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All requests will be forwarded to the appropriate orga- Joint Summer nizing committee for consideration. In late April all ap- plicants will receive formal invitations (including specific Research Conferences offers of support if applicable), a brochure of conference information, program information known to date, along with information on travel and dormitories and other local in the Mathematical housing. All participants will be required to pay a nomi- nal conference fee. Sciences Questions concerning the scientific program should be University of Colorado addressed to the organizers. Questions of a nonscientific Boulder, Colorado nature should be directed to the Summer Research Con- June 13–July 1, 1999 ferences coordinator at the address provided above. Please watch http://www.ams.org/meetings/ for future de- The 1999 Joint Summer Research Conferences will be held velopments about these conferences. at the University of Colorado, Boulder, Colorado, from Lectures begin on Sunday morning and run through June 13–July 1, 1999. The topics and organizers for the Thursday. Check-in for housing begins on Saturday. No seven conferences were selected by a committee repre- lectures are held on Saturday. senting the AMS, the Institute of Mathematical Sciences (IMS), and the Society for Industrial and Applied Math- ematics (SIAM), whose members at the time were Alejan- From Manifolds to Singular Varieties dro Adem, David Brydges, Percy Deift, James W. Demmel, Sunday, June 13–Thursday, June 17, 1999 Dipak Dey, Tom Diciccio, Steven Hurder, Alan F. Karr, Bar- bara Keyfitz, W. Brent Lindquist, Andre Manitius, and Bart Sylvain E. Cappell, Courant Institute Ng. Ronnie Lee, Yale University It is anticipated that the conferences will be partially Wolfgang Lück, Westfälische Wilhelms-Universität funded by a grant from the National Science Foundation Münster and perhaps others. Special encouragement is extended to junior scientists to apply. A special pool of funds expected Recently, researchers in topology, geometry and global from grant agencies has been earmarked for this group. analysis have been encountering some related issues in at- Other participants who wish to apply for support funds tempts to extend classical methods and results from man- should so indicate; however, available funds are limited, ifolds to more general settings of singular varieties. These and individuals who can obtain support from other sources are needed for applications because many of the natural are encouraged to do so. spaces that are the focus of current investigations are usu- All persons who are interested in participating in one ally singular. Examples include: representation spaces, of the conferences (women and minorities are especially such as those of fundamental groups of Riemann surfaces encouraged) should request an invitation by sending the arising in the study of low dimensional manifolds; mod- following information to: Summer Research Conferences uli space constructions in algebraic geometry; the singu- Coordinator, American Mathematical Society, P.O. Box lar spaces which arise in trying to study and classify gen- 6887, Providence, RI 02940, or by e-mail to [email protected] eral finite or compact group actions on manifolds; and no later than March 3, 1999. spaces produced in a variety of contexts by natural com- Please type or print the following: pactification procedures. 1. Title and dates of conference. The object of this conference will be to describe some 2. Full name. recent advances in this general area and to discuss and com- 3. Mailing address. pare questions, methods, and applications from a variety 4. Phone numbers (including area code) for office, home, of perspectives. Among other subjects, the conference will and fax. include talks on the following topics: 5. E-mail address. 1. Topological invariants and classifications of singular 6. Your anticipated arrival/departure dates. varieties. A toplogical motivation for studying singular va- 7. Scientific background relevant to the Institute topics; rieties is the study of group actions, because their please indicate if you are a student or if you received your spaces are stratified spaces. A goal is to make some pow- Ph.D. on or after 7/1/93. erful new classification methods more accessible and ap- 8. The amount of financial assistance requested (or in- plicable by considering spaces arising from natural con- dicate if no support is required). texts. A natural question, which can be approached from

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several viewpoints, is: what can be said about topological Atiyah conjecture, the Singer Conjecture and the zero-in- classification of algebraic varieties? the-spectrum conjecture which are related to algebraic K- Important recent developments in the theory of singu- theory, global analysis, and topological rigidity conjec- lar varieties are related to the study of their characteris- tures. tic classes. There are, in fact, several closely related theo- ries of characteristic classes developed by a number of workers. Among questions to be addressed are: What are Computability Theory and Applications the relations between different theories of characteristic Sunday, June 13–Thursday, June 17, 1999 classes and what do they reveal about geometric and an- alytic structures of the varieties? When can such classes, Peter Cholak, University of Notre Dame which for singular varieties generally take values in ho- Steffen Lempp, University of Wisconsin (co-chair) mology, be lifted back at least partially towards cohomol- Manuel Lerman, University of Connecticut (co-chair) ogy or to other theories; and what additional structures are Richard Shore, Cornell University (co-chair) necessary for such refinements? Intersection homology theory has been a powerful tool Computability theory (or recursion theory) is an area of for the investigation of intersection theories and numbers mathematical logic dealing with the theoretical bounds on in singular varieties. But its functoriality, analytical inter- and structure of computability and with the interplay be- pretation, e.g., its relation to L2 cohomolgy, as well as other tween computability and definability in mathematical lan- issues are still not fully understood. Moreover, it doesn't guages and structures. The field started in the 1930s with cover all the now needed settings, e.g., in symplectic geom- ground-breaking work of Gödel and Turing and has de- etry. This meeting will be a forum for such new issues in veloped into a rich theory with applications and connec- intersection theory. tions to areas ranging from computer science to descrip- 2. 3-manifold invariants and moduli spaces. Some sub- tive set theory as well as more traditional branches of tle invariants of 3-manifolds are related to representation , including algebra, analysis, and combina- spaces. But extensions of Casson's SU(2) ideas to more gen- torics. The meeting will focus on classical computability eral Lie groups encounter difficulties due to the more sin- theory, an area in which many recent advances have been gular nature of the representation varieties and require in- made, and those applications which currently seem most vestigations of Lagrangian subvarieties in singular directly connected to and most likely to benefit from these symplectic varieties. These invariants shoud be compared advances. In particular, applications in algebra, model the- with combinatorial invariants, e.g., those related to Vas- ory, and proof theory will be highlighted. Lectures will siliev's perspective on knot theory and to “finite type in- stress open problems, their relationship to some of the re- variants” of 3-manifolds. cent advances, and further obstacles which need to be Representation varieties have been investigated by al- overcome to solve the problems. Problems, primarily from gebraic geometers as moduli spaces of holomorphic G-bun- the following areas, will be discussed. dles over a Riemann surface. Related moduli objects (e.g., 1. Classical computability theory. There have been a of Higgs bundles, of the moduli spaces of stable k-pairs, number of major advances in the understanding of sub- etc.) arose through interactions with physics. Such mod- structures of the Turing degrees in the past few years, in- uli spaces exhibit similar symplectic and Kahler structures cluding a phenomenal number of solutions to diverse prob- as well as gauge theory interpretations. It will be desirable to compare different treatments of singularities. lems that had been open for decades and had always been 3. L2-Betti numbers and L2-torsions. L2-invariants such considered very hard. In addition, substantial progress as L2-Betti numbers, Novikov-Shubin invariants, L2-tor- has been made towards the solution of other problems. Re- sions can be defined analytically in terms of the heat ker- sults have been obtained about automorphisms of these nel of the universal covering as well as topologically. This structures, characterizing definable sets and relations, and gives fruitful links between analysis and topology with decidability and undecidability of fragments of elementary applications in differential geometry, topology, group the- theories of the structures. These results will be discussed ory and algebraic K-theory. with an eye towards the limitations of the methods and the Connections to the first topic come from the relation obstacles which need to be overcome in order to solve other of intersection homology and L2-cohomology. The Cheeger- problems of a similar nature. Goresky-MacPherson Conjecture and the Zucker Conjecture 2. Computable mathematics. The area of applied com- link the L2-cohomology of the regular part with the inter- putability theory on which we propose to concentrate is section homology of an algebraic variety. For applications computable mathematics. Generally speaking, one wishes in group theory and algebraic K-theory it was necessary to investigate the effective content of mathematical to define L2-Betti numbers for “very singular” spaces. This constructions and theorems, that is, to determine which suggests extending results from actions of finite groups procedures or relations are computable and the relative to (proper) actions of discrete groups. of those that are not. The problems which we L2-Betti numbers, Novikov-Shubin invariants and L2-tor- will address deal with determining properties of com- sion have been studied for 3-manifolds and are linked to putable structures which can be decided effectively from other invariants, e.g., volume of hyperbolic manifolds and their presentations and, if not, on the possible limits on Gromov's simplicial volume. Open conjectures include the their complexity.

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Reverse mathematics is a proof-theoretic and founda- New Approaches to Homotopy Theory tional investigation into the axiom systems needed to A closed model category is a category with structure al- prove standard theorems of classical mathematics, but lowing one to do homotopy theory satisfying expected many of its arguments and results can also be viewed as properties. Voevodsky, Morel, and others have led major belonging to computable mathematics. There is an almost developments in the construction of such structures in al- perfect translation between the proof-theoretic systems gebro-geometric settings. One expects that this ongoing used and the levels of complexity in computability. Each work will feed back into classic topology in much the way approach contributes its own techniques, which often pro- that the development of étale homotopy did two decades duce results with overlapping but supplementary content. before. An important foundational issue is the existence of clas- Related to this is recent work on the foundations of sta- sical theorems not equivalent to any of the standard sys- ble homotopy. Jeff Smith and his collaborators have de- tems. It seems likely that further computability theory veloped the theory of symmetric spectra, and the Chicago school, led by J. Peter May, has used modern operadic analyses using more delicate techniques can shed light on methods to develop the theory of S-modules. Both of these this area. underlie the stable category, and both are influencing, and are influenced by, nearby areas of mathematics. Homotopy Methods in Algebraic Topology Goodwillie’s theory of polynomial resolutions of ho- motopy functors, originally developed to answer ques- Sunday, June 20–Thursday, June 24, 1999 tions in algebraic K-theory, is now being employed in clas- sical homotopy with ever more success, including Robert Bruner, Wayne State University (co-chair) applications to vn-homotopy and Hopf invariants and to Anthony Elmendorf, Purdue University, Calumet general homological algebra. John Greenlees, Sheffield University (England) Cobordism and Homotopy Theory Nicholas J. Kuhn, University of Virginia (co-chair) By connecting the geometry of manifolds, homotopy the- James McClure, Purdue University, West Lafayette ory, and the study of formal groups, cobordism theory plays a pivotal and beautiful role in topology. The MIT school has Algebraic topology has continually developed sophisti- been using algebro-geometric methods to explain and gen- cated new homological and homotopical methods, which eralize new invariants of manifolds (e.g., Witten’s elliptic have then been exported to algebraic settings such as rep- genus) while simultaneously illuminating algebraic results resentation theory, group theory, ring theory, and algebraic in the theory of modular forms. The demands of this work geometry. Topology currently seems to be going through also provided significant motivation for the new results on a period in which much of the most striking work involves the foundations of the stable category described above. such interfacing with algebra. The primary purpose of the There are ongoing fundamental questions about good geo- conference will be to present a broad range of current metric models for cobordism theories, such as elliptic co- work in this direction, with talks in each of five general homology, which have height greater than one. There has areas. been slow but steady progress towards answering these, Localization Methods and Group Cohomology involving work in disparate areas: equivariant formal group There has been a renaissance of interest in the axiomatics laws, the effect of Tate cohomology on periodic theories, of triangulated categories and their localizations. Results and connections with differential geometry. familiar in algebraic topology such as Brown representability and Bousfield localization have been refined and extended Historical Talks and then fruitfully transported into new algebraic settings There will be a few historical talks by mathematicians with such as algebraic geometry and group cohomology. Cor- broad interests who have been major contributors to al- responding computational results include the recent com- gebraic topology during the last two decades. putation of the cohomology of the Steenrod algebra up to Preliminary List of Speakers F-isomorphism and new work on the cohomology of var- Jeanne Duflot, Colorado State University; William Dwyer, ious arithmetic groups. University of Notre Dame; Michael Hopkins, MIT; Igor Kriz, Homology Operations, Combinatorics, and Ring Theory University of Michigan, Ann Arbor; Ib Madsen, Aarhus Uni- An ongoing theme in algebraic topology has been the de- versity (Denmark); Randy McCarthy, University of Illinois, velopment of the algebraic machinery necessary to effi- Urbana; James McClure, Purdue University, West Lafayette; ciently encode information about various sorts of homol- Fabien Morel, École Polytechniques (France); John Palmieri, ogy operations. Recently this machinery has been used in University of Notre Dame; Nigel Ray, University of Man- a number of distinctly purely algebraic settings. Examples chester (England); Jeremy Rickard, University of Bristol include the use of Steenrod algebra technology to prove (England); Hal Sadofsky, University of Oregon; Stephan the Stong-Landweber Conjecture about the depth of rings Stolz, University of Notre Dame; and Saïd Zarati, Univer- of invariants, the Manchester school’s work relating alge- sity of Tunis (Tunisia). bras of operations arising in topology to fundamental com- binatorial structures, and new studies on deformations of Hopf algebras arising in topology.

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propriate materials exhibiting localization of light and Wave Phenomena in Complex Media other classical waves. Sunday, June 20–Thursday, June 24, 1999 Another focus of the conference will be Schrödinger op- erators in complex media. Localization of Schrödinger op- Michael Aizenman, Princeton University erators in disordered media has been an area of intensive Alexander Figotin, University of North Carolina, mathematical research. Recently, progress was made in ap- Charlotte plying the mathematical ideas developed in that context Svetlana Jitomirskaya, University of California, Irvine to classical waves. Maxwell’s and acoustic equations can Abel Klein, University of California, Irvine (chair) be rewritten as first-order conservative systems, resembling Stephanos Venakides, Duke University Schrödinger’s equation. Spectral theory becomes relevant, and the methods developed for Schrödinger operators can The conference will bring together researchers with inter- be transported to the study of Maxwell and acoustic op- ests and experience in classical and quantum mechanical erators. These methods are also relevant to elastic waves. waves and in related aspects of complex media and make But the analogy is not complete, and there are important contact with recent developments in the theory of random technical differences. In particular, the analogy is with lo- matrices. calization of Schrödinger operators in spectral gaps, not The topics discussed will include: at the bottom of the spectrum. This analogy will be one of • Localization of classical and quantum mechanical the main themes of the conference. waves in disordered systems In addition to localization, the phenomenon of con- • Extended states in the presence of disorder duction through extended states continues to present a • Wave localization in nonlinear media major mathematical challenge. An interesting possibility Wave phenomena in composite media • is that relevant insight can be obtained by analyzing the Localization and gap statistics • related issues in the context of random matrices. Differ- Insights from random matrices • ent methods are available in that field; however, it seems Wave phenomena in complex media is a subject of great possible that both areas could be enriched through cross- interest. Periodic, perturbed periodic, or random media can comparisons. be used to filter or amplify waves in a prescribed range of The preliminary list of speakers includes Michael Aizen- frequencies, to discriminate waves propagating in chosen man, Princeton University; Carlo Beenakker, Leiden Uni- directions, and more. Phenomena as the existence of spec- versity; Percy Deift, Courant Institute; Alexander Figotin, tral gaps for periodic composite materials or Anderson lo- University of North Carolina, Charlotte; Rainer Hempel, calization for disordered materials are expected to have Technische Universität Braunschweig; Peter Hislop, Uni- important new applications, especially in the case of clas- versity of Kentucky;, Svetlana Jitomirskaya, University of sical waves, which have not been studied as intensively in California, Irvine; Yulia Karpeshina, University of Alabama, these types of medium. The utilization of these properties Birmingham; Abel Klein, University of California, Irvine; in composite dielectric materials such as photonic crystals Werner Kohler, Virginia Polytechnic Institute; Peter Kuch- can lead to new optical devices: optical transistors, high- ment, Wichita State University; Ken McLaughlin, University efficiency lasers, laser diodes, mirrors, antennas, switches, memories, and more. The quantitative mathematical the- of Arizona; Graeme Milton, University of Utah; George Pa- ory of photonic band structures and disordered dielectrics panicolaou, Stanford University; Leonid Pastur, University ought to be advanced in order to fully utilize the poten- of Paris VII; Israel Sigal, University of Toronto; Alexander tial suggested by the rather qualitative arguments devel- Soshnikov, CALTECH; Stephanos Venakides, Duke Univer- oped up to now. The mathematical theory will play an im- sity; Eugene Wayne, Boston University; and Michael Wein- portant role in the existing effort to tailor the optical stein, University of Michigan. properties of the materials to our technological needs. One focus of the conference will be classical waves in Groupoids in Physics, Analysis and complex media. Classical waves can exhibit phenomena nor- mally associated with quantum mechanical electron waves. Geometry The quantum mechanical wave nature of electrons has Sunday June 20–Thursday, June 24, 1999 been studied based on an analogy with the scattering and interference of classical waves. But only recently has the Jerome Kaminker, Indiana University-Purdue analogy been reversed and such phenomena as the local- University at Indianapolis (co-chair) ization of classical waves started to be investigated. There Arlan B. Ramsay, University of Colorado (chair) is now intensive research on this subject, driven in part by Jean Renault, Université d’Orleans (co-chair) the interest in optical materials that would lead to “lightron- Alan D. Weinstein, University of California, Berkeley ics” (optical transistors, etc.), a development with tremen- (co-chair) dous technological applications. But while electronic bound states, the simplest example of a localized electron wave, Groupoids have recently been used in essential ways in sev- are ubiquitous in nature, the analogous phenomena for light eral areas, providing a unifying theme for seemingly diverse are not commonly observed, even if theoretically possible. topics. It is the goal of this Joint Summer Research Con- Mathematics plays an important role in the design of ap- ference to bring together a broad group of researchers to

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discuss the ways in which the use of groupoids in each of The proposed conference seeks to explore the infusion of their areas leads to interesting results. The title tells the differential geometric methods into the analysis of control focus areas. theory problems for partial differential equations (P.D.E.s). The uses of groupoids in physics come from two main Very recent research supports the expectation that sources. The first is Alain Connes' theory of noncommu- Bochner techniques in differential geometry, when brought tative geometry, in which groupoids are a main source of to bear on the classes of P.D.E.s modelling and control problems discussed below, will yield significant examples of noncommutative spaces. This theory is being mathematical advances. These include: studied very actively by physicists, and by mathemati- cians. Bellisard's work studying the quantum Hall effect via (a) Intrinsic, coordinate-free models of (nonlinear) shells noncommutative geometry has led to the study of con- equations, more suitable for mathematical investiga- nections between solid state physics and noncommutative tion than present, exceedingly complicated, coordinate- geometry models associated with tilings. based models, mostly derived in the mechanical litera- The second major source of the use of groupoids in ture. (b)A priori direct (trace regularity) and reverse (continu- physics is the general theory of quantization in math- ous observability) inequalities for mixed problems for ematical physics. A theory of quantization has been in- second order hyperbolic P.D.E.s, Maxwell equations, troduced by V. Maslov and A. Karasev, and a version due plate-like equations, Schrodinger P.D.E.s (Petrowski- to Alan Weinstein has been actively developed by him and type) etc., defined on a multidimensional Euclidean do- his collaborators. One step in this program is to associate main, with emphasis on the variable coefficient case. In a symplectic groupoid to a given Poisson manifold. the case of dissipative systems, reverse inequalities, Differentiable groupoids and their associated Lie alge- which are generally more challenging to achieve, yield broids have proved to be a valuable ingredient in the work energy decay (stabilization) results. of Richard Melrose and Victor Nistor on partial differen- (c) Establishment of direct and reverse a priori inequalities tial equations on manifolds with corners. Further, the ap- for highly coupled systems of P.D.E.s arising in modern proach to quantization based on deformations of sym- technological applications, such as: shell models and plectic manifolds, introduced by Fedosov and developed thermoelastic models defined on 2-dimensional sur- by Ryszard Nest and Boris Tzygan, makes strong use of Lie face-like domains; structurally acoustic models defined on acoustic 3-dimensional chambers with curved walls, algebroids. possibly subject to thermoelastic effects; etc. An early use of groupoids was in the approach to group These three groups of results are of fundamental impor- representations due to George Mackey. This provided one tance, because they constitute necessary prerequisites for link to and von Neumann algebras. A dif- well-posedness and solvability of control and stabiliza- ferent viewpoint appears in the fundamental paper of tion problems for the P.D.E.s systems described above. Alain Connes on noncommutative integration, which in- The class of problems on shells listed in (a) is entirely fluences much of the work on groupoids in analysis. open; and so are, as a consequence, many of those listed Another connection between groupoids and von Neu- in (c), which depend on the solution of shell problems in mann algebras appears in recent work by Masamichi Take- (a). As to the problems listed in (b), while a wealth of re- saki and his collaborators. Groupoids have played a major sults has already been obtained, however, they refer so far role in C∗-algebras since the thesis of Jean Renault. An im- either to the constant coefficient case or, in the available portant current direction is the connection between dy- cases of variable coefficients, the conditions are not read- ily verifiable and the proofs are highly technical. Very re- namics and C∗-algebras as seen in the work of Ian Putnam and Jean Renault and their collaborators. cent research indicates that, in the general case of (space) variable coefficients, differential (Riemann) geometric Finally, the work of Jean-Luc Brylinski and his collabo- methods have the potential to enhance and simplify the rators using groupoids to understand Deligne cohomology presently available theory, as well as to extend it by over- from a geometric point of view has substantial connections coming remaining difficulties. The time is ripe and pro- and roots in problems in . pitious, therefore, to further explore in a systematic way new advances along this line of research. Thus, the aim is to organize an exploratory, focused, research conference, Differential Geometric Methods in the which involves control theory and P.D.E. experts as well as Control of Partial Differential Equations differential geometers interested in P.D.E.s. As to (a), the role of Riemann geometry is expected to Sunday, June 27–Thursday, July 1, 1999 be paramount in capturing geometric features of general shells, both static and dynamic; to express, in intrinsic form, Robert D. Gulliver University of Minnesota, the correct boundary conditions; and to establish the re- Minneapolis (co-chair) quired estimates. Walter Littman, University of Minnesota, Minneapolis As to (b), very recent research has indicated that Riemann (co-chair) geometric methods can profitably be used to complement Roberto Triggiani, University of Virginia (co-chair) and extend known analysis-based methods of proving the

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a priori reverse inequalities in the general case of variable be “control of P.D.E.s opened to geometry, and geometry coefficients. Riemann geometric methods appear to bring infused into P.D.E.s control”. a few advantages: (i) they essentially reduce the analysis to the constant coefficient principal part case, where strate- gies are well understood; (ii) they ultimately provide eas- Structured Matrices in Operator Theory, ier-to-verify conditions, with a distinct geometric flavor in- Numerical Analysis, Control, Signal and volving notions such as convexity in the Riemann metric Image Processing and gaussian curvature; (iii) they require only a finite, nat- ural degree of smoothness, rather than high smoothness Sunday, June 27–Thursday, July 1, 1999 as in pseudo-differential analysis. As to (c), the overall system may consist either of two Richard Brualdi, University of Wisconsin, Madison P.D.E.s of the same type (hyperbolic/hyperbolic coupling), Gene Golub, Stanford University or else of different type (hyperbolic/parabolic coupling), Franklin Luk, Rensellaer Polytechnical Institute possibly defined on different contiguous domains, and Vadim Olshevsky, Georgia State University (chair) with strong, possibly boundary, coupling. For example the elastic wall of an acoustic chamber may be subject to high Introduction internal damping, whereby the original plate equation be- Many important problems in pure and applied mathematics comes parabolic-like. This is a vastly open research topic and engineering can be reduced to linear algebra problems. for basic P.D.E.s theory in general, and for control and op- Unfortunately, practical circumstances impose limitations timization theory in particular. Because of their original on the use of available standard linear algebra methods. description on curved domains (manifolds), these problems For example, in many applications the size of the associ- appear particularly well suited for differential geometric ated matrices is prohibitively large, so the available stan- methods to supplement analytic approaches, at the level dard methods often require an extremely large amount of of both modeling and analysis, including notions such as arithmetic operations. operators on manifolds, forms, gaussian curvature of the This is one reason why one seeks in various applications domain, etc. Here the difficulties are compounded over to identify special/characteristic structures that may be as- those described in point a) above for a single shell, since sumed in order to speed-up computations. Such additional the shell may be just one component of a composite, highly assumptions are often provided by particular physical coupled system. properties leading to various structured matrices, such as The proposed exploratory conference will show once Toeplitz, Hankel, Vandermonde, Cauchy, Pick matrices, more that mathematical research knows no boundaries be- Bezoutians, and others. The structure of these dense ma- tween specific disciplines—analysis versus differential trices is understood in the sense that their n2 entries are geometry; and that potentially productive interactions may defined by a smaller number O(n) of parameters. So ex- take place between P.D.E. control theory and differential ploiting such structures allows one to obtain nice solutions geometry. These will be well served by expanding the tra- for many applied problems as well as to design efficient ditionally analysis-based P.D.E.s approaches into fields fast algorithms to compute these solutions. such as Riemannian (and, in the time variable case, Lorent- Structured matrices are encountered in a surprising va- ian) geometry. riety of areas and algorithms, including Pade approxima- Of course, the introduction and use of differential geo- tions; continuous fractions; classical algorithms of Euclid, metric methods in more general P.D.E.s theory has long been Schur, Nevanlinna, Lanzcos, Levinson; and their general- established. However, the use and role of differential geo- izations and applications. metric methods toward the solution of any of the model- Two Examples ing, control, and optimization problems mentioned in 1. Operator theory. In the classical Nevanlinna-Pick in- points a), b), c) above is largely unexplored. It opens up a terpolation problem one seeks a rational interpolant highly promising area of research. whose norm is bounded by unity in the right half-plane. By contrast, the introduction of differential geometric For this problem the well-known Pick solvability con- methods in the study of control problems (such as exact dition (1916) and the Nevanlinna algorithm (1919) both controllability, feedback stabilization, optimization, fil- involve a certain structured matrix called the Pick ma- tering, etc.) for dynamical systems modeled by ordinary dif- trix. ferential equations (O.D.E.s) dates as far back as the early 2. Electrical engineering. In the now classical N. Wiener 1970s. monograph Extrapolation, Interpolation and Smoothing In keeping with the stated character and goals, the pro- of Stationary Time Series a linear prediction problem was posed conference is intended to be highly focused and ex- reduced to recursive, solving the so-called Yule-Walker ploratory. The conference will feature high caliber speak- equations whose coefficient matrix has the Toeplitz ers from both fields, geometry and P.D.E.s, and seeks structure. likewise to attract a mixed audience of geometers and P.D.E. control theorists. A particular effort will be made to Further Progress include a representative group of young mathematicians These problems were among those seeds that grew into from both fields. The conference's distinctive theme will deep studies of structured matrices in linear algebra, op-

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erator theory, numerical analysis, theoretical computer Preliminary List of Invited Speakers science, and electrical engineering. Dario Bini, University of Pisa; Patrick Dewilde, Delft Uni- Interpolation versity; Israel Gohber, Tel Aviv University; Georg Heinig, There is a vast operator theory literature on far-reaching Kuwait University; Rien Kaashoek, Vrije University, Ams- generalizations of passive interpolation of the Nevanlinna- terdam; Tom Kailath, Stanford University; Franklin Luk, Pick type; we mention only that deep results were obtained Rensselaer Polytechnical Institute; Vadim Olshevsky, Geor- in the frameworks of several “languages”, including the gia State University; Haesun Park, University of Minnesota; band extension method, the Buerling-Lax-theorem ap- Bob Plemmons, Wake Forest University; Philip Regalia, In- proach, the state-space approach, and lifting-of-commutants stitut National des Telecommunications; Lothar Reichel, method. Kent State University; and Leiba Rodman, College of William Electrical Engineering and Mary. For further information, call for papers, application In the framework of system and circuit theories, inter- deadline, and updates see the conference page polants arise as transfer functions, so passivity is naturally http://www.cs.gsu.edu/~matvro/JSRC99.html. imposed by the conservation of energy. Thus, it is not sur- prising that fruitful connections to many applied areas were discovered. Many applications such as model reduction, sen- sitivity minimization, and robust stabilization have been 1999 Summer addressed in this way. Matrix Analysis Research Institute It turns out that many nice results and especially many im- portant fast algorithms that were initially obtained for spe- Smooth Ergodic Theory and Applications cific patterns of structure can be naturally carried over to the more general important classes of matrices having University of Washington what is now called displacement structure. Seattle, Washington July 26–August 13, 1999 Numerical Analysis In floating point arithmetic, where the roundoff errors are The forty-fifth Summer Research Institute sponsored by present, the crucial factor that makes an algorithm prac- the American Mathematical Society will be held at the Uni- tical is its numerical accuracy. Unfortunately, many fast al- versity of Washington, Seattle, Washington. The topic was gorithms suffer from often catastrophic propogation of selected by the Committee on Summer Institutes and Spe- roundoff errors, so one can say that they are often efficient cial Symposia, whose members at the time included Michael ways to compute “garbage solutions”. Moreover, these two D. Fried (chair), Robert Osserman, Jeffrey B. Rauch, Leon targets have even been incorrectly regarded as being un- Takhtajan, and Ruth J. Williams. attainable simultaneously, thus leading to the folk con- Organizing Committee jecture “one has to sacrifice accuracy for speed”. It is remarkable that the results of recent years reveal Anatole Katok, Pennsylvania State University (co- that the above two targets not only do not conflict with each chair) other but in fact do just the opposite: proper and careful Rafael De La Llave, University of Texas (co-chair) use of structure allows one to design more accurate fast Yakov Pesin, Pennsylvania State University (co-chair) algorithms that can be even better than the standard nu- Howard Weiss, Pennsylvania State University (co- merically stable algorithms. chair)

Conference Scope and Topics Smooth ergodic theory deals with the study of invariant Though special sessions and minisymposia on structured measures under differentiable mappings or flows. The rel- matrices are usually included in the programs of the ILAS, evance of invariant measures is that they describe the fre- SIAM, SPIE, and MTNS conferences, their narrow frameworks quencies of visits for an orbit and hence they give a prob- usually allow us to focus on one specific application only. abilistic description of the motion of a deterministic system. The purpose of this conference is to foster integration be- The fact that the system is differentiable allows one to use tween different areas and to bring together leading re- techniques from analysis and geometry. Much of the par- searchers working on all aspects of structured matrices. ticular fascination of the field stems from the interaction There will be several invited tutorial lectures. Con- between measure theory and geometry and the possibil- tributed talks will focus on recent advances in the follow- ity of drawing geometric conclusions from measure theo- ing areas: fast algorithms for structured matrices, dis- retic facts and vice versa. placement structure, abstract interpolation, computer The study of transformations and their long-term be- arithmetic, numerical accuracy, applications of structured havior is ubiquitous in mathematics and the sciences. They matrices in system theory, circuits, signal processing, adap- arise not only in applications to the real world—the name tive filtering, control, image processing, and precondi- “ergodic” was introduced by Boltzmann, who intended to tioning. use his “Ergodic Hypothesis” as a justification of thermo-

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dynamics—but also to diverse mathematical disciplines, in- There will be three types of talks catering to the different cluding number theory, Lie groups, algorithms, Riemann- possible audiences: (i) in-depth minicourses (taught by ian geometry, etc. Hence smooth ergodic theory is the leading experts in the respective subjects) on both core ma- meeting ground of many different ideas in pure and ap- terial, recent exciting developments, open problems, and plied mathematics. applications; (ii) survey lectures on the current state of the The core of the theory includes the study of stochastic subject; and (iii) research talks on topics of current inter- properties of the invariant measures of various kinds of est. Minicourses will usually be about a week long. Pre- systems with hyperbolic behavior (i.e., exponential growth liminary plans for the program are to have two or three of instabilities). The principal role of smooth ergodic the- minicourse lectures in the morning and two or three sur- ory is twofold: vey and research talks in the afternoon. 1. It provides a paradigm for the rigorous study of com- We plan to run minicourses in the following subjects (in- plicated or chaotic behavior in deterministic systems. structors’ names in parentheses): Smooth ergodic theory is an essential tool in the study of 1. Ergodic Theory of Nonuniformly Hyperbolic Systems specific classes of dynamical systems, such as some low- (Katok, Pesin). dimensional systems with strange , Hamiltonian 2. Decay of Correlations and Zeta Functions (Baladi, systems, geodesic flows, and actions of higher-rank Abelian Dolgopyat, Pollicott). and semisimple Lie groups by diffeomorphisms. 3. Ergodic and Geometric Properties of Partially Hyper- 2. It serves as the basis for applications both inside and bolic Systems (Shub, Wilkinson). outside the theory of dynamical systems. There are many 4. Geometrical and Statistical Properties of Chaotic Sys- striking applications that use this machinery to prove deep tems (Chernov, Young). theorems in Riemannian geometry (solution of Klingenberg 5. KAM Theory and Applications (Eliasson, De La Llave). conjecture on the existence of infinitely many closed geo- 6. Applications of Ergodic Theory to Geometry of Man- ifolds of Nonpositive Curvature (Brin, Knieper, Ballmann). desics on every convex surface, classification of manifolds 7. Dimension Theory and Dynamics (Schmelling, Weiss). of nonpositive curvature), number theory (solution of Op- 8. Applications of Ergodic Theory to Number Theory penheim conjecture on the values of quadratic forms at (Eskin, Kleinbock). primitive integral points), Lie groups (rigidity of smooth Other minicourses may include Lyapunov Exponents actions of many large groups, homogeneity of closures and and Their Estimation and Henon Attractors and Related equidistribution of individual orbits of one-parameter Topics; others are still being developed. unipotent subgroups), statistical physics (substantial Among the confirmed survey speakers are Ledrappier, progress on the Boltzman hypothesis on the of Parry, Xia, and Yorke. Other confirmed participants in- the hard ball gas), and partial differential equations (ergodic clude Afraimovish, Burns, Eberlein, Fathi, Feres, Flaminio, properties of coupled map lattices, dimension estimates Fornaess, Forni, Foulon, Gerber, Hammenstadt, Hassel- of the maximal in certain partial differential equa- blatt, Hunt, Jakobson, Kifer, Levi, Liverani, Przytcki, Rugh, tions, including the Navier-Stokes equation). Rychlik, Schmelling, Schmidt, Smillie, Spatzier, Swiatek, Because smooth ergodic theory overlaps many areas of Szasz, Viana, and Young. There are still pending invitations. mathematics and its arguments often cross category lines, Proceedings of the Institute will be published in the it is difficult for a student, or even an established math- AMS series Proceedings of Symposia in Pure Mathematics. ematician, to acquire a working knowledge of smooth er- It is anticipated that the Institute will be partially godic theory and to learn how to use its tools. Given the funded by a grant from the National Science Foundation very rapid development of smooth ergodic theory in recent and perhaps others. The organizers hope to be able to pro- years, it has been difficult, even for specialists in the field, vide some support for subsistence for most of the par- to keep up with the many significant advances in the field. ticipants. Special encouragement is extended to junior sci- The five main purposes of our proposed summer in- entists to apply. A special pool of funds expected from stitute are: (i) to train young researchers (especially students grant agencies has been earmarked for this group. Other and postdocs) interested in learning smooth ergodic the- participants who wish to apply for support funds should ory, (ii) to teach mathematicians working in other fields the so indicate; however, available funds are limited, and in- core machinery of smooth ergodic theory to use in their dividuals who can obtain support from other sources are applications, (iii) to keep specialists in smooth ergodic encouraged to do so. theory knowledgeable of the current state of the subject All persons who are interested in participating in the In- and some of its new applications, (iv) to discuss major open stitute should request an invitation. Applicants should problems and to help chart a course of research in the area send the following information to: Summer Research In- for the early twenty-first century, and (v) to serve as an ef- stitute Coordinator, American Mathematical Society, P.O. fective meeting ground for mathematicians at all stages in Box 6887, Providence, RI 02940, or by e-mail to their careers, including experts and beginning graduate stu- [email protected] no later than March 3, 1999. dents. Please type or print the following: Our proposed institute will focus on three broad areas 1. Full name. of smooth ergodic theory: structural theory (both con- 2. Mailing address. servative and dissipative), dimension theory, and appli- 3. Phone numbers (including area code) for office, home, cations. and fax.

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4. E-mail address. 5. Your anticipated arrival/departure dates. Geometry and 6. Scientific background relevant to the Institute topics; please indicate if you are a student or if you received your Ph.D. on or after 7/1/93. Topology 7. The amount of financial assistance requested (or in- dicate if no support is required). All requests will be forwarded to the organizing com- EMOIRS Lie Groups and M of the mittee for consideration. In late April all applicants will re- American Mathematical Society Subsemigroups with Volume 130 Number 618 ceive formal invitations (including specific offers of sup- Surjective Lie Groups and Subsemigroups port if applicable), a brochure of conference information, with Surjective Exponential program information known to date, along with informa- Function Exponential Function Karl H. Hofmann tion on travel and dormitories and other local housing. All Wolfgang A. F. Ruppert Karl H. Hofmann, Technische

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M A 8 Hochschule Darmstadt, Germany, participants will be required to pay a nominal conference F 8 O 8 UNDED 1 fee. and Wolfgang A. F. Ruppert, Questions concerning the scientific program should be American Mathematical Society University of Vienna, Austria addressed to [email protected]. Questions of a non- In the structure theory of real Lie scientific nature should be directed to the Summer Insti- groups, there is still information lacking about the exponen- tute coordinator at the address provided above. tial function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjec- tive. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under natural reductions setting aside the “group part” of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protago- nists on the scene are SL(2,R) and its universal covering group, almost abelian solvable Lie groups (i.e., vector groups extended by homotheties), and compact Lie groups. Memoirs of the American Mathematical Society, Volume 130, Number 618; 1997; 174 pages; Softcover; ISBN 0-8218-0641-6; List $45; Individual member $27; Order code MEMO/130/618NA

MEMOIRS Two Classes of American Mathematicalof the Society

Volume 130 Number 619 Riemannian

Two Classes of Riemannian Manifolds Whose Manifolds Whose Geodesic Flows Are Integrable Geodesic Flows Are Kazuyoshi Kiyohara Integrable HEM AT AT M IC A N ΤΡΗΤΟΣ ΜΗ L

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F O U 8 NDED 188 Kazuyoshi Kiyohara, The American Mathematical Society Mathematical Society of Japan, Tokyo Two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kähler-Liouville manifolds respec- tively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow. Memoirs of the American Mathematical Society, Volume 130, Number 619; 1997; 143 pages; Softcover; ISBN 0-8218-0640-8; List $41; Individual member $25; Order code MEMO/130/619NA

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