Updated: 4-15-2009
APRIL 2009
Department Colloquium
Topic: Invariant distributions and scaling in parabolic dynamics
Presenter: Giovanni Forni, University of Maryland
Date:  Wednesday, April 15, 2009, Time: 4:30 p.m., Location: Fine Hall 314
Abstract: A smooth dynamical system is often called parabolic if nearby orbits diverge with power-like (polynomial) speed. There is no general theory of parabolic dynamics and a few classes of examples are relatively well-understood: area-preserving flows with saddle singularities on surfaces (or, equivalently, interval exchange transformations) and to a lesser extent 'rational' polygonal billiards; SL(2,R) unipotent subgroups (horocycle flows on surfaces of constant negative curvature) and nilflows. In all the above cases, the typical system is uniquely ergodic, hence ergodic averages of continuous functions converge unformly to the mean. A fundamental question concerns the speed of this convergence for sufficiently smooth functions. In many cases it is possible to prove power-like (polynomial) upper bounds. A unified approach to this problem consists in constructing invariant distributions (in the sense of L. S. Sobolev or L. Schwartz) usually by methods of harmonic analysis and studying how they rescale under an appropriate 'renormalization' scheme. This approach yields quite precise bounds for many of the above examples but often cannot be implemented for lack of an (effective) renormalization. In this talk, after a review of some of the main known results for renormalizable systems, we will present a quantitative equidistribution result for some non-renormalizable nilflows and we will discuss some new ideas we have introduced (in joint work with L. Flaminio) to deal with this problem. Bounds on Weyl sums that can be derived from our results will be discussed.
Graduate Student Seminar
Topic: Equivariant Cohomology
Presenter: Iman Setayesh, Princeton University
Date:  Thursday, April 16, 2009, Time: 12:30 p.m., Location: Fine Hall 314

A cohomology theory is a way of associating a complex of abelian groups to a topological space. Study of such complexes gives geometric information about the structure of the space. Moreover if we have an action of a suitable group on a space we can associate another such complex to that space which carries more information. I will sketch main ideas of this construction and provide some examples.

Ergodic Theory and Statistical Mechanics Seminar
Topic: Existence/Non-existence of limiting distributions for horocycle flows on compact surfaces of constant negative curvature
Presenter: Giovanni Forni, University of Maryland
Date:  Thursday, April 16, 2009, Time: 2:00 p.m., Location: Fine Hall 401

A few years ago we have proved in collaboration with L. Flaminio that some non-trivial limit distributions for the horocycle flow must have compact support. In this talk we will refine that result and describe an existence/non-existence result for limiting distributions. In fact it turns out that whether limiting distributions exist or not depends on the geometry of the surface (via the eigenvalues of the Laplace operator) and on the observable under consideration. The main new idea is to express the precise results on the asymptotics of ergodic averages for the horocycle flow in terms of a dynamically defined cocycle which has the correct scaling property under the dynamics of the geodesic flow. Such cocycles are closely related to the invariant distributions of Flaminio-Forni and are analogous to the coycles constructed by A. Bufetov for asymptotic foliations of a Markov compactum (and in particular for area-preserving flows on higher genus surfaces).

This is joint work with A. Bufetov.

Discrete Mathematics Seminar
Topic: Geometric selection theorems
Presenter: Boris Bukh, Princeton University and UCLA
Date:  Thursday, April 16, 2009, Time: 2:15 p.m., Location: Fine Hall 224

In combinatorial geometry one frequently wants to select a point or a set of points that meets many simplices of a given family. The two examples are choosing a point in many simplices spanned by points of some P in R^d, and choosing a small set of points which meets the convex hull of every large subset of P (the weak epsilon-net problem). I will present a new class of constructions that yield the first nontrivial lower bound on the weak epsilon-net problem, and improve the best bounds for several other selection problems. Joint work with Jiří Matoušek and Gabriel Nivasch.

Joint Princeton and IAS Number Theory Seminar
Topic: Stable topology of Hurwitz spaces and arithmetic counting problems
Presenter: Jordan Ellenberg, University of Wisconsin - Madison
Date:  Thursday, April 16, 2009, Time: 4:30 p.m., Location: Fine Hall 214

We will discuss some arithmetic counting problems, ranging from the antique (how many squarefree integers are there in [0..N]?) to the au courant (conjectures of Bhargava and Cohen-Lenstra about the distributions of discriminants and of class groups.) When considered over function fields, these conjectures reveal themselves as having to do with stabilization of cohomology of moduli spaces of covers of curves, or Hurwitz spaces. We will report on progress on the topological study of Hurwitz spaces, which leads to information about arithmetic counting problems over function fields over finite fields; for instance, a version of Cohen-Lenstra "correct up to the constant" for F_q(t). If time permits I will try to give a picture of the rather general ensemble of arithmetic counting conjectures suggested by the method (e.g. -- for how many squarefree integers in [0..N] is there a totally real quintic extension of Q with discriminant N?) and explain how to prove versions of these conjectures in the much easier regime where "q goes to infinity first."

(joint work with Akshay Venkatesh and Craig Westerland)

Topology Seminar
Topic: Heegaard Floer homology and pants decompositions
Presenter: Zoltan Szabo, Princeton University
Date:  Thursday, April 16, 2009, Time: 4:30 p.m., Location: Fine Hall 314
Applied Mathematics Seminar
Topic: Recent results on critical nonlinear schrödinger equations
Presenter: Dong Li, IAS
Date:  Friday, April 17, 2009, Time: 1:00 p.m., Location: Fine Hall 224
Abstract: Dr. Li will review some recent progress on critical nonlinear schrödinger equations. This talk will focus on the scattering conjecture and the solitary wave conjecture for both the mass-critical case and the energy-critical case. If time permits, Dr. Li will also discuss some related results for other types of dispersive equations.
Differential Geometry and Geometric Analysis Seminar
Topic: Lagrangian Mean Curvature flow for entire Lipschitz graphs
Presenter: Jingyi Chen, UBC
Date:  Friday, April 17, 2009, Time: 3:00 p.m., Location: Fine Hall 314
Abstract: We prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. As an application of the estimates for the solution, we establish a Bernstein type result for translating solitons. The results are from joint work with Albert Chau and Weiyong He.
Analysis Seminar
Topic: From Boltzmann equation to the incompressible Navier-Stokes-Fourier system with long-range interactions
Presenter: Diogo Arsenio, Courant Institute
Date:  Monday, April 20, 2009, Time: 4:00 p.m., Location: Fine Hall 110

Boltzmann's equation is known to converge, under a certain hydrodynamic regime, to an incompressible Navier-Stokes-Fourier system. It is only recently that the final steps to a mathematically rigorous and complete justification of this hydrodynamic convergence were provided. However, only certain types of intermolecular interactions, still physically unsatisfying, were considered.

We establish this hydrodynamic limit for the physically relevant case of long range intermolecular interactions. In this situation, the difficulty comes from the fact that the Boltzmann collision operator exhibits a rather complex nature due to a non-integrable singularity in the collision kernel.

PACM Colloquium
Topic: Interdisciplinarity in the Age of Networks
Presenter: Jennifer Chayes, Microsoft Corporation
Date:  Monday, April 20, 2009, Time: 4:00 p.m., Location: Fine Hall 214
Abstract: Everywhere we turn these days, we find that networks have become increasing appropriate descriptions of relevant interactions. In the high tech world, we see the Internet, the World Wide Web, mobile phone networks, and a variety of online social networks. In economics, we are increasingly experiencing both the positive and negative effects of a global networked economy. In epidemiology, we find disease spreading over our ever growing social networks, complicated by mutation of the disease agents. In problems of world health, distribution of limited resources, such as water resources, quickly becomes a problem of finding the optimal network for resource allocation. In biomedical research, we are beginning to understand the structure of gene regulatory networks, with the prospect of using this understanding to manage the many diseases caused by gene mis-regulation. In this talk, I look quite generally at some of the models we are using to describe these networks, processes we are studying on the networks, algorithms we have devised for the networks, and finally, methods we are developing to indirectly infer network structure from measured data. In particular, I will discuss models and techniques which cut across many disciplinary boundaries.
Ergodic Theory and Statistical Mechanics Seminar***Pleaset note special date and time
Topic: An explicit approach to the control of Lyapunov exponents
Presenter: Ilya Goldsheid, Queen Mary, University of London
Date:  Tuesday, April 21, 2009, Time: 4:30 p.m., Location: Fine Hall 401
Abstract: I shall discuss a new approach to the proof the exponential growth of products of random matrices. The classical Furstenberg's analysis relies on properties of infinite-dimensional unitary representations. The method I am going to discuss uses finite-dimensional representations and allows one to have a more explicit control over Lyapunov exponents.
Algebraic Geometry Seminar
Topic: Automorphisms mapping a point into a subvariety
Presenter: Bjorn Poonen, MIT
Date:  Tuesday, April 21, 2009, Time: 4:30 p.m., Location: Fine Hall 322
Abstract: Given a variety X, a point x in X, and a subvariety Z of X, is there an automorphism of X mapping x into Z? We prove that this problem is undecidable.
Automorphic Forms and Galois Representations Seminars
Topic: Phi-modules and coefficient spaces for Galois representations
Presenter: George Pappas, Michigan State
Date:  Wednesday, April 22, 2009, Time: 1:30 p.m., Location: Fine Hall 314
Department Colloquium
Topic: TBA
Presenter: Jean Michel Bismut, Universite Paris-Sud
Date:  Wednesday, April 22, 2009, Time: 4:30 p.m., Location: Fine Hall 314
Ergodic Theory and Statistical Mechanics Seminar
Topic: Local entropy and projections of dynamically defined fractals
Presenter: Michael Hochman, Princeton University
Date:  Thursday, April 23, 2009, Time: 2:00 p.m., Location: Fine Hall 401

If a closed subset X of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of X (since the projection is Lipschitz), and also no larger than 1 (since it is a subset of a line). A classical theorem of Marstrand says that for any such X, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to min(1,dim(X)). In general, there can be uncountably many exceptional directions.

An old conjecture of Furstenberg is that if A, B are subsets of [0,1] invariant respectively under x2 and x3 mod 1, then for their product, X=AxB, the only exceptional directions in Marstrand's theorem are the two trivial ones, namely the projections onto the x and y axes. Recently, Y. Peres and P. Shmerkin proved that this is true for certain self-similar fractals, such as regular Cantor sets. I will discuss the proof of the general case, which relies on a method for computing dimension using local entropy estimates. I will also describe some other applications. This is joint work with Pablo Shmerkin.

Discrete Mathematics Seminar
Topic: Packing seagulls in graphs with no stable set of size three
Presenter: Maria Chudnovsky, Columbia University
Date:  Thursday, April 23, 2009, Time: 2:15 p.m., Location: Fine Hall 224

Hadwiger's conjecture is a well known open problem in graph theory. It states that every graph with chromatic number k, contains a certain structure, called a "clique minor" of size k. An interesting special case of the conjecture, that is still wide open, is when the graph G does not contain three pairwise non-adjacent vertices. In this case, it should be true that G contains a clique minor of size t where t >= |V(G)|/2. This remains open, but Jonah Blasiak proved it in the subcase when |V(G)| is even and the vertex set of G is the union of three cliques. Here we prove a strengthening of Blasiak's result: that the conjecture holds if some clique in G contains at least |V(G)|/4 vertices.

This is a consequence of a result about packing ``seagulls''. A seagull in G is an induced three-vertex path. It is not known in general how to decide in polynomial time whether a graph contains k pairwise disjoint seagulls; but we answer this for graphs with no stable sets of size three.

This is joint work with Paul Seymour.

Joint Princeton and IAS Number Theory Seminar
Topic: Toroidal compactifications of certain Kuga families
Presenter: Kai-Wen Lan, Princeton University
Date:  Thursday, April 23, 2009, Time: 4:30 p.m., Location: Fine Hall 214

We will explain how toroidal compactifications of certain Kuga families of abelian varieties over integral models of PEL-type Shimura varieties, including for example all those products of universal abelian schemes, can be constructed by a uniform method. We will also explain some of their applications to the cohomology theories of automorphic bundles.

Topology Seminar
Topic: Annulus open book decompositions and the self linking number
Presenter: Kekiko Kawamuro, IAS
Date:  Thursday, April 23, 2009, Time: 4:30 p.m., Location: Fine Hall 314
Abstract: We introduce a construction of an immersed surface for a null-homologous braid in an annulus open book decomposition. This is hinted by the so called Bennequin surface for a braid in R3. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. Then we compute a self-linking number formula using this embedded surface and observe that the Bennequin inequality is satisfied if and only the contact structure is tight. We also prove that our self-linking formula is invariant (changes by 2) under a positive (negative) braid stabilization which preserves (changes) the transverse knot class.
Analysis Seminar
Topic: Stefan Problem with Surface Tension
Presenter: Yan Guo, Brown University
Date:  Monday, April 27, 2009, Time: 4:00 p.m., Location: Fine Hall 110
PACM Colloquium
Topic: State-of-the-art Computer Simulations of Supernova Explosions
Presenter: Adam Burrows, Astrophysics, Princeton University
Date:  Monday, April 27, 2009, Time: 4:00 p.m., Location: Fine Hall 214

To simulate supernova explosions, one must solve simultaneously the non-linear, coupled partial differential equations of radiation hydrodynamics. What's more, due to a variety of instabilities and asymmetries, this must eventually be accomplished in 3D. The current state-of-the-art is 2D, plus rotation and magnetic fields (assuming axisymmetry). Nevertheless, with the current suite of codes, we have been able to explore the evolution of the high-density, high-temperature, high-speed environment at the core of a massive star at death. It is in this core that the supernova explosion is launched. However, the complexity of the problem has to date obscured the essential physics and mechanisms of the phenomenon, making it indeed one of the "Grand Challenges" of 21st century astrophysics. Requiring forefront numerical algorithms and massive computational resources, the resolution of this puzzle awaits the advent of peta- and exa-scale architectures and the software to efficiently use them. In this talk, I will review the current state of the science and simulations as we plan for the fully 3D, multi-physics capabilities that promise credibly to crack open this obdurate astrophysical nut.

Algebraic Geometry Seminar
Topic: Calabi-Yau threefolds with vanishing third Betti number
Presenter: Chad Schoen, Duke University
Date:  Tuesday, April 28, 2009, Time: 4:30 p.m., Location: Fine Hall 322
Abstract: Smooth, projective, three dimensional, algebraic varieties with trivial canonical sheaf and vanishing third etale Betti number do not exist over fields of characteristic zero. In the past few years a number of examples have been found in positive characteristic. Some of these examples and questions they raise will be discussed.
Mathematical Physics Seminar
Topic: Eigenvalue Statistics for Random CMV Matrices
Presenter: Mihai Stoiciu, Williams College
Date:  Tuesday, April 28, 2009, Time: 4:30 p.m., Location: Jadwin 343
Abstract: CMV matrices are the unitary analogues of one dimensional discrete Schrodinger operators. We consider CMV matrices with random coefficients and we study the statistical distribution of their eigenvalues. For slowly decreasing random coefficients, we show that the eigenvalues are distributed according to a Poisson process. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (clock distribution). For a certain critical rate of decay we obtain the circular beta distribution. This is a joint work with Rowan Killip.
Ergodic Theory and Statistical Mechanics Seminar
Topic: Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics
Presenter: Mikhail Lyubich, State University of New York at Stony Brook
Date:  Thursday, April 30, 2009, Time: 2:00 p.m., Location: Fine Hall 401
Abstract: In a classical work of 1950's, Lee and Yang proved that zeros of the partition functions of the Ising models on graphs always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for a special ``Diamond Hierarchical Lattice". In this case, it can be described in terms of the dynamics of an explicit rational map in two variables. We prove partial hyperbolicity of this map on an invariant cylinder, and derive from it that the Lee-Yang zeros are organized asymptotically in a transverse measure for the central foliation. From the global complex point of view, the zero distributions get interpreted as slices of the Green (1,1)-current on the projective space. It is a joint work with Pavel Bleher and Roland Roeder.
Joint Princeton and IAS Number Theory Seminar
Topic: Symplectic Galois representations over totally real fields.
Presenter: Claus Sorensen, Princeton University
Date:  Thursday, April 30, 2009, Time: 4:30 p.m., Location: Fine Hall 214

We associate p-adic Galois representations to globally generic cusp forms on GSp(4), over a totally real field, with a Steinberg component at some finite place. At places v not dividing p one has local-global compatibility, the local correspondence being that defined by Gan and Takeda. In particular, the rank of the monodromy operator at such a place v is determined by the level of the v-component of the cusp form. Moreover, the Swan conductor is essentially the depth.

Topology Seminar
Topic: On the geometry of space-time
Presenter: Thierry Barbot, Universite d'Avignon
Date:  Thursday, April 30, 2009, Time: 4:30 p.m., Location: Fine Hall 314
Abstract: In the relativistic point of view, the geometry of space should evolve with time, in a manner directed by Einstein equations. I will briefly summarize two interesting aspects with open questions:

-- Bianchi cosmologies: these are 3+1 dimensional lorentzian manifolds
satisfying Einstein equation (for this talk, in the void) and admitting a
locally free isometric spacelike action by a 3-dimensional Lie group. The space
of Bianchi cosmologies, as a whole, admits a very rich and interesting dynamical
feature which has not yet been fully investigated.

-- Constant curvature case: interesting and paradigmatic cases of solutions of
Eintein equations (even if physically questionable) are space-times with
constant curvature (i.e locally modeled on Minkowski, de Sitter or anti-de
Sitter space). In the 2+1 dimensional case, G. Mess gave a very nice
description of these space-times and a close connection with Teichmüller space.
In the higher dimensional case, they give rise to a proof of the following

Let Gamma be a cocompact lattice in SO(1,n) (n >= 2). Then, in the space
Rep(Gamma, SO(2,n)) of representations of Gamma into SO(2,n), every
representation contained in the connected component containing the inclusion
Gamma subset SO(1,n) subset SO(2,n) is faithfull and discrete.
MAY 2009
Analysis Seminar ***Please note special date and time
Topic: An Extension of the Stability Theorem of the Minkowski Space in General Relativity
Presenter: Lydia Bieri, Harvard University
Date:  Wednesday, May 6, 2009, Time: 5:00 p.m., Location: Fine Hall 110
Abstract: We present a generalization of the celebrated results by D. Christodoulou and S. Klainerman for solutions of the Einstein vacuum equations in General Relativity. In 'The global nonlinear stability of the Minkowski space', they showed that every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is no longer bounded in $L^{\infty}$. As a major result and as a consequence of our relaxed assumptions, we encounter in our work borderline cases, which we discuss in this talk as well. The main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime.
Ergodic Theory and Statistical Mechanics Seminar
Topic: TBA
Presenter: Ilya Vinogradov and Francesco Cellarosi, Princeton University
Date:  Thursday, May 7, 2009, Time: 2:00 p.m., Location: Fine Hall 401