| APRIL 2012 |
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| Department Colloquium |
| Topic: |
Near-optimal mean value estimates for Weyl sums |
| Presenter: |
Trevor Wooley, University of Bristol |
| Date: |
Wednesday, April 25, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring?s problem. Recent applications from the 21st century include sum-product estimates in additive combinatorics, and the investigation of the geometry of moduli spaces. Over the past 75 years, estimates for the moments underlying Vinogradov?s mean value theorem have failed to achieve those conjectured by a factor of roughly log k in the number of implicit variables required to successfully analyse exponential sums of degree k. In this talk we will sketch out some history, several applications, and the ideas underlying our recent work which comes within a stone?s throw of the best possible conclusions. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Invariant Measures, Conjugations and Renormalizations of Circle Maps with Break points |
| Presenter: |
Akhtam Dzhalilov, Samarkand State University |
| Date: |
Thursday, April 26, 2012, Time: 2:00 p.m., Location: Fine Hall 601 |
| Abstract: |
An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle diffeomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite different. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point $b$ and satisfying the Katznelson-Ornsteins smoothness condition i.e. $Df$ is absolutely continuous on $[b, b + 1]$ and $D^2f \in L^p(S^1, dl)$, $p > 1$. We will formulate some results concerning the renormalization behavior of such circle maps. |
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| Discrete Mathematics Seminar |
| Topic: |
Points, lines, and local correction of codes |
| Presenter: |
Avi Wigderson, IAS |
| Date: |
Thursday, April 26, 2012, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and related ones), which are motivated from questions about locally correctable codes and matrix rigidity. The proofs use an interesting combination of combinatorial, algebraic and analytic tools. The talk is self contained. Joint work with Boaz Barak, Zeev Dvir and Amir Yehudayoff |
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| Algebraic Topology Seminar |
| Topic: |
Loop products and dynamics |
| Presenter: |
Nancy Hingston, IAS and the College of New Jersey |
| Date: |
Thursday, April 26, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
| Abstract: |
A metric on a compact manifold M gives rise to a length function on the free loop space LM whose critical points are the closed geodesics on M in the given metric. Morse theory gives a link between Hamiltonian dynamics and the topology of loop spaces, between iteration of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of LM. Geometry reveals the existence of a related product on the cohomology of LM. A number of known results on the existence of closed geodesics are naturally expressed in terms of nilpotence of products. We use products to prove a resonance result for the loop homology of spheres. I will not assume any prior knowledge of loop products. Mark Goresky, Hans-Bert Rademacher, and (work in progress) Ralph Cohen and Nathalie Wahl are collaborators. |
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| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
Deligne-Lusztig theory for unipotent groups and the local Langlands correspondence |
| Presenter: |
Dmitriy Boyarchenko, University of Michigan |
| Date: |
Thursday, April 26, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
(1) A (very) special case of Deligne-Lusztig theory yields a construction of cuspidal irreducible representations of the finite group GL_n(F_q) in the cohomology of an algebraic variety equipped with an action of GL_n(F_q). There is also a well known relationship between cuspidal representations of GL_n(F_q) and depth 0 supercuspidal representations of GL_n(F), where F is a local field with residue field F_q. (2) On the other hand, thanks to the work of Boyer, Carayol, Deligne, Harris, Henniart, Laumon, Rapoport, Stuhler, Taylor..., it is known that the local Langlands correspondence for GL_n(F) is realized in the cohomology of the Lubin-Tate tower of rigid analytic spaces over F. There is a direct geometric link between (1) and (2): the first level of the Lubin-Tate tower contains an open affinoid with good reduction, whose special fiber is isomorphic to a Deligne-Lusztig variety for GL_n(F_q). I will explain a similar picture for certain supercuspidal representations of GL_n(F) of positive depth. In particular, I will describe the construction of an open affinoid (with good reduction) in a higher level of the Lubin-Tate tower, which has the following properties. On the one hand, its cohomology gives an explicit geometric realization of the local Langlands correspondence for a certain class of positive depth supercuspidal representations of GL_n(F). On the other hand, its special fiber is related to a certain unipotent group over F_q in a way that is similar to one of the known approaches to Deligne-Lusztig theory for reductive groups over F_q. |
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| Topology Seminar |
| Topic: |
Nonorientable four-ball genus can be arbitrarily large |
| Presenter: |
Joshua Batson, MIT |
| Date: |
Thursday, April 26, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
A classical problem in low-dimensional topology is to find a surface of minimal genus bounding a given knot K in the 3-sphere. Of course, the minimal genus will depend on the class of surface allowed: must it lie in S3 as well, or can it bend into B4? must the embedding be smooth, or only locally flat? must the surface admit an orientation, or can it be nonorientable? Our ability to bound or compute these genera varies dramatically between classes. Orientable surfaces form homology classes, so are amenable to algebraic topology (cf Alexander polynomial), and they admit complex structures, so can be understood using gauge theory (cf Ozsvath-Szabo's \tau). In contrast, the largest lower bound on the genus of a nonorientable surface smoothly embedded in B4 bounding any knot K was, until recently, the integer 3. We will construct a better bound using Heegaard-Floer d-invariants and the Murasugi signature. In particular, we will show that the minimal b_1 of a smoothly embedded, nonorientable surface in B4 bounding the torus knot T(2k,2k-1) is k-1. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Gluing for Nonlinear PDEs, and Self-Shrinking Solitons in Mean Curvature Flow |
| Presenter: |
Niels Martin Møller, MIT |
| Date: |
Friday, April 27, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
I will discuss some recent gluing constructions from minimal surface theory that yield complete, embedded, self-shrinking soliton surfaces of large genus g in R^3 (as expected from numerics by Tom Ilmanen and others in the early 90's), by fusing known low-genus examples. The analysis in the case of non-compact ends (joint w/ N. Kapouleas & S. Kleene), is complicated by the unbounded geometry, where Schrödinger operators (of Ornstein-Uhlenbeck type) with fast growth of the coefficients need to be understood well via Liouville-type results, which in turn enable construction of the resolvent of the stability operator and closing the PDE system. |
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| Analysis Seminar |
| Topic: |
The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet) |
| Presenter: |
Didier Pilod, Universidade Federal do Rio de Janeiro / University of Chicago |
| Date: |
Monday, April 30, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
The Benjamin-Ono equation models the unidirectional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system, one being in finitely deep. The Cauchy problem associated to this equation presents interesting mathematical difficulties and has been extensively studied in the recent years. In a recent work (2007), Ionescu and Kenig proved well-posedness for real-valued initial data in L^2(R). In this talk, we will give another proof of Ionescu and Kenig's result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in H^s(R), for s > 1/4 . Note that our approach also permits to simplify the proof of the global well-posedness in L^2(T) by Molinet (2008) and yields unconditional well-posedness in H^{1/2}(T). Finally, it is worthwhile to mention that our technique of proof also apply for a higher-order Benjamin-Ono equation. We prove that the associated Cauchy problem is globally well-posed in the energy space H^1(R). |
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| Joint Princeton-Rutgers Seminar on Geometric PDEs |
| Topic: |
Min-max theory and the Willmore Conjecture |
| Presenter: |
Andre Neves, Imperial College |
| Date: |
Monday, April 30, 2012, Time: 4:30 p.m., Location: Fine Hall 110 |
| Abstract: |
In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of any torus immersed in Euclidean three-space is at least 2 \pi^2. I will talk about my recent joint work with Fernando Marques in which we prove this conjecture using the min-max theory of minimal surfaces. |
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| PACM Colloquium |
| Topic: |
Nonlinear Expectation, Nonlinear PDE and Stochastic Calculus under Knightian Uncertainty |
| Presenter: |
Shige Peng, Shandong University |
| Date: |
Monday, April 30, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
A. N. Kolmogorov's "Foundations of the Theory of Probability" published in 1933, has established the modern axiomatic foundations of probability theory. Since then this theory has been profoundly developed and widely applied to situations where uncertainty cannot be neglected. But in 1921 Frank Knight has been already clearly classified two types of uncertainties: the first one is for which the probability is known; the second one, now called Knightian uncertainty, is for cases where the probability itself is also uncertain. The situation with Knightian uncertainty has become one of main concerns in the domain of data processing, economics, statistics, and specially in measuring and controlling financial risks. A long time challenging problem is how to establish a theoretical framework comparable to the Kolmogorov's one, to treat these more complicated situations with Knightian uncertainties. Tthe objective of the theory of nonlinear expectation rapidly developed in recent years is to solve this problem. This is an important program. Some fundamental results have been established such as law of large numbers, central limit theorem, martingales, G-Brownian motions, G-martingales and the corresponding stochastic calculus of Itˆo's type, nonlinear Markov processes, as well as the calculation of measures of risk in finance. But still so many deep problems are still to be explored. This new framework of nonlinear expectation is naturally and deeply linked to nonlinear partial differential equations (PDE) of parabolic and elliptic types. These PDEs appear in the law of large numbers, central limit theorem, and nonlinear diffusion processes in the new theory, and inversely, almost all solutions of linear, quasilinear and/or fully nonlinear PDEs can be expressed in term of the nonlinear expectation of a function of the corresponding (nonlinear) diffusion processes. Moreover, a new type of 'path-dependent partial differential equations' have been introduced which provide a PDE tool to study a stochastic process under a nonlinear expectation. Numerical calculations of these path dependent PDE will provide the corresponding backward stochastic calculations. |
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| MAY 2012 |
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| Algebraic Geometry Seminar |
| Topic: |
Comparison theorems in p-adic Hodge theory |
| Presenter: |
Bhargav Bhatt, University of Michigan |
| Date: |
Tuesday, May 1, 2012, Time: 4:30 p.m., Location: Fine Hall 322 |
| Abstract: |
A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine's conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry --- specifically, derived de Rham cohomology --- and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way. |
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| Department Colloquium |
| Topic: |
Approximate groups and Hilbert's fifth problem |
| Presenter: |
T. Tao, University of California - Los Angeles |
| Date: |
Wednesday, May 2, 2012, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Approximate groups are, roughly speaking, finite subsets of groups that are approximately closed under the group operations, such as the discrete interval {-N,...,N} in the integers. Originally studied in arithmetic combinatorics, they also make an appearance in geometric group theory and in the theory of expansion in Cayley graphs. Hilbert's fifth problem asked for a topological description of Lie groups, and in particular whether any topological group that was a continuous (but not necessarily smooth) manifold was automatically a Lie group. This problem was famously solved in the affirmative by Montgomery-Zippin and Gleason in the 1950s. These two mathematical topics initially seem unrelated, but there is a remarkable correspondence principle (first implicitly used by Gromov, and later developed by Hrushovski and Breuillard, Green, and myself) that connects the combinatorics of approximate groups to problems in topological group theory such as Hilbert's fifth problem. This correspondence has led to recent advances both in the understanding of approximate groups and in Hilbert's fifth problem, leading in particular to a classification theorem for approximate groups, which in turn has led to refinements of Gromov's theorem on groups of polynomial growth that have applications to the study of the topology of manifolds. We will survey these interconnected topics in this talk. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Reducibility for the quasi-periodic linear Schrödinger and wave equations. |
| Presenter: |
L. H. Eliasson, Université Paris Diderot |
| Date: |
Thursday, May 3, 2012, Time: 2:00 p.m., Location: Fine Hall 601 |
| Abstract: |
We shall discuss reducibility of these equations on the torus with a small potential that depends quasi-periodically on time. Reducibility amounts to "reduce" the equation to a time-independent linear equation with pure point spectrum in which case all solutions will be of Floquet type. For the Schrödinger equation, this has been proven in a joint work with S. Kuksin, and for the wave equation we shall report on a work in progress with B. Grebert and S. Kuksin. |
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| Discrete Mathematics Seminar |
| Topic: |
Edge-coloring 8-regular planar graphs |
| Presenter: |
Maria Chudnovsky, Columbia University |
| Date: |
Thursday, May 3, 2012, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
In 1974 Seymour made the following conjecture: Let G be a k-regular planar (multi)graph, such that for every odd set X of vertices of G, at least k edges of G have one end in X and the other in V(G) \ X. Then G is k-edge colorable. For k=3 this is equivalent to the four-color theorem. The cases k=4,5 were solved by Guenin, the case k=6 by Dvorak, Kawarabayashi and Kral, and the case k=7 by Edwards and Kawarabayashi. In joint work with Edwards and Seymour, we now have a proof for the case k=8, and that is the topic of this talk. |
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| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
Eisenstein series on exceptional groups, graviton scattering amplitudes, and the unitary dual |
| Presenter: |
Steven D. Miller, Rutgers University |
| Date: |
Thursday, May 3, 2012, Time: 4:30 p.m., Location: IAS - Room S-101 |
| Abstract: |
I will describe the appearance of special values of Eisenstein series on E6, E7, and E8 that arose in studying the low energy expansion of the 4-graviton scattering amplitude in string theory (see arxiv:1004.0163 and 1111.2983). Through methods to handle the combinatorics of Langlands' constant term formulas we were able to exactly identify some correction terms beyond the main term predicted by Einstein general relativity. In some cases string theory predicts cancellations of terms in these formulas, while in others derives information from them. Some of the correction terms are proven to be automorphic realizations of small, real unitary representations of split real groups; this is used to limit the instanton contributions to these terms (i.e., verifying their fractional BPS properties). As a consequence of the combinatorial methods we prove a conjecture of Arthur concerning the spherical unitary dual of split real groups. (Joint work with Michael Green and Pierre Vanhove) |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Bo Guan, Ohio State University |
| Date: |
Friday, May 4, 2012, Time: 3:00 p.m., Location: Fine Hall 314 |
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| Analysis Seminar |
| Topic: |
BBM: a statiscal point of view |
| Presenter: |
Anne-Sophie de Suzzoni, Universite de Cergy-Pontoise |
| Date: |
Monday, May 7, 2012, Time: 3:15 p.m., Location: Fine Hall 314 |
| Abstract: |
After presenting the BBM equation and some of its properties, we will try to understand which kind of statistics have a chance to remain invariant by its flow and produce one of them. The stability of this statistics will be studied then : to do so, we will sketch the parallelism between properties of equations and the statistics whose laws remain invariant by their flow. |
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| Joint PACM & Analysis Seminar |
| Topic: |
The 2D Boussinesq equations with partial dissipation |
| Presenter: |
Jiahong Wu, Oklahoma State University |
| Date: |
Monday, 7, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial dissipation has attracted considerable attention in the last few years. In this talk we will summarize recent results on various cases of partial dissipation, present the work of Cao and Wu on the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion and explain the work of Chae and Wu on the logarithmically supercritical Boussinesq equations. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Effective discreteness of the 3-dimensional Markov spectrum |
| Presenter: |
Han Li, Yale University |
| Date: |
Thursday, May 10, 2012, Time: 2:00 p.m., Location: Fine Hall 601 |
| Abstract: |
Let the set O={non-degenerate, indefinite, real quadratic forms in 3-variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{|Q(v)|: v is a non-zero integral vector in $R^3$}. The set M={m(Q): Q is in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis' proof of the Oppenheim conjecture asserts that, for every a>0 $M \intersect (a, \infty)$ is a finite set. In this lecture we will show that #{M \intersect (a, \infty)}<< a^{-26}. This is a joint work with Prof. Margulis, and our method is based on dynamics on homogeneous spaces. |
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| Joint IAS and Princeton University Number Theory Seminar |
| Topic: |
TBA |
| Presenter: |
Jack Thorne, Harvard University |
| Date: |
Thursday, May 10, 2012, Time: 4:30 p.m., Location: Fine Hall 214 |
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| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Allison Gilmore, Princeton University |
| Date: |
Thursday, May 17, 2012, Time: 3:00 p.m., Location: Fine Hall 214 |
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