

NOVEMBER 2008 


Statistical Mechanics Seminar 
Topic: 
Current large deviations in stochastic systems 
Presenter: 
Thierry Bodineau, Ecole Normale, Paris 
Date: 
Wednesday, November 5, 2008, Time: 2:00 p.m., Location: Jadwin 343 
Abstract: 
Using the hydrodynamic limit theory, we will review the large deviations of the heat current through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. In particular, we will discuss the occurrence of dynamical phase transitions which may occur for some models and the structure of the long range correlations in systems maintained off equilibrium. 


Department Colloquium 
Topic: 
Cremona transformations and homeomorphisms of topological surfaces 
Presenter: 
János Kollár, Princeton University 
Date: 
Wednesday, November 5, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
See http://www.math.princeton.edu/colloq/co_kollar.pdf 


Graduate Student Seminar 
Topic: 
Embedded Surfaces in 4Manifolds 
Presenter: 
Josh Green, Princeton University 
Date: 
Thursday, November 6, 2008, Time: 12:30 p.m., Location: Fine Hall 314 
Abstract: 
Given a 2dimensional homology class in a closed 4manifold, you can try to represent it by an embedded, closed, orientable surface. What is the minimum genus of such a surface? Come find out. 


Discrete Mathematics Seminar 
Topic: 
On the Singular Probability of Random Discrete Matrices 
Presenter: 
Philip Matchett Wood, Rutgers University 
Date: 
Thursday, November 6, 2008, Time: 2:15 p.m., Location: Fine Hall 224 
Abstract: 
Let $n$ be a large integer and $M_n$ be an $n$ by $n$ random matrix whose entries are independent (but not necessarily identically distributed) random variables. The main goal of this paper is to prove a general upper bound for the probability that $M_{n}$ is singular.
For a constant $0< p< 1$ and a constant positive integer $r$, we will define a property called $p$bounded of exponent $r$. Our main result shows that if the entries of $M_n$ satisfy this property, then the probability that $M_n$ is singular is at most $(p^{1/r} + o(1))^n$. Here are a few sample corollaries of this theorem:
(1) If the distribution of each entry has mass at most p on a single point, then the singular probability is at most $(p+o(1))^{n/2}$. In the special case of random Bernoulli matrices, this improves the previous bound $(3/4+o(1))^n$ due to Tao and Vu.
(2) If the entries are iid with distribution which puts mass 1/2 on 0 and 1/4 on 1 and 1, then the singular probability is at most $(1/2+o(1))^n$. This bounds is sharp, as the probability of having a zero row is (1/2+o(1))^n.
The proof refines the approach from KahnKomlosSzemeredi and TaoVu and makes a critical use of TaoVu inverse theorem. (Joint work with J. Bourgain and V. Vu). 


Number Theory Seminar 
Topic: 
Weight Cycling and Serretype Conjectures 
Presenter: 
Florian Herzig, Northwestern University 
Date: 
Thursday, November 6, 2008, Time: 4:30 p.m., Location: Fine Hall 214 
Abstract: 
Suppose that rho is a threedimensional modular mod p Galois representation whose restriction to the decomposition groups at p is irreducible and generic. If rho is modular in some (Serre) weight, then a representationtheoretic argument shows that it also has to be modular in certain other weights (we can give a short list of possibilities). This goes back to an observation of Buzzard for GL_2. Previously we formulated a Serretype conjecture on the possible weights of rho. Under the assumption that the weights of rho are contained in the predicted weight set, we apply the above weight cycling argument to show that rho is modular in precisely all the nine predicted weights. This is joint work with Matthew Emerton and Toby Gee. 


Topology Seminar **** CANCELLED **** 
Topic: 
Congruence subgroup problem for mapping class groups 
Presenter: 
Ben McReynolds, University of Chicago

Date: 
Thursday, November 6, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
I will discuss the congruence subgroup problem for mapping class groups, a problem that generalizes the classical one for arithmetic groups. I will discuss an unpublished proof by W. Thurston for an affirmative answer to this problem for genus zero mapping class groups. Time permitting, I will discuss the current state of this problem. 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
The singular set of C^{1} smooth surfaces in the Heisenberg group 
Presenter: 
JihHsin, Academica Sinica Taipei 
Date: 
Friday, November 7, 2008, Time: 3:00 p.m., Location: Fine Hall 314 


Analysis Seminar 
Topic: 
Almost global wellposedness of the 2D full water wave problem 
Presenter: 
Sijue Wu, University of Michigan, Ann Arbor 
Date: 
Monday, November 10, 2008, Time: 4:00 p.m., Location: Fine Hall 110 
Abstract: 
We consider the problem of global in time existence and uniqueness of solutions of the 2D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of form $\epsilon\Psi$, where $T$ depends only on $\Psi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/{\epsilon}}]$. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. 


Geometry, Representation Theory, and Moduli Seminar 
Topic: 
Universal equations for GromovWitten invariants 
Presenter: 
X. Liu, Notre Dame 
Date: 
Monday, November 10, 2008, Time: 4:00 p.m., Location: Fine Hall 314 
Abstract: 
It is well known that relations in tautological rings of moduli spaces of curves produce differential equations for generating functions of GromovWitten invariants for all compact symplectic manifolds. We call such equations universal equations. These equations are very helpful in understanding structures of GromovWitten theory and played an important role in the so called Virasoro conjecture. However for finding explcit universal equations seems to be a hard problem especially when genus is large. I will talk about a genus3 topological recursion relation found together with T. Kimura and some recently discovered universal equations conjectured by K. Liu and H. Xu and proved in a joint work with R. Pandharipande. 


PACM Colloquium 
Topic: 
Symmetric functions of a large number of variables 
Presenter: 
PierreLouis Lions, College de France and Ecole Polytechnique 
Date: 
Monday, November 10, 2008, Time: 4:00 p.m., Location: Fine Hall 214 


Special Topology Seminar ***Please note special date, time, and location 
Topic: 
On the Number of Solutions to Asymptotic Plateau Problem 
Presenter: 
Baris Coskunuzer, Koc University, Turkey

Date: 
Tuesday, November 11, 2008, Time: 3:30 p.m., Location: Fine Hall 401 
Abstract: 
We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyperbolic nspace, is dense in the space of all codimension1 closed submanifolds at infinity. In dimension 3, we also prove that the set of uniqueness curves in asymptotic sphere for area minimizing planes is generic in the set of Jordan curves at infinity. We also give a nonuniqueness result by showing existence of simple closed curves in the sphere at infinity which are the asymptotic boundaries of more than one area minimizing surfaces. 


Algebraic Geometry Seminar 
Topic: 
Vanishing and torsionfree theorems for the log minimal model program 
Presenter: 
Osamu Fujino, Nagoya University 
Date: 
Tuesday, November 11, 2008, Time: 4:30 p.m., Location: Fine Hall 322 
Abstract: 
We will discuss Ambro's formulation of Kollár's injectivity, torsionfree, and vanishing theorems. It is indispensable for the study of the log minimal model program for log canonical pairs. 


Statistical Mechanics Seminar 
Topic: 
On quantum, stationary, nonequilibrium states 
Presenter: 
Michael Sigal, University of Toronto 
Date: 
Wednesday, November 12, 2008, Time: 2:00 p.m., Location: Jadwin 343 
Abstract: 
In this talk I will describe recent results on existence and dynamical stability of stationary, nonequilibrium states in certain models of quantum statistical mechanics. This is a joint work with Marco Merkli and Matthias Mueck. 


Department Colloquium 
Topic: 
Fundamental lemma and Hitchin fibration 
Presenter: 
Bao Châu Ngô, Institute for Advanced Study 
Date: 
Wednesday, November 12, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
The fundamental lemma is an identity of orbital integrals on padic reductive groups which was stated precisely by Langlands and Shelstads as a conjecture in the 80's. We now have a proof due to the efforts of many peoples with many ingredients. I will only explain how a certain particular type of geometry like affine Springer fibers and Hitchin were helpful in this proof. 


Number Theory Seminar 
Topic: 
Faltings' height of CM cycles and Derivative of $L$functions 
Presenter: 
Tonghai Yang, University of Wisconsin at Madison 
Date: 
Thursday, November 13, 2008, Time: 4:30 p.m., Location: Fine Hall 214 
Abstract: 
In this talk, we first describe a systematic way to construct `automorphic Green functions' for Kudla's special divisors on a Shimura variety of orthogonal type $(n, 2)$. We then give an explicit formula for their values at a CM cycle. This formula suggests a direct relation between the Faltings' height of these CM cycles with the central derivative of some RankinSelberg $L$function. As an application, we also give an `analytic proof' of the GrossZagier formula without computing the local intersection numbers at finite primes. This is a joint work with Jan Bruinier. 


Topology Seminar 
Topic: 
Minimal intersection and selfintersection of curves on surfaces 
Presenter: 
Moira Chas, SUNY Stony Brook 
Date: 
Thursday, November 13, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
Consider the set of free homotopy classes of directed closed curves on an oriented surface and denote by V the Zmodule generated by this set. Goldman discovered a Lie algebra structure on this module, obtained by combining the geometric intersection of curves with the usual loop product. Later on, Turaev found a Lie coalgebra structure on the quotient of V by the one dimensional subspace generated by the trivial loop. Moreover, the Goldman Lie bracket passes to this quotient and both operations satisfy the identities of a Lie bialgebra. This Lie bialgebra has a purely combinatorial presentation. When the surface has nonempty boundary, one can use this presentation to prove it is possible to compute the minimal number of selfintersection points of representatives of a free homotopy class A by means of Lie bialgebra in two different ways: firstly, counting (with multiplicity) the number of terms of the cobracket of powers of A and secondly, counting (with multiplicity) the number of terms of the bracket betweem different powers of A. From this Lie bialgebra structure one can recover the minimal intersection number of two free homotopy classes, provided that one of these classes contains a simple representative. The tools used to prove this result suggest that it would be possible to prove analogous results for the String bracket on certain closed three manifolds. Some of these results are joint work with Fabiana Krongold. 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space 
Presenter: 
Xiaodong Wang, Michigan State University 
Date: 
Friday, November 14, 2008, Time: 3:00 p.m., Location: Fine Hall 314 
Abstract: 
Complete Riemannian manifolds with nonnegative Ricci curvature have been well studied. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a rigidity theorem which characterizes hyperbolic manifolds. The proof uses idea from potential theory and Brownian motion on Riemannian manifolds 


PACM Colloquium 
Topic: 
Multiscale Methods for Hydrodynamics of Polymer Chains in Solution 
Presenter: 
Aleksandar Donev, Lawrence Livermore National Laboratory 
Date: 
Monday, November 17, 2008, Time: 4:00 p.m., Location: Fine Hall 214 
Abstract: 
The hydrodynamics of complex fluids, such as polymer solutions and colloidal suspensions, has attracted great interest due to recent advances in fabrication of micro and nanofluidic devices. I will first review recent advances in mesoscopic numerical methods for simulating the interaction between complex fluid flow and suspended macro molecules and structures. Computational issues at play include coarsegraining to bridge the large gap in timescales and length scales, coupling between disparate methods such as molecular dynamics and NavierStokes solvers, the inclusion of thermal fluctuations.
I will then present my recent work at LLNL to develop novel particle methods for modeling polymer chains in flow. Typically, Molecular Dynamics (MD) is used for the polymer chains, and the solvent is modeled with a mesoscopic method. In our algorithm, termed Stochastic EventDriven Molecular Dynamics (SEDMD) [A. Donev and A. L. Garcia and B. J. Alder, J. Comp. Phys., 227(4), 26442665, 2008], polymers are modeled as chains of hard spheres and the solvent is modeled using a densefluid generalization of the Direct Simulation Monte Carlo (DSMC) method [Phys. Rev. Lett., 101, 075902, 2008]. Even with all of the speedup compared to bruteforce MD the algorithm is still timeconsuming due to the large number of solvent particles necessary to fill the computational domain. It is natural to restrict the particle model only to regions close to a polymer chain and use a lowerresolution continuum model elsewhere. I will present a hybrid method that couples an explicit fluctuating compressible NavierStokes solver with the particle method. The coupling is fluxbased and generalizes previous work [J. B. Bell and A. L. Garcia and S. A. Williams, SIAM Multiscale Modeling and Simulation, 6, 12561280, 2008] to dense fluids as appropriate for polymer problems.
I will conclude with a look into the challenges of developing a simulation methodology capable of simulating macroscopic flows of complex fluids with atomistic fidelity. 


Algebraic Geometry Seminar 
Topic: 
Equivalences from geometric sl_2 actions 
Presenter: 
Sabin Cautis, Rice University 
Date: 
Tuesday, November 18, 2008, Time: 4:30 p.m., Location: Fine Hall 322 
Abstract: 
We explain how sl_2 actions on derived categories of coherent sheaves can be used to construct new derived equivalences. The example I will describe in detail is an sl_2 action via correspondences on the cotangent bundles of Grassmannians which generalizes the basic Mukai flop. More generally we can construct an action on the derived category of coherent sheaves on quiver varieties which lifts Nakajima's action on their cohomology. (joint with Joel Kamnitzer and Anthony Licata) 


Statistical Mechanics Seminar 
Topic: 
Bosons in rapid rotation 
Presenter: 
Jakob Yngvason, University of Vienna 
Date: 
Wednesday, November 19, 2008, Time: 2:00 p.m., Location: Jadwin 343 
Abstract: 
One of the most remarkable manifestations of superfluidity in BoseEinstein condensates is the way the condensate responds to rotation. In a superfluid the rotation of the container confining the fluid leads to the formation of vortices with quantized circulation. This phenomenon can be studied through the solutions of a nonlinear Schrodinger equation, the GrossPitaevskii equation. In the lecture recent results concerning the appearance and disappearance of vortex lattices in rapidly rotating BoseEinstein condensates will be discussed. 


Department Colloquium 
Topic: 
TBA 
Presenter: 
Valery Alexeev, University of Georgia 
Date: 
Wednesday, November 19, 2008, Time: 4:30 p.m., Location: Fine Hall 314 


Discrete Mathematics Seminar 
Topic: 
TBA 
Presenter: 
Melvyn B. Nathanson, CUNY 
Date: 
Wednesday, November 20, 2008, Time: 2:15 p.m., Location: Fine Hall 224 


Topology Seminar 
Topic: 
TBA 
Presenter: 
Peter Ozsvath, Columbia University 
Date: 
Thursday, November 20, 2008, Time: 4:30 p.m., Location: Fine Hall 314 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
An Exotic Sphere with Positive Sectional Curvature 
Presenter: 
Fred Wilhelm, UCR 
Date: 
Friday, November 21, 2008, Time: 3:00 p.m., Location: Fine Hall 314 
Abstract: 
I'll discuss joint work with Peter Petersen that shows that the GromollMeyer exotic 7sphere admits a metric of positive sectional curvature. I'll discuss the history of the problem and give a coarse outline of the proof. 


Analysis Seminar 
Topic: 
TBA 
Presenter: 
Soonsik Kwon, Princeton University 
Date: 
Monday,November 24, 2008, Time: 4:00 p.m., Location: Fine Hall 110 


PACM Colloquium 
Topic: 
Emissions Market Models 
Presenter: 
René Carmona, PACM & ORFE, Princeton University 
Date: 
Monday, November 24, 2008, Time: 4:00 p.m., Location: Fine Hall 214 
Abstract: 
The main goal of the talk is to introduce a new capandtrade scheme design for the control and the reduction of atmospheric pollution. The tools developed for the purpose of the study are intended to help policy makers and regulators understand the pros and cons of the emissions markets at a quantitative level.
We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits by the regulator in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from those firms anticipating that they will not need them, creating a financial market for pollution credits.
Our model captures most of the features of the European Union Emissions Trading Scheme. We show existence of an equilibrium and uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the optimal production and trading strategies of the firms. We choose the electricity market in Texas to illustrate numerically the qualitative properties observed during the implementation of the first phase of the European Union capandtrade CO2 emissions scheme, comparing the results of capandtrade schemes to the Business As Usual benchmark. In particular, we confirm the presence of windfall profits criticized by the opponents of these markets. We also demonstrate the shortcomings of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which, despite its ease of implementation, leads to smaller windfall profits than the standard scheme. 


Algebraic Geometry Seminar 
Topic: 
Finiteness theorems for algebraic groups over function fields 
Presenter: 
Brian Conrad, Stanford University 
Date: 
Tuesday, November 25, 2008, Time: 4:30 p.m., Location: Fine Hall 322 
Abstract: 
If X is a smooth variety over a global field k, G is an algebraic group over k equipped with an action on X, and x is a point in X(k) then it is natural to ask how the property of x' in X(k) being in the G(k)orbit of x compares with being in the G(k_v)orbit of x for all places v of k. In general there is a nontrivial "localtoglobal" obstruction space, but one can ask if it is finite. Even when G is semisimple, this finiteness problem leads to the consideration of the isotropy group G_x that is generally not connected or reductive (or even smooth when char(k) > 0). In the number field case the finiteness of these obstruction spaces was proved by Borel and Serre long ago, but their method used characteristic 0 in an essential way. Recently in joint work with Gabber and G. Prasad we have developed a theory of "pseudoreductive groups" which is a very useful tool to prove results for general affine algebraic groups in the function field case that were previously known only in the reductive case. In particular, this work makes it possible to prove the analogue of the BorelSerre finiteness result over function fields (away from char. 2 for now). The first part of the talk will explain a bit about the theory of pseudoreductive groups, and the rest of the talk will show how it is used to establish the finiteness of the localtoglobal obstruction spaces in the function field case (in char. > 2). If time permits we will also discuss an application to the problem of whether the kisomorphism class of a projective kvariety is determined (up to "finite ambiguity") by its k_visomorphism class for all places v of k (a problem solved by Mazur over number fields, once again making essential use of characteristic 0). 


DECEMBER 2008 


Analysis Seminar 
Topic: 
TBA 
Presenter: 
Alessio Figalli, Université de Nice SophiaAntipolis and Ecole Polytechnique 
Date: 
Monday, December 1, 2008, Time: 4:00 p.m., Location: Fine Hall 110 


PACM Colloquium 
Topic: 
TBA 
Presenter: 
Ingrid Daubechies, PACM,, Princeton University 
Date: 
Monday, December 1, 2008, Time: 4:00 p.m., Location: Fine Hall 214 


Statistical Mechanics Seminar 
Topic: 
Local and Global Structure of Stationary States of Macroscopic Systems 
Presenter: 
Joel Lebowitz, Rutgers University 
Date: 
Wednesday, December 3, 2008, Time: 2:00 p.m., Location: Jadwin 343 
Abstract: 
The microscopic structure of a macroscopic system in a steady state is described locally, i.e. at a suitably scaled macroscopic point $x$, by a time invariant measure of the corresponding infinite system with translation invariant dynamics. This measure may be extremal, with good decay of correlations, or a superposition of extremal measures, with weights depending on $x$ (and possibly even on the way one scales).
I will illustrate the above by some exact results for 1D lattice systems with two types of particles (plus holes) evolving according to variants of the simple asymmetric exclusion process, in open or closed systems. Somewhat surprisingly, the spatially asymmetric local dynamics satisfy (in some cases) detailed balance with respect to a global Gibbs measure with long range pair interactions. 


Department Colloquium 
Topic: 
TBA 
Presenter: 
Bruce Kleiner, Yale University 
Date: 
Wednesday, December 3, 2008, Time: 4:30 p.m., Location: Fine Hall 314 


Number Theory Seminar 
Topic: 
Mock modular forms 
Presenter: 
Sandors Zwegers, University College Dublin 
Date: 
Thursday, December 4, 2008, Time: 4:30 p.m., Location: IAS SH101 
Abstract: 
The main motivation for the theory of mock modular forms comes from the desire to provide a framework in which we can understand the mysterious and intriguing mock theta functions, as well as related functions, like Appell functions and theta functions associated to indefinite quadratic forms.
In this talk, we will describe the nature of the modularity of the original mock theta functions, formulate a general definition of mock modular forms, and describe further examples. We will also consider a generalization to higher depth mock modular forms 


Topology Seminar 
Topic: 
TBA 
Presenter: 
Elisenda Grigsby, Columbia University 
Date: 
Thursday, December 4, 2008, Time: 4:30 p.m., Location: Fine Hall 314 


PACM Colloquium 
Topic: 
Computational Astrophysics and the Dynamics of Accretion Disks 
Presenter: 
James M. Stone, PACM & Astrophysical Sciences 
Date: 
Monday, December 8, 2008, Time: 4:00 p.m., Location: Fine Hall 214 
Abstract: 
he ever increasing performance of computer hardware and improvements to the accuracy of numerical algorithms are revolutionizing scientific research in many disciplines, but perhaps none more so than astrophysics. I will begin by describing why computation is crucial for the solution of a variety of problems at the forefront of research in astronomy and astrophysics, with particular emphasis on understanding accretion flows onto black holes. I will outline the challenge of developing, testing, and implementing numerical algorithms for the investigation of these problems. Finally, I will present results that demonstrate how computation can help us understand the basic physics of magnetized accretion disks. 


Geometry, Representation Theory, and Moduli Seminar 
Topic: 
TBA 
Presenter: 
Kai Behrend, University of British Columbia 
Date: 
Monday, December 8, 2008, Time: 4:00 p.m., Location: Fine Hall 314 


Algebraic Geometry Seminar 
Topic: 
TBA 
Presenter: 
David Smyth, Harvard University 
Date: 
Tuesday, December 9, 2008, Time: 4:30 p.m., Location: Fine Hall 322 


Department Colloquium 
Topic: 
TBA 
Presenter: 
Kai Behrend, University of British Columbia 
Date: 
Wednesday, December 10, 2008, Time: 4:30 p.m., Location: Fine Hall 314 


Discrete Mathematics Seminar 
Topic: 
TBA 
Presenter: 
Paul Wollan, University of Hamburg 
Date: 
Wednesday, December 11, 2008, Time: 2:15 p.m., Location: Fine Hall 224 


Number Theory Seminar 
Topic: 
Langlands functoriality and the inverse problem in Galois theory 
Presenter: 
Gordon Savin, University of Utah 
Date: 
Thursday, December 11, 2008, Time: 4:30 p.m., Location: IAS SH101 
Abstract: 
In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type B_n, C_n and G_2 as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with ladic representations attached to selfdual automrophic representations of GL(n). 



