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Integral 2012.Pdf

Integral 2012.Pdf

Autumn 2012 Volume 7 Institute of Technology 1ntegral n e w s f r o m t h e d e p a r t m e n t a t m i t

Annual Retreat Inside The first Mathematics Department Retreat took place late September, organized by • New Faculty 2 our graduate students. Over 150 depart- • Faculty and Student Awards 3 ment members, families, and guests trav- • Donor Profile and Building 2 4 eled to Purity Spring Resort in the White Mountains in New Hampshire for a splen- • Letter from Haynes Miller 5 did weekend of hiking, canoeing, picking • Department Retreat 6 mushrooms, doing mathematics and relax- • 2012 Doctorates 7 ing. Late nights laughing around the bon- • Alumni Corner, Outreach 8 fire and playing board games in the lodge made for a memorable time. We expect the Retreat to become an annual tradition. Building 2 Renovation MITx and edX Dear Friends, Preparations for the Building 2 renovation In other activities, the Department has been Greetings from MIT Mathematics! gathered speed over the summer. We’ve considering ways to participate in edX, the Faculty recruiting was tremendous last worked closely with Ann Beha Architects initiative for online education. The edX en- year and brought us four fantastic new and MIT planners to arrive at an excit- terprise is building technology to enhance individuals: Professors Alice Guionnet ing scheme that adds common spaces and the learning experience for MIT students (probability), Larry Guth (geometry and offices, mezzanines and skylights, and and to reach thousands or millions of stu- harmonic analysis), Bill Minicozzi (geo- takes good advantage of our outlooks over dents worldwide. The potential is extraordi- metric analysis and PDEs), and Assistant the Charles River. The schedule calls for nary, but the uncertainties are many. We are Professor Aaron Naber (Ricci solitons, construction to begin this summer. We’ll excited about the possibilities and we will collapsing theory). Jörn Dunkel (physical pack up our things and move to temporary work on overcoming the challenges. applied mathematics) will join us next year quarters in Building E17/18, which is being Visiting Committee refurbished too. Mathematics faculty and as Assistant Professor. I’ll close by thanking the members of our staff recently saw the E17/18 space during Visiting Committee, which convened this We filled two key staff positions with out- a lunch we served there. It met with general past spring under the leadership of its new standing people: Barbara Peskin joined approval—proximity to the Kendall Square chair, Art Samberg. I also thank my col- the department in January as Academic restaurants and the T station being seen as leagues for their excellent presentations and Administrator, with primary responsibil- a plus, as noted by several people. We will participation. The meeting was successful, ity for running MAS, the Mathematics be sharing E17/18 with Economics, whose with broad enthusiasm expressed for the Academic Services office. This office, and regular building will be renovated at the results of our recent faculty recruiting ef- hence Barbara herself, is our nexus for all same time. When we return to Building 2 forts and lots of discussion about how the things educational. With the growth in the after the work is done, Building E17/18 will Department could and should participate in mathematics major, now at over 350 stu- serve as swing space for other departments edX. The renovation was a major agenda dents the third largest at MIT, and assorted as their turn comes for major renovation. new technologies and other changes, we item. Lead architect Ann Beha came and are most fortunate to have Barbara, one of We are deeply grateful to Jim and Marilyn inspired us with her vision for new spaces our former PhDs, here with us. Barbara’s Simons, and the , for at MIT. She knows the campus well from extensive mathematical experience as well the leadership gift that is helping make this her days as an architecture graduate student as managerial experience from leader- project possible. We are in the process of here. We appreciate the time invested by ship positions at Dragon Systems and the raising additional funds to cover as much everyone, as well as their helpful ideas and International Computer Science Institute of the remaining renovation cost as we can. their support. The MIT Administration and the Facilities will serve us well. Cynthia Shen arrived Have a good year! two months ago as Administrative Officer. Department have worked with us intensively Cynthia runs Headquarters and is respon- to produce a design that meets our needs. sible for the oversight of all department Simons Lectures finances and staff. She brings a great depth This year’s Simons Lectures will take place Michael Sipser of knowledge from her prior MIT positions in May, given by Emmanuel Candès and Department Head in finance at CSAIL and RLE. Raphaël Rouquier. 2 New Faculty

Alice Guionnet, Professor of Mathematics, comes to MIT Larry Guth, Professor of Mathematics, arrives from NYU’s from École Normale Supérieure Lyon, where she was on the Courant Institute of Mathematical Sciences. Guth works faculty since 2000. Guionnet is a probabilist, specializing in systolic and harmonic analysis. He’s made major break- in random matrices, large deviations, free probability, and throughs on several long-standing questions including the the statistical mechanics of disordered systems. As Direc- endpoint multilinear Kakeya conjecture and the Erdös dis- tor of Research at ENS Lyon, she built a top-ranking prob- tinct distances problem in combinatorial geometry. ability group. Her distinctions include the Miller Institute Guth received his PhD at MIT in 2005 under Tom Mrowka. Fellowship, the Loève Prize, the Silver Medal of CNRS, and Following appointments at Stanford and the University of Simons Investigator. She received her PhD from ENS Paris Toronto, he joined the Courant Institute as Professor of under the guidance of Gérard Ben Arous. Mathematics in 2011.

Aaron Naber, , Assistant Professor Professor of Math- of Mathematics, has ematics, arrives from been a CLE Moore Johns Hopkins Uni- Instructor here since versity, where he has 2009. Naber is a been on the faculty geometric analyst since 1994. Mini- working on the large- cozzi’s field is geo- scale structure of metric analysis. His Riemannian ge- groundbreaking work ometry, , with Toby Colding singularity theory of settled several ma- harmonic maps, and jor problems in the Kähler geometry. theory of embedded Naber completed minimal surfaces his PhD at Princeton of 3 . For University in 2009 these contributions, under Gang Tian. they shared the 2010 Veblen Prize in Geometry. Mini- cozzi received his PhD from in 1994 under . 3

Faculty Achievements Student Awards Igor Rodnianski was awarded the ten lectures on determinantal point Graduate students Sheel Ganatra, 2011 Fermat Prize by the Toulouse processes and representation theory. Hoeskuldur Halldorsson, and Alejan- Mathematics Institute, “for his funda- gave the 2012 Mordell dro Morales received the Charles and mental contributions to the study of Lecture at Cambridge University and Holly Housman Award for excellence the equations of and the 2011-2012 Distinguished Lecture in undergraduate teaching. Steven Sam of the propagation of light on curved Series at UCLA. David Vogan was received the Charles W. and Jennifer C. space-times.” Bonnie Berger was elected President of the American Johnson Prize for an outstanding paper elected fellow of the American Acad- Mathematical Society, starting Feb- accepted for publication. emy of Arts and Sciences and fellow ruary 2013. Undergraduate Fan Wei ’12 received the Jon A. Bucsela Prize in Math- of the International Society for Com- Ju-Lee Kim was promoted to Professor. putational Biology. was ematics for distinguished scholastic elected fellow of the American Acade- Abhinav Kumar and Jonathan Kelner achievement, professional promise, my of Arts and Sciences. Alice Guion- were promoted to Associate Professor. and enthusiasm for mathematics. She net and Paul Seidel were chosen to be also received the Alice T. Schafer Prize Simons Investigators by the Simons Research Staff Awards for excellence in mathematics by an Foundation. Victor Kac was selected Alejandro Rodriguez and Andrew undergraduate woman in mathematics, given by the Association for Women to be a Simons Fellow by the Simons Sutherland each received the School in Mathematics. Undergraduate Amol Foundation. Mark Behrens received of Science Infinite Mile Award. Aggarwal ’15 and his mentor, gradu- the 2011 School of Science Prize Andrew was also awarded the Selfridge ate student Guozhen Wang, shared the for Excellence in Graduate Teaching. Prize for the top paper at the Algo- Hartley Rogers Jr. Prize for the best Pavel Etingof was selected to be the rithmic Symposium. SPUR paper. Jacob Steinhardt ’12 re- next Robert E. Collins Distinguished He was recently promoted to Principal ceived a Hertz fellowship to support his Scholar. Jacob Fox was selected to be Research Scientist. graduate studies. George Arzeno ’14, the next recipient of the Edmund F. Jon Schneider ’13, Shawn Tsosie ’12, Kelly Research Award. Michael Sipser Staff Award and Fan Wei ’12 were Poster Session was selected to be the next holder of Erin McGrath, Director of Development Winners at the Joint Meetings of the the Barton L. Weller Professorship. for the Mathematics and Physics de- AMS and MAA. Alexei Borodin gave the 2012 London partments, received the 2012 MIT Mathematical Society Lectures at Excellence Award in the category of Another Putnam Record the University of Glasgow, a series of Serving the Client. We had a record-breaking year at the 2011 William Lowell Putnam Math- ematical Competition. An astonishing 36% of all high scorers in this North MLK Visiting Assistant Professor Terrence Blackman American competition were MIT stu- dents. Precisely speaking, of the 81 Terrence Blackman high scorers (Honorable Mention and is spending this year higher), MIT had 29, more than the with us as Dr. Martin next five schools combined (Prince­ton 9, Stanford 6, Carnegie-Mellon 5, Luther King, Jr., Visiting Harvard 4, and Caltech 3). Assistant Professor. Congratulations to the fabulous 29: Terrence is on the Noah Arbesfeld, Paul Christiano, Cesar mathematics faculty of Cuenca, Akashnil Dutta, Vlad Firoiu, Medgar Evers College, Whan Ghang, Benjamin Gunby, Brian CUNY, in Brooklyn, Hamrick, Travis Hance, Yangzhou Hu, NY. He works in the Jiaoyang Huang, Hyun Hwang, Tian- Jacquet-Langlands Yi Jiang, Supanat Kamtue, Vladislav correspondence in the Kontsevoi, Reed LaFleche, Holden Lee, Zipei Nie, Timothy Reynolds, David Rol- Langlands program, and more broadly in representation theory, number nick, Colin Sandon, Jonathan Schnei- theory, and automorphic forms. He’s also interested in promoting diversity der, Jeffrey Shen, Xiaolin Shi, Warut in mathematics and science, specifically in the teaching of mathematics Suksompong, Bogdan Veklych, Mark in largely African-American settings. Velednitsky, Tianqi Wu, Kerry Xing. 4

Lead Gift from the Simons Foundation Catalyzes Building 2 Renovation

The Simons Foundation, founded by Jim and Marilyn Simons, has made the lead gift for the renovation of Building 2, creating a new home for our faculty, students, staff, and visitors. Jim is well known as a great mathema- tician and legendary hedge fund man- ager, but he wasn’t always so success- ful. When he was 14, Jim got a job at a garden supply store. He was first sent to the stock room, and reports “I was terrible at it.” He couldn’t re- member what went where. The couple who owned the store demoted him to floor sweeper. That suited him more. “I loved it,” Jim remembers. “I got to walk and think—and I got paid.” When He went on to Berkeley for a PhD and Jim’s area of research is in geometry the job was over, the owners asked Jim taught at both MIT and Harvard. He and topology. He received the Ameri- about his future plans. He told them then got a job with the Institute for De- can Mathematical Society’s 1976 he intended to study mathematics fense Analysis, from which he was not Veblen Prize in Geometry for work that at MIT. “My bosses thought this was just demoted but fired, this time for involved a recasting of the subject hilarious, the kid who couldn’t remem- criticizing the war in Vietnam. of area-minimizing multidimensional ber where to put the sheep manure surfaces. A consequence was the set- wanting to study math.” Jim went on to become head of the Department of Mathematics at Stony tling of two classical questions, the Jim completed his undergraduate Brook. He was only 30 years old. Bernstein conjecture and the Plateau degree in math at MIT in three years. problem. His most influential research

The Walls of Building 2

Building 2 has long been our home, since 1916 when Mathematics was called “General Studies.” If only these corridors could speak, what stories they could tell. Farewell Building 2. We’ll be back! 5

From Associate Head Haynes Miller involved the discovery and applica- tion of certain geometric measure- Barbara Peskin heads the new ments, now called the Chern-Simons Mathematics Academic Services office invariants. Leadership of the newly merged undergraduate and graduate math- Then Jim decided to make a change. ematics offices passed in February He started an investment business 2012 to Dr. Barbara Peskin. Barbara that deployed sophisticated propri- etary models. These generated aston- holds a 1980 PhD from this depart- ishing returns. The New York – based ment, under the direction of Mike company, Renaissance Technologies, Artin. Her academic career included employs mathematics and physics six years at Mt. Holyoke and several PhDs to analyze the behavior of mar- years at Harvard. She joined Dragon kets. Its Medallion Fund is one of the Systems, moved on to the Interna- initiative known as edX, including most successful hedge funds of all tional Computer Science Institute, Harvard University as an equal part- time. Jim is now retired from the day- then directed work on clinical trial ner. Since then, UC Berkeley and to-day management of Renaissance simulations with Kaiser Permanente. and is concentrating on mathematics We are lucky to have her back in the the University of Texas system joined and philanthropy. A few years ago, MIT fold as Mathematics Academic the consortium. This initiative offers Jim gave a guest lecture at MIT on Administrator. an exciting prospect of enhancing gauge theory and topology. More re- our residential education leading What’s new at OCW? to an MIT degree, while providing cently, he gave the Dean of Science The launch of MIT OpenCourseWare Colloquium entitled, “Mathematics, state-of-the-art distance education a decade ago changed the landscape to the world. Common Sense, and Good Luck: My of online education. The Mathemat- Life and Careers.” It filled every seat ics Department listings are among Unlike OpenCourseWare, edX courses in lecture hall 10-250. The video of OCW’s most popular courses. Gilbert will be interactive. Students will sub- that lecture is at Strang’s Linear Algebra class, for mit homework and do exams. We rec- http://math.mit.edu/simons-video. example, will record its two millionth ognize that producing first-rate edX In addition to their magnificent hit this fall! courseware will take time. EdX has support of our new home, Jim and hired former Moore Instructor Grace Marilyn have endowed three profes- OCW publications are essentially Lyo to serve as mathematics liaison sorships in mathematics: the Simons archives. Over the summers of 2010 and content development manager. Chair, held by Richard Melrose; the and 2011, OCW launched OCW Our current objective is to launch Norman Levinson Chair, held by Toby Scholar, which offers complete MITx versions of one or two courses Colding; and the Isadore Singer Chair, chronologically sequenced online in fall 2013. held by Tom Mrowka. courses for several science subjects by augmenting the basic OCW mate- Probability and Statistics As Mathematics Department Head rial with recitation videos and text. The Mathematics Department has Michael Sipser says, “Jim and Marilyn The OCW Scholar projects in math- won a two-year grant from the Davis have been instrumental in changing ematics were led by Associate Pro- Educational Foundation to build a the landscape of mathematics by fessor Benjamin Brubaker (18.01), new curriculum and pedagogy for their extraordinary support at MIT and Lecturer Jeremy Orloff (18.02 and 18.05 Probability and Statistics. throughout the country, and by chang- 18.03), and (18.06), This subject will introduce a new ing the perception of mathematics and involved video recorded “recita- degree of active classroom engage- from ivory tower to an instrument of ment in MIT mathematics. We will be extraordinary power. We are all deeply tions” led by a dozen graduate stu- dents and postdocs. focusing on techniques and examples grateful to them.’’ from the life sciences. The Principal The Mathematics Department is Investigator for this project is Haynes For information on making working with MITx to create online Miller; the lead content developer is a gift to the Mathematics versions of these basic courses. Jeremy Orloff, with assistance from Department, please contact MITx, edX, and Math Moore Instructor Jonathan Bloom Director of Development for In December, 2011, MIT announced and consultancy by Professor Sanjoy Mathematics, Erin McGrath, an initiative in online education Mahajan of Olin College of Engineer- at [email protected] or under the name MITx. Five months ing and formerly of MIT’s Teaching 617-452-2807. later this was expanded to a broader and Learning Laboratory. 6

Department Retreat

Huge thanks to our graduate students for organizing our department retreat. Chief organizer Rosalie Belanger-Rioux coordinated the planning and partici- pated at every level. Others involved include: Michael Donovan – sign-up, buses, Dana Mendelson – rooming list, food and drinks, John Binder – venue finding, buses,Yasha Berchenko-Kogan – venue, food and drinks, Anand Oza – food and drinks, Xuwen Zhu – survey, campfire,Efrat Engel-Shaposhnik – rooming list, Hannah Alpert – minutes, hikes, Ailsa Keating – venue finding. Activity planners: Jonathan Bloom, Ben Elias, Pavel Etingof, Katrin Wehrheim, Roberto Svaldi, Yasha Berchenko-Kogan, and others. 7

2012 Doctorates

Degrees awarded September 2011 through September 2012

Martina Balagovic, “On Representations of Quan- Matrix Factorizations, and Results on the Higher Struc- tum Groups and Cherednik Algebras,” under Pavel tures of the Hochschild Invariants,” under Jacob Lurie. Etingof. Martina is now a postdoc at York University. Anatoly is now a postdoc at UC Berkeley. Nadia Benbernou, “Geometric Algorithms for Re- Steven Sam, “Free Resolutions in Combinatorics and configurable Structures,” under Erik Demaine. Nadia Geometry,” under Richard Stanley. Steven is now a is now at Google. postdoc at the Miller Institute, UC Berkeley. Tsao-Hsien Chen, “Geometric Langlands in Prime Nikhil Savale, “Spectral Asymptotics for Coupled Characteristic,” under Roman Bezrukavnikov. Tsao- Dirac Operators,” under Tom Mrowka. Nikhil is now a Hsien is a postdoc at and at IAS. postdoc at the University of Paris-Sud. 2012 Sheel Ganatra, “Symplectic Cohomology and Duality Nicholas Sheridan, “Homological Mirror Symmetry for the Wrapped Fukaya Category,” under Denis Au- for a Calabi-Yau Hypersurface in Projective Space,” roux. Sheel is a now a postdoc at Stanford. under Paul Seidel. Nicholas is now a postdoc at Hila Hashemi, “Geometric Manipulation of Light: Princeton. From Nonlinear Optics to Invisibility Cloaks,” under David Shirokoff, “I. A Pressure Poisson Method for Steven Johnson. the Incompressible Navier-Stokes Equations: II. Long YoonSuk Hyun, “On Affine Embeddings of Reductive Time Behavior of the Klein-Gordon Equations,” under Groups,” under James McKernan. YoonSuk is now a Ruben Rosales. David is now a postdoc at McGill postdoc at KAIST. University. Nikola Kamburov, “A Free Boundary Problem In- Peter Speh, “A Classification of Real and Complex spired by a Conjecture of De Giorgi,” under David Nilpotent Orbits of Reductive Groups in Terms of Jerison. Nikola is now a postdoc at the University of Complex Even Nilpotent Orbits,” under David Vogan. Arizona. Peter is now at Jane Street Capital. Joel Lewis, “Pattern Avoidance for Alternating Per- Olga Stroilova, “The Generalized Tate Construction,” mutations and Reading Words of Tableaux,” under under Haynes Miller. Olga is now at Vecna Technologies. Alexander Postnikov. Joel is now a postdoc at the Fucheng Tan, “Families of p-adic Galois Repre- University of Minnesota. sentations,” under . Fucheng is now a Niels Moeller, “Mean Curvature Flow Self-Shrinkers postdoc at McMaster University. with Genus and Asymptotically Conical Ends,” under Roman Travkin, “Quantum Geometric Langlands Toby Colding. Niels is now a postdoc at Princeton. Correspondence in Positive Characteristic: the GL(N) Alejandro Morales, “Combinatorics of Colored Case,” under Roman Bezrukavnikov. Roman is now a Factorizations, Flow Polytopes and of Matrices over research scholar at the Clay Mathematics Institute. Finite Fields,” under Alexander Postnikov. Alejandro Jethro van Ekeren, “Modular Invariance for Vertex is now a postdoc at the University of Quebec. Operator Superalgebras,” under Victor Kac. Jethro is Ramis Movassagh, “Eigenvalues and Low Energy now a postdoc at IMPA, Brazil. Eigenvectors of Quantum Many-Body Systems,” Kartik Venkatram, “Birational Geometry of the under Peter Shor. Ramis is now an instructor at Space of Rational Curves in Homogeneous Variet- Northeastern. ies,” under James McKernan. Kartik is now a CCR at Su Ho Oh, “Combinatorics Related to the Totally Non- the US Government. negative Grassmannian,” under Alexander Postnikov. Inna Zakharevich, “Scissors Congruence as K- Su Ho is now a postdoc at the University of Michigan. theory,” under Michael Hopkins. Inna is now a post- Anatoly Preygel, “Thom-Sebastiani and Duality for doc at the University of Chicago. 8

Alumni Corner RSI, SPUR, PRIMES, Memories of Dirk J. Struik and and PRIMES Circle I was an undergraduate at MIT during the mid-fifties, where my bachelor’s thesis PRIMES (Program for Research in Math- advisor was the geometer and historian of ematics, Engineering and Science) is mathematics, Dirk J. Struik. Struik was starting its third year with tremendous then sixty-two, recently reinstated after momentum and success. This program his refusal to cooperate with the House gives local area high school students an The Legacy of Daniel Quillen Un-American Activities Committee. I opportunity to work with MIT researchers K-theory and Homotopical Algebra studied elementary Saturday–Monday, October 6-8, 2012 on exciting unsolved problems in math- Massachusetts Institute of Technology, Cambridge, MA with him for a semester, wrote my bach- ematics, computer science, and compu- math.mit.edu/quillen elor’s thesis under his tutelage, but never tational biology. RSI, SPUR, and PRIMES sniffed a whiff of his Marxist ideology. His students have been recognized with SPEAKERS Luchezar Avramov love of knowledge and appreciation of its numerous honors for their accomplish- Joachim Cuntz Eric Friedlander unity and beauty, on the other hand, was Tom Goodwillie ments. PRIMES students David Ding and Lars Hesselholt Michael Hopkins a fragrance I could sense. His teaching Jean-Louis Loday Xiaoyu He and RSI student Sitan Chen Graeme Segal* Christophe Soule combined mathematical precision with an won Davidson Fellowships. All eight 2012 Douglas Ravenel

appreciation for intuitive foundations. I Ulrike Tillmann Application information is available on the conferenceweb-site. Siemens regional finalists in Massachu- Matthai Varghese *To be confirmed Limited travel funding is available for graduate students and postdocs. once asked him to explain the concept of setts and New Hampshire were PRIMES an osculating plane, which came up in his students. Thanks to Chief Research discussion of curves in space. “I like to Advisor Pavel Etingof, Program Director hike on Mt. Washington,” Struik replied. Slava Gerovitch, and Head Mentor Tanya “The trail I take has large flat rocks. Since Khovanova for these wonderful results. In Memory of Daniel Quillen I am an old man, I choose to walk along The world lost a great a path—a curve—for which the rocks are In the fall of 2012, PRIMES opened a on April 30, 2011, with the passing osculating planes, because it is easiest.” new section, PRIMES Circle, for talented of Daniel Quillen. Quillen began at I took elementary calculus from Norbert sophomores and juniors from public MIT as a Moore Instructor in 1964 Wiener. One day he asked for questions high schools of , Cambridge, and and served as Professor of Mathemat- about the homework. A student raised his Somerville. The Circle students will work ics from 1971 to 1984, when he hand and said that he knew the answer to under the guidance of MIT undergraduate left for Oxford University. In 1978 a particular problem because it was in the student mentors to study mathematics he won the Fields Medal for his work back of the book, but “would Professor beyond the high school curriculum. The on the cohomology of groups and the Wiener please work the problem” because students will practice problem solving, creation of higher algebraic K-theory. the student could not see how to do it. expository writing, and presentation skills. While at MIT, he was known for his Wiener nodded, paced back and forth The goal of this program is to increase beautifully crafted courses. He en- at the front of the classroom for a mo- diversity in the mathematics community joyed teaching 18.075 Mathematical ment, turned to the board, and wrote the by helping strong students with disadvan- Methods for Scientists and Engineers, answer. “I know the answer,” the student taged backgrounds develop their interest using Francis Hildebrand’s book. reminded him. “But how do you work the in mathematics and set them on a path MIT hosted a conference in memory problem?” Wiener nodded apologetically, toward pursuing a math major in college. of Dan Quillen over the Columbus briefly paced back and forth again, then Day weekend: http://math.mit.edu/ turned to the board and wrote the answer Dr. Chelsea quillen/. Thirteen speakers and about a second time. The student became up- Walton, an NSF 150 participants discussed current set, whereupon Wiener, out of patience, postdoctoral research in some of the areas deeply exploded. “What do you want? I’ve worked researcher and influenced by Quillen’s work. the problem two different ways for you!” Mathematics Department Both Struik and Wiener appreciated the Upcoming deep role mathematics has played in the Moore Instructor, evolution of civilization. One day I was in is the PRIMES Gelfand Centennial Conference Struik’s office when Wiener tapped on the Circle Program A View of 21st Century Mathematics, door and Struik invited him in. They sat Coordinator. MIT August 28 –September 2, 2013. on the only two chairs in the small office, and I sat on the floor in a corner. The two great men launched into a wide-ranging m i t d e p a r t m e n t o f mathematics discussion about scientific developments in twelfth-century Europe, ignoring me completely. Finally Wiener ended a mono- Department of Mathematics logue with the expression “Cœur de Lion.” He then turned to me, bent down and said, “Richard the Lionhearted.” Massachusetts Institute of Technology Building 2, Room 236 Telephone: 617-253-4381 —H. L. Resnikoff 77 Massachusetts Ave. Fax: 617-253-4358 Cambridge, MA 02139-4307 Web: math.mit.edu PSB 09.08.0423 Editor Joel Segel Designer∫ ByteGraphics Photographer Bryce Vickmark Printed on recycled paper by Arlington Lithograph

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