Mathematics Colloquium

This colloquium program is supported in part by a generous donation from Neil Zimmerman.

February 23, 2012
Wilhelm Schlag

Wilhelm Schlag is a professor at the University of Chicago. Upon completing his Ph.D. at Cal Tech under the direction of Professor Thomas Wolff in 1996, Prof. Schlag spent time at the Institute for Advanced Study and Princeton before returning to Cal Tech from 2001-2005, from where he moved to Chicago. He has been named a Sloan Fellow (2001-03), a Guggenheim Fellow (2009-10) and twice been a plenary speaker at the International Congress of Mathematical Physics (2004,2012). He is regarded as an expert in the areas of harmonic analysis, spectral theory and nonlinear partial differential equations, though he has also done some work in probability theory. See for instance his works "Time decay for solutions of Schrödinger equations with rough and time-dependent potentials" with I. Rodnianski from 2004, "Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions" with Yuval Peres from 2000 and his groundbreaking work "Stable manifolds for an orbitally unstable nonlinear Schrödinger equation" that appeared in 2009.

March 1, 2012
Rick Durrett

Richard T. Durrett is a professor at Duke University. Upon completing his Ph.D. at Stanford under the direction of Professor Donald Iglehart in 1976, Prof. Durrett ended up at Cornell University from 1985-2010, where he founded and still runs the well-respected Cornell Probability Summer School. He is most known for his many contributions to the fields of probability and statistics, in particular those with applications in ecology and genetics. Prof. Durrett has both produced an incredible amount of new scientific work and written many books about his field, for instance his heavily cited book "Probability: theory and examples" most recently published in 1996 and his highly influential paper on mathematical biology "The importance of being discrete (and spatial)" from 1994 among many others. His work in these areas and others led to him being elected as a member of the National Academy of Sciences in 2007, where the citation particularly references the importance of his work to develop "mathematical models to study the evolution of microsatellites, impacts of selective sweeps on genetic variation, genome rearrangement, gene duplication, and gene regulation."

March 15, 2012
Stephan Garcia

Stephan Garcia is an Associate Professor of Mathematics at Pomona College. After completing his Ph. D at Berkeley in 2003 under the direction of Don Sarason, he took a visiting position at Santa Barbara prior to accepting his current position in 2006. Most of Stephan’s research has been in Operator Theory and Linear Algebra. He has over 30 published papers and has had NSF funding since 2006. He has made important contributions in Operator Theory, both on function spaces and on abstract Hilbert space. Stephan also has an exceptional record as a teacher and expositor. He has earned campus-wide teaching awards at Berkeley, Santa Barbara, and Pomona.

March 22, 2012
Jean-Christophe Yoccoz

Jean-Christophe Yoccoz is a professor at the prestigious College de France in Paris. Upon completing his Doctorat d'Etat under the direction of Professor Michael Herman in 1985, Prof. Yoccoz quickly established himself as a leading expert in dynamical systems, which he continues to be to this day. He has produced groundbreaking results in the theory of small divisors and the structural and dynamic stability of dynamical systems, let us mention "Siegel theorem, Bryuno numbers and quadratic polynomials" from 1995 and "Stable intersections of regular Cantor sets with large Hausdorff dimensions" from 2002 with C. de A. Moreira among others. His work in dynamical systems led to the Salem Prize in 1988 and subsequently the Fields medal in 1994. He has been twice an invited speaker at the ICM (plenary speaker in 1994). He is a member of the French Academy of Sciences (1994) and the Brazilian Academy of Sciences (1994). Other awards include the IBM prize of mathematics in 1985 and the Jaffe prize of the Academy of Sciences in 1991. Prof. Yoccoz's lecture will occur in a special location as he is visiting us as the plenary speaker in the Ergodic theory workshop. Please see the conference web-site for further details.

March 29, 2012
Xiaoyi Zhang

Xiaoyi Zhang is an Assistant Professor at the University of Iowa. Upon completing her Ph.D. at the China Academy of Engineering Physics in 2003, Prof. Zhang spent time at MSRI and the Institute for Advanced Study at Princeton before starting her position at Iowa in 2009. She has been named a Sloan Fellow (2010-2013) and a Von Neumann Early Career Fellow at IAS. She has produced extremely sharp results in the areas of harmonic analysis and nonlinear dispersive partial differential equations, as well as doing some work in fluid dynamics. See for instance her works "Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions" with M. Visan and T. Tao from 2007, as well as "The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher" with R. Killip and M. Visan from 2008.

April 12, 2012
Jared Bronski

Jared Bronski is a professor at the University of Illinois, Urbana-Champaign. Upon completing his Ph.D. at Princeton with David McLaughlin in 1994, Prof. Bronski was a Szëgo postdoc at Stanford and a postdoctoral fellow at the IMA in Minneapolis before arriving in Illinois. An expert in the intersection of spectral theory, dynamical systems and semiclassical analysis, Prof. Bronski's work has helped rigorously develop stability theory in nonlinear partial differential equations with applications to the mathematical studies of soliton theory in optics and Bose-Einstein condensates, as well as other Hamiltonian models in mathematical physics. He has also done work on uncertainty and randomness. In addition, Prof. Bronski won the Sloan Research Fellowship in 2001 and currently serves as an editor of the journal Physica D, which has published many important developments in mathematical physics. Among his notable works are "Nonlinear scattering and analyticity of solitons" in 1998, "Soliton dynamics in a potential" with R. Gerrard in 2000, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem" in 1996 and "Uncertainty estimates and L2 bounds for the Kuramoto-Sivashinsky equation" with T. Gambill in 2006.

April 19, 2012
Gunther Uhlmann

Gunther Uhlmann is currently a professor at both the University of California, Irvine and the University of Washington. Upon completing his Ph.D. under the direction of Professor Victor Guillemin at MIT in 1976, Prof. Uhlmann did postdoctoral work at the Courant Institute and was an assistant professor at MIT before moving to the University of Washington, where he has been from 1984-present.   In 2010, he took up the Excellence in Teaching Chair in Mathematics Professorship at UC-Irvine. Prof. Uhlmann has won numerous awards for his work on inverse problems and partial differential equations using microlocal analysis, including the Kleinman Prize (2011), Bocher Prize (2011), election to the American Academy of Arts and Sciences (2009), a Guggenheim Fellowship (2001-02) and a SIAM Fellowship (2010). He has delivered many plenary talks throughout the world, among them at several ICIAM and AMS meetings. He was also an invited ICM speaker in Berlin in 1998. Among his notable works are "A global uniqueness theorem for an inverse boundary value problem" with J. Sylvester in 1987, "The Calderón problem with partial data" with C. Kenig and J. Sjöstrand in 2007 and a groundbreaking work on cloaking "Full-wave invisibility of active devices at all frequencies" with A. Greenleaf, Y. Kurylev and M. Lassis in 2007.

December 1, 2011
Alexei Oblomkov

Alexei Oblomkov is a promising young mathematician working at the intersection of algebraic geometry, representation theory, 'quantum' topology and physics. He got his Ph.D. at M.I.T. under the direction of Pavel Etingof. Currently he is a Sloan Fellow at the UMass Amherst.
He is one of the best experts in double affine Hecke algebras; his papers on quantization of cubic and del Pezzo surfaces (with Etingof and Rains) are among the best achievements of this theory. He is engaded now in the study of cohomology of Hilbert schemes in collaboration with Maulik, Okounkov and Pandharipande. It led him towards exploring the relation between Hilbert schemes of a singular curve and HOMFLY polynomial of the torus knot.
Working with Shende, Oblomkov extends the relation between the Euler characteristic of a Hilbert scheme and the HOMFLY polynomial to that between the cohomology of Hilbert schemes and the triply graded homology of torus knots. This new theory attracts a lot of attention now of mathematicians and physicists.

November 3, 2011
Ken McLaughlin

Ken McLaughlin is one of the leading experts in random matrix theory, approximation theory, orthogonal polynomials and the analytical theory of integrable systems. He contributed significantly to the intricate ties between these fields, which have been identified and exploited with very interesting results.
Among quite a few his major invited lectures, he was a speaker of the 14th Congress on Mathematical Physics (Lisbon, 2003); Percy Dieft reported his joint work with Kriecherbauer, McLaughlin, Venakides and Zhou at the ICM-98 in Berlin (an invited lecture).
Ken is currently working on universality in partial differential equations and random matrix theory. The lecture, centering on rough periodic solutions to the linear (yes, linear!) Schrödinger equation, will also discuss the history of the Gibbs phenomenon, the strange correspondence and slow pace of the mail which led to Michelson not receiving credit where credit was due, the Talbot effect and the precision of experimentation during the 1840s.
Ken moved to Arizona University 7 years ago from our dept, one of the best our researches and a great colleague; we are very happy to see him!

October 27, 2011
Boris Khesin

Boris Khesin is a well-known scientist with truly universal interests in mathematics, from hydrodynamics and global analysis to the infinite-dimensional Lie theory; not very surprising for a student of Vladimir Arnold! Indisputably, he is one of the best lecturers in mathematics with great international connections; France and Germany are among his favorite places when working abroad (10+7 visiting positions there from 1992). He was a Clay fellow and an invited speaker of Royal Society (London) among other honors; 3 Canadian awards must be mentioned: Premier Research Excellence Award (Ontario), the McLean Award (University of Toronto) and the Andre-Aisenstadt Mathematics Prize (Montreal).

October 13, 2011
Michele Vergne

Michele Vergne is a prominent scholar, a member of the French Academy of Sciences and the American Academy of Arts and Sciences. Académie des Sciences is one of the oldest and the most respected such bodies in the world (only 150 full members and 120 foreign associates); Prix Ampère preceded her election in 1998.
She was a professor at MIT during 1977-86 and Directeur de Recherches, CNRS from 1981 till 2008 (promoted to Classe Exceptionelle in 1997). She was a plenary speaker at International and European Congresses of Mathematicians (2006, 1992), delivered Emmy Noether Lecture series in Goettingen (2008) among many others talks and lectures.
Her fundamental contributions to geometry, representation theory and harmonic analysis are well known, including important books "The Weil representation, Maslov index and theta series" (with G. Leon) and "Heat Kernels and Dirac Operators" (with Berline and Getzler).

September 8, 2011
Andras Szenes

The lecturer is a professor of mathematics from the Geneve Univesity, a well-known specialist in a variety of fields including topology, algebraic geometry, mathematical physics and related representation theory. His lecture will be about one of the most interesting interdisciplinary topics in mathematical physics and beyond, the Y-system, which is connected, among quite a few other lines, with the Hirota identities, the Zamolodchikovs conjecture, proven via the cluster algebras, and the AdS-CFT correspondence (the previous colloquium by Vladimir Kazakov). We are sure that the lecture will be understandable and helpful.

ABSTRACT: In 1991, Zamolodchikov discovered an intriguing rational recursion called the Y-system. The recursion has some remarkable periodicity properties, and relations to number theory, representation theory, and several other fields of Mathematics. The Y-systems gave rise to identities among the values of the dilogarithm function, and also served as one of the first examples of an important new notion of modern algebra: the cluster algebras. In this talk, we will review recent results and developments in the field.

April 21, 2011
Vladimir Kazakov

The lecturer, a brilliant physicist, is one of the best experts in Matrix Models and integrability of physics theories. He will speak about the breakthrough development called the AdS/CFT correspondence , which resulted in the integrability of important "really" 4-dimensional Super-Yang Mills theories (in the planar limit). Counting zillions of Feynman graphs needed for finding anomalous dimensions and similar important physics quantaties can be replaced by an essentially algebraic and very elegant theory based on the quantum Hirota identities, Bethe equations and the Y-system for certain super Lie algebras of non-compact type. The lecture is expected understandable for mathematicians; certain experience with classical "integrability" is desirable.

ABSTRACT: I will speak about the recently found solution of the problem of spectrum of anomalous dimensions in a four dimensional conformal field theory - the Yang-Mills theory with N=4 super-symmetries at any strength of the interaction. Due to the AdS/CFT correspondence, this gauge theory can be mapped to a two-dimensional string sigma model, which appeared to be integrable classically, as well as quantum mechanically. This allowed to apply the full set of the integrability tools, such as the thermodynamic Bethe ansatz and the so called Y-system (equivalent to the famous Hirota bi-linear finite difference equation). The lowest anomalous dimension, for the Konishi operator, was successfully found by numeric solution of the resulting equations, and their analytic analysis allowed to easily reproduce the results of summation of Feynman diagrams.

March 31, 2011
Alexander Braverman

Alexander Braverman is one of the best specialists in the homogeneous spaces of loop groups, the geometric Langlands program and related algebraic geometry. His interests are truly universal, in the range from quantum cohomology to double arithmetic. Whatever his current research field is, he always obtains advanced sharp results. Undoubtedly, he is one of the best lecturers in this highly developed and quickly progressing area. His lecture will be devoted to the Satake isomorphism, which is one of the key topics in the classical and modern representation theory.

ABSTRACT: In the first part of the talk I am going to recall the formulation of the so called Satake isomorphism for a group G over a p-adic field K and explain why and how it can be regarded as the starting point for the notion of automorphic L-function and for the Langlands conjectures. I will also explain some interesting combinatorics around the Satake isomorphism.
In the second part of the talk I am going to describe the generalization of the Satake isomorphism to (infinite-dimensional) loop groups groups as well as a (much more involved) generalization of the above combinatorics (joint work with David Kazhdan and Manish Patnaik).

March 17, 2011
Yakov Sinai

The speaker is known virtually to everyone in the world of professional mathematics and physics. His major groundbreaking results are in the theory of dynamical systems, mathematical physics and in probability theory; Kolmogorov-Sinai entropy, Sinai's billiards, Sinai's random walk, Sinai-Ruelle-Bowen measures, just to mention some. Among his awards are the Boltzmann Medal (1986), Dirac Medal (1992), the Wolf Prize in Mathematics (1997), most recently, his election to the Royal Society of London (2009) and Henri Poincare Prize (2009). We are honored by his visit initiated by the Ergodic Theory Workshop (March 17-20).
The talk will be on a new approach proposed in a work by Cellarosi and Sinai that allows to study random properties of the classical Moebius function.

February 24, 2011
Andrei Caldararu

Andrei Caldararu is a relatively young mathematician; he got his PhD from Cornell University in 2000. However he is already a recognized expert in algebraic geometry, including the algebraic aspects of string theory and homological algebra. Many of his works fit into the derived algebraic geometry, a new branch of commutative and non-commutative algebraic geometry that provides powerful tools for the deformation theory and intersection theory. Maxim Kontsevich, Carlos Simpson and, recently, Jacob Lurie are among the major contributors to this field. It brings the idea of homotopy equivalence into the core of scheme theory, with the goal of explaining such phenomena as stacks that really should have a tangent space that is not a vector space, but a complex of vector spaces (up to homotopy). Understanding derived categories of sheaves, fundamental invariants of complex manifolds, is one of the key objectives of Andrei Caldararu's research.

ABSTRACT: String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students.

February 17, 2011
Michael Weinstein

Michael Weinstein is a professor in the Applied Physics and Mathematics Department at Columbia University. He has rare deep insights both into the physically interesting models to study, as well as the mathematical techniques required to analyze them. His contributions to the theory of nonlinear waves are well known, including modulational stability for solitary wave solutions of the equations like KdV and NLS, dispersive PDE, cloaking, fluid dynamics and geophysics, in addition to holding two patents related to theoretical optical engineering devices. His lecture will introduce problems from nonlinear waves and discuss some of the mathematical ideas needed to understand the nature of the fascinating underlying nonlinear dynamics.

ABSTRACT: This talk will overview results and open problems on the dynamics of coherent structures for certain nonlinear dispersive PDEs, a class of infinite dimensional Hamiltonian systems. Many such systems have spatially localized solutions, describing coherent structures (hydrodynamic, electrodynamic, quantum...) such as soliton pulses or vortices, with remarkable stability properties. The general PDE dynamics can be viewed in terms of the nonlinear interaction of such coherent structures with linear dispersive waves. The infinite-time behavior is the subject of nonlinear scattering. Intermediate but very long-time transients, e.g. metastable states, play an important role in the analysis and a central role in applications. A detailed understanding involves ideas from dynamical systems (Hamiltonian theory of normal forms,...) and scattering theory (wave operators, non-self adjoint spectral theory,...), variational and harmonic analysis. We will consider these questions in the context of the nonlinear Schroedinger - Gross Pitaevskii equation, a class of PDEs having wide applications to classical and quantum systems. Applications to the control of soliton-like states in nonlinear optical and quantum systems will be discussed.

February 10, 2011
Raman Parimala

Raman Rarimala, a speaker at the latest ICM in Hyderabad and the one in 1994, is a fellow of all three Indian academies of science. She received Srinivasa Ramanujan Birth Centenary Award in 2003 and the 2005 prize in mathematics from the Academy of Sciences for the Developing World (TWAS) among her other awards. She was the first woman working in the fields of mathematics and physics honored with the TWAS prize during its 20 years of history for her work on the quadratic analogue of Serre's conjecture, the triviality of principal homogeneous spaces of classical groups over fields of cohomological dimensions 2 and the m-invariant of p-adic function fields. In 2005, Raman Parimala was appointed the Asa Griggs Candler Professor of Mathematics at Emory University in Atlanta.

ABSTRACT: Hasse-Minkowski's theorem says that a quadratic form over a number field is isotropic if it is isotropic over completions at all places of the number field. One could look for a Hasse principle in the function field setting with respect to all discrete valuations of the function field. This is particularly interesting for Q_p(t); a Hasse principle for isotropy of quadratic forms would lead to the fact: every quadratic form over Q_p(t) in at least nine variables has a nontrivial zero. We explain the Hasse principle over function fields of nondyadic p-adic curves with respect to a set of divisorial discrete valuations of the field.

January 27, 2011
Nicolas Burq

Nicolas Burq, a member of Institut universitaire de France, made fundamental contributions to a wide range of modern problems in analysis and physics, from scattering theory and eigenfunction properties to control theory and nonlinear dispersive equations. His works on dispersive estimates for the Schrödinger equation on manifolds, eigenfunction restriction properties, defect measures in various geometric settings, scattering theory and resonances are well known. Impressive diversity of his research interests is recognized; three (!) times he spoke at Séminaire Bourbaki. His lecture will be based on joint works with N. Tzvetkov (Université de Cergy).

ABSTRACT: The strating point of my talk will be a result by Paley and Zygmund (1930): Consider any l² sequence, (α_n), nÎN. Then of course Parseval's theorem shows that the trigonometric series on the torus ∑_n α_ne^inx is in L²(T). It is also known (and quite easy to prove) that generically (for the l² topology, i.e. for (α_n) in a dense Gδ set), the series is in no L^p(T) space (p>2). However, Paley-Zygmund's theorem ensures that if one simply changes the signs of the coefficients α_n randomly and independently, then the series is almost surely in every L^p(T) space! This striking phenomenon was later widely studied in the context of harmonic analysis, giving rise to a huge amount of results (works by Kahane, Pisier and many others). However, surprisingly, this phenomenon was until recently not exploited in the context of partial differential equations. The purpose of this talk is precisely to present some occurences of this phenomenon in non linear partial differential equations. For simplicity, I will focus my talk on one of the simplest models: the cubic non linear wave equation, and give examples where it is possible to exhibit much better behaviour for solutions to such a PDE when the initial data are randomly chosen than the behaviour predicted by the classical deterministic theory.

November 11, 2010
Zhiwei Yun

This colloquium lecture will be mainly devoted to the Fundamental Lemma in finite characteristic, an important part of the Langlands program, verified recently by Bao-Chau Ngo in complete generality. It resisted all attempts to prove it for almost 30 years.
The speaker is a (very) young mathematician, an absolute winner of the International Mathematics Olympiad in 2000, a recognized expert in a bunch of highly regrarded directions around the Fundamental Lemma. We are delighted that he agreed to deliver an introductory lecture for us on this topic, which attracts a lot of attention (to be continued at "Geometric Methods in RT", Nov. 12, 4:15pm, Ph367).

ABSTRACT: This summer, Bao-Chau Ngo got a Fields medal for his proof of the Fundamental Lemma, FL. I will try to explain what FL is about and the idea of his proof from a geometric point of view. The FL as formulated by Langlands and Shelstad was an identity between integrals. We will first see that these integrals actually count lattices. Then, in the function field case, one can "globalize" the picture to count vector bundles. Towards the end of the talk, Hitchin moduli spaces will show up (hopefully) naturally.

November 4, 2010
Vikram Mehta

Vikram Mehta is a professor of Tata Institute of Fundamental Research. Jointly with A. Ramanathan, he introduced the notion of Frobenius Splitting, which is a tool of fundamental importance in algebraic geometry and representation theory. It guaranties that the cohomology (i>0) vanishes for all ample line bundles on an algebraic variety (with a possibility to use it in characteristic zero); many Schubert varieties are of this type. His colloquium lecture is expected understandable to non-specialists; some familiarity with algebraic manifolds and the cohomology is assumed.

ABSTRACT: The talk will be mainly about the local criterion for splitting and compatible splitting, with special emphasis on Schubert varieties in G/B. We will discuss the diagonal splitting, the splitting of the cotangent bundle, the splitting of the moduli spaces of bundles on curves and applications to varieties in characteristic p>0 with trivial tangent bundle.

October 28, 2010
Mikhail Khovanov

A special lecture by the creater of Khovanov cohomology, a striking discovery, which goes well beyond the low-dimensional topology. The quantum SL(2) and the related categorification constructions will be the key theme; certain familiarity with simple Lie algebras is assumed.

ABSTRACT: Quantum groups, discovered by Drinfeld and Jimbo, are Hopf algebra deformations of the universal enveloping algebras of simple Lie algebras. Various forms of quantum groups can be realized as Grothendieck groups of certain rings given by planar diagrammatical generators and relations. These constructions will be explained in the talk.

List of past colloquia.