36, Issue 1 Issue of Abstracts: Volume 38, Issue 1 Deadlines Deadlines For organizers: April 1, 2014 For organizers: April 1, 2016 For consideration of contributed papers in Special Ses- For consideration of contributed papers in Special Ses- sions: To be announced sions: To be announced For abstracts: To be announced For abstracts: To be announced 326 NOTICES OF THE AMS VOLUME 57, NUMBER 2 Meetings and Conferences of the AMS Associate Secretaries of the AMS Eastern Section: Steven H. Weintraub, Department of Math- Western Section: Michel L. Lapidus, Department of Math- ematics, Lehigh University, Bethlehem, PA 18105-3174; e-mail: ematics, University of California, Surge Bldg., Riverside, CA [email protected]; telephone: 610-758-3717. 92521-0135; e-mail: [email protected]; telephone: Southeastern Section: Matthew Miller, Department of Math- 951-827-5910. ematics, University of South Carolina, Columbia, SC 29208-0001, Central Section: Georgia Benkart (after January 31, 2010), e-mail: [email protected]; telephone: 803-777-3690. University of Wisconsin-Madison, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706-1388; e-mail: benkart@ math.wisc.edu; telephone: 608-263-4283. The Meetings and Conferences section of the Notices 2012 gives information on all AMS meetings and conferences January 4–7 Boston, Massachusetts p. 325 approved by press time for this issue. Please refer to the page Annual Meeting numbers cited in the table of contents on this page for more 2013 detailed information on each event. Invited Speakers and January 9–12 San Diego, California p. 325 Special Sessions are listed as soon as they are approved by the Annual Meeting cognizant program committee; the codes listed are needed for electronic abstract submission. For some meetings the list may 2014 be incomplete. Information in this issue may be dated. Up-to- January 15–18 Baltimore, Maryland p. 326 date meeting and conference information can be found at www. Annual Meeting ams.org/meetings/. 2015 January 10–13 San Antonio, Texas p. 326 Meetings: Annual Meeting 2010 2016 March 27–28 Lexington, Kentucky p. 314 January 6–9 Seattle, Washington p. 326 April 10–11 St. Paul, Minnesota p. 315 Annual Meeting April 17–18 Albuquerque, New Mexico p. 318 2017 May 22–23 Newark, New Jersey p. 320 January 4–7 Atlanta, Georgia p. 326 June 2–5 Berkeley, California p. 321 Annual Meeting October 2–3 Syracuse, New York p. 321 Important Information Regarding AMS Meetings October 9–10 Los Angeles, California p. 322 Potential organizers, speakers, and hosts should refer to October 29–31 Notre Dame, Indiana p. 322 page 92 in the January 2010 issue of the Notices for general November 6–7 Richmond, Virginia p. 323 information regarding participation in AMS meetings and December 15–18 Pucon, Chile p. 323 conferences. Abstracts 2011 Speakers should submit abstracts on the easy-to-use interac- January 5–8 New Orleans, Louisiana p. 324 tive Web form. No knowledge of is necessary to submit Annual Meeting an electronic form, although those who use may submit March 12–13 Statesboro, Georgia p. 324 abstracts with such coding, and all math displays and simi- March 18–20 Iowa City, Iowa p. 324 larily coded material (such as accent marks in text) must April 9–10 Worcester, Massachusetts p. 324 be typeset in . Visit http://www.ams.org/cgi-bin/ April 30–May 1 Las Vegas, Nevada p. 324 abstracts/abstract.pl. Questions about abstracts may be October 14–16 Lincoln, Nebraska p. 325 sent to [email protected]. Close attention should be paid to October 22–23 Salt Lake City, Utah p. 325 specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated. Conferences: (see http://www.ams.org/meetings/ for the most up-to-date information on these conferences.) Co-sponsored conferences: February 18–22, 2010: AAAS Meeting in San Diego, CA (please see www.aaas.org/meetings for more information). March 18-21, 2010: First International Conference on Mathematics and Statistics, AUS-ICMS ’10, American University of Sharjah, Sharjah, United Arab Emirates (please see http://www.aus.edu/conferences/icms10/ for more information). May 24-29, 2010: From Carthage to the World, the Isoperimetric Problem of Queen Dido and its Mathematics Ramifi ca- tions, Carthage, Tunisia (for more information please see http://math.arizona.edu/~dido/welcome.html). June 17-19, 2010: Coimbra Meeting on 0-1 Matrix Theory and Related Topics, University of Coimbra, Portugal (for more information please see http://www.mat.uc.pt/~cmf/01MatrixTheory). FEBRUARY 2010 NOTICES OF THE AMS 327 Outstanding Titles in Mathematics! Bayesian Nonparametrics Now in Paperback! Edited by Nils Lid Hjort, An Algebraic Introduction Chris Holmes, Peter Müller, to K-Theory and Stephen G. Walker Bruce A. Magurn Cambridge Series in Statistical and Probabilistic Mathematics “This volume is a very useful graduate $59.00: Hardback: 978-0-521-51346-3: 270 pp. algebra text with an orientation towards algebraic K-theory....This volume will form an excellent basis for several types of one-and two-semester graduate algebra Second Edition! courses.” Smooth Compactifi cations of –Mathematical Reviews Locally Symmetric Varieties Encyclopedia of Mathematics and its Applications Avner Ash, David Mumford, $80.00: Paperback: 978-0-521-10658-0: 692 pp. Michael Rapoport, and Yung-sheng Tai Cambridge Mathematical Library Stochastic Control $50.00: Paperback: 978-0-521-73955-9: 250 pp. and Mathematical Modeling Applications in Economics Hiroaki Morimoto Logical Foundations Encyclopedia of Mathematics of Proof Complexity and its Applications Stephen Cook $110.00: Hardback: 978-0-521-19503-4: 344 pp. and Phuong Nguyen $80.00: Hardback: 978-0-521-51729-4: 492 pp. Locally Convex Spaces over Non-Archimedean Valued Fields Geometry of Riemann C. Perez-Garcia Surfaces and W. H. Schikhof Edited by Frederick P. Gardiner, Cambridge Studies in Gabino González-Diez, Advanced Mathematics and Christos Kourouniotis $110.00: Hardback: 978-0-521-19243-9: 505 pp. London Mathematical Society Lecture Note Series $78.00: Paperback: 978-0-521-73307-6: 415 pp. Epidemics and Rumours in Complex Networks Zariski Geometries Moez Draief Geometry from the and Laurent Massoulié Logician’s Point of View London Mathematical Society Boris Zilber Lecture Note Series London Mathematical Society $45.00: Paperback: 978-0-521-73443-1: 130 pp. Lecture Note Series $60.00: Paperback: 978-0-521-73560-5: 230 pp. Prices subject to change. XXXDBNCSJEHFPSHVTNBUIFNBUJDTt AMERICAN MATHEMATICAL SOCIETY A Selection of the Best-selling AMS Titles in 2009 What’s Markov Chains and Mixing The Geometry of Heisenberg Happening in the Times Groups Mathematical David A. Levin, University of Oregon, With Applications in Signal Theory, Sciences Eugene, OR, Yuval Peres, Microsoft Optics, Quantization, and Field Dana Mackenzie Research, Redmond, WA, and University of Quantization California, Berkeley, CA, and Elizabeth L. Ernst Binz and Sonja Pods, University of Wilmer, Oberlin College, OH What’s Happening in the Mannheim, Germany Mathematical Sciences, 2009; 371 pages; Hardcover; ISBN: 978-0-8218- Volume 7; 2009; 127 pages; Softcover; ISBN: 978- Mathematical Surveys and Monographs, 4739-8; List US$65; AMS members US$52; Order 0-8218-4478-6; List US$19.95; AMS members Volume 151; 2008; 299 pages; Hardcover; ISBN: 978- code MBK/58 US$15.95; Order code HAPPENING/7 0-8218-4495-3; List US$85; AMS members US$68; Order code SURV/151 Introduction to Analysis Quantum Mechanics for Fifth Edition Mathematicians Pioneering Women in American Edward D. Gaughan, New Mexico State Leon A. Takhtajan, Stony Brook University, Mathematics University, Las Cruces, NM NY The Pre-1940 PhD’s Pure and Applied Undergraduate Texts, Graduate Studies in Mathematics, Volume Judy Green, Marymount University, Volume 1; 1998; 240 pages; Hardcover; ISBN: 978- 95; 2008; 387 pages; Hardcover; ISBN: 978-0- Arlington, VA, and Jeanne LaDuke, DePaul 0-8218-4787-9; List US$62; AMS members US$50; 8218-4630-8; List US$69; AMS members US$55; University, Chicago, IL Order code GSM/95 Order code AMSTEXT/1 Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS Lectures on Quantum Tools of the Trade is registered with the Charity Commissioners. Mechanics for Mathematics Introduction to Advanced Mathematics History of Mathematics, Volume 34; 2009; Students Paul J. Sally, Jr., University of Chicago, IL 345 pages; Hardcover; ISBN: 978-0-8218-4376-5; List US$79; AMS members US$63; Order code L. D. 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