

APRIL 2009 


Department Colloqium 
Topic: 
TBA 
Presenter: 
Igor Rodnianski, Princeton University 
Date: 
Wednesday, April 8, 2009, Time: 4:30 p.m., Location: Fine Hall 314 


Graduate Student Seminar 
Topic: 
MathaiQuillen's Thom form and AtiyahHirzebruch's RiemannRoch theorem 
Presenter: 
Guangbo Xu, Princeton University 
Date: 
Thursday, April 9, 2009, Time: 12:30 p.m., Location: Fine Hall 314 
Abstract: 
After Hirzebruch's generalization of the classical RiemannRoch formula, Grothendieck extended this result to a relative version. Then Atiyah and Hirzebruch gave a "differentiable analogue" of Grothendieck's theorem, which is called AtiyahHirzebruch's RiemannRoch theorem. In spite of the original topological proof, I will present another one, in a more differentialgeometric flavor, using MathaiQuillen's construction of the Thom form. 


Ergodic Theory and Statistical Mechanics Seminar 
Topic: 
On mixing properties of locally Hamiltonian flows on surfaces 
Presenter: 
Corinna Ulcigrai, University of Bristol 
Date: 
Thursday, April 9, 2009, Time: 2:00 p.m., Location: Fine Hall 401 
Abstract: 
We consider areapreserving flows on surfaces which are locally given by smooth Hamiltonians. It turns out that the presence or absence of mixing depends on the type of fixed points. We proved in our PhD thesis that the presence of centers is generically enough to create mixing. Recently we showed that if such flows have only saddles, they are generically not mixing, but weakly mixing. The results use the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges. 


Discrete Mathematics Seminar 
Topic: 
Linear Programming Relaxations for the TSP 
Presenter: 
William Cook, Georgia Tech. 
Date: 
Thursday, April 9, 2009, Time: 2:15 p.m., Location: Fine Hall 224 
Abstract: 
The most successful solution approaches to the traveling salesman problem are based on lower bounds obtained from linear programming relaxations. We will discuss a number of open problems concerning this approach, with emphasis on topics that could improve the practical performance of LPbased algorithms. 


Topology Seminar ***Please note special time and location 
Topic: 
Existence and rigidity of pseudoAnosov flows transverse to Rcovered foliations 
Presenter: 
Sergio Fenley, Princeton University and Florida StateUniversity 
Date: 
Thursday, April 9, 2009, Time: 4:00 p.m., Location: Fine Hall 401 
Abstract: 
PseudoAnosov flows are extremely common in three manifolds and they are very useful. How many pseudoAnosov flows are there in a manifold up to topological conjugacy? We analyse this question in the context of flows transverse to a given foliation F. We prove that if F is Rcovered (leaf space in the universal cover is the real numbers) then there are at most two pseudoAnosov flows transverse to F. In addition if there are two, then the manifold is hyperbolic and the the foliation F blows down to a foliation topologically conjugate to the stable foliation of a particular type of an Anosov flow. The results use the topological theory of pseudoAnosov flows, the universal circle for foliations and the geometric theory of Rcovered foliations. We also discuss the existence of transverse pseudoAnosov flows in this setting. 


Joint Princeton University and IAS Number Theory Seminar 
Topic: 
Eigenvarieties and $p$adic families of finite slope automorphic representations 
Presenter: 
Eric Urban, Columbia University 
Date: 
Thursday, April 9, 2009, Time: 4:30 p.m., Location: Fine Hall 214 


Applied Mathematics Seminar 
Topic: 
Stochastic finite element approximations of elliptic problems of higher stochastic order 
Presenter: 
Xiaoliang Wan, Princeton University  PACM 
Date: 
Friday, April 10, 2009, Time: 1:00 p.m., Location: Fine Hall 224 
Abstract: 
In this talk we will address numerical methods for two stochastic elliptic models, where the coefficients are perturbed by colored noise or white noise. An overview of the development of numerical methods will be given for the first model. We will focus on a stochastic Galerkin finite element method for the second model. Such a model is unbiased in that the expectation of the solution solves the same equation with statistically averaged coefficients. The developed numerical algorithms are based on finite element discretization in the physical space, and Wiener chaos expansion in the probability space. Since in many practically important examples solutions of the stochastic elliptic SPDEs have infinite variance, we investigate the convergence of our algorithms in appropriately weighted Wiener chaos spaces. The convergence is studied both theoretically and numerically. We also provide a comparison of the two aforementioned stochastic elliptic models through a numerical study. 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
A Second Boundary Value Problem for special Lagrangian submanifolds 
Presenter: 
Micah Warren, Princeton University 
Date: 
Friday, April 10, 2009, Time: 3:00 p.m., Location: Fine Hall 314 
Abstract: 
Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle. 


Joint Princeton University and Institute for Advanced Study Analysis Seminar 
Topic: 
$h$Principle and fluid dynamics 
Presenter: 
Camillo De Lellis, Universitaet Zuerich 
Date: 
Monday, April 13, 2009, Time: 4:00 p.m., Location: Fine Hall 110 
Abstract: 
In the early nineties Scheffer produced a complicated example of a nontrivial weak solution to the incompressible Euler equations, having compact support in space and time. Subsequent papers by Shnirelman produced other examples of quite irregular solutions by different, yet complicated, methods.
In a recent joint work with L\'aszl\'o Sz\'ekelyhidi we have used a suitable ``$h$principle'' to produce solutions with the same behavior in a relatively simple way. Our approach answers to further questions left open by the works of Scheffer and Shnirelman and might be relevant in understanding a longstanding conjecture of Onsager. The same kind of analysis has relevant applications also to the theory of hyperbolic systems of conservation laws and shares some surprising similarities with aspects of the theory of fully developed turbulence.
***Please note that there will be an additional talk by the speaker at IAS 


PACM Colloquium 
Topic: 
Surface Correspondence via Discrete Uniformization 
Presenter: 
Yaron Lipman, PACM/Computer Scienc 
Date: 
Monday, April 13, 2009, Time: 4:00 p.m., Location: Fine Hall 214 
Abstract: 
Many appliedscience fields like medical imaging, computer graphics and biology use meshes to model surfaces. It is a challenging problem to determine whether, how and to what extent such surfaces correspond to each other, e.g. to see whether they are differently parametrized views of one object, or whether they indicate movement of part of an object with respect to its other parts. In this talk we will show how the Uniformization theory can be used to establish correspondences between simplyconnected surfaces. We will present an algorithm for automatically finding corresponding points between two discrete surfaces (meshes). The algorithm is based on the observation that the correspondence problem between nearly isometric surfaces is a low dimensional problem in practice, which is well characterized by the Mobius group of fractional linear transformations. 


Analysis Seminar ***Please note special time 
Topic: 
New results for reactiondiffusion equations arising from reversible chemistry 
Presenter: 
Laurent Desvillettes, ENS Cachan 
Date: 
Monday, April 13, 2009, Time: 5:00 p.m., Location: Fine Hall 110 
Abstract: 
Entropy/entropy dissipation methods have been used with success lately in the study of the large time behavior of kinetic equations, nonlinear diffusions, etc., and have led to the development of the concept of hypocoercivity. They are also very useful in the context of reactiondiffusion equations (especially when they are derived from reversible chemistry), where they lead to new results of convergence to equilibrium as well as new results of existence of weak and strongs solutions. We shall detail some of those results, together with their links with recent works on coagulationfragmentation models, and the use of results of regularity for singular parabolic problems. 


Algebraic Geometry Seminar 
Topic: 
Vector bundles with sections 
Presenter: 
Brian Osserman, UC Davis 
Date: 
Tuesday, April 14, 2009, Time: 4:30 p.m., Location: Fine Hall 322 
Abstract: 
Classical BrillNoether theory studies, for given g, r, d, the space of line bundles of degree d with r+1 global sections on a curve of genus g. We will review the main results in this theory, and the role of degeneration techniques in proving them, and then we will discuss the situation for higherrank vector bundles, where even the most basic questions remain wide open. 


Mathematical Physics Seminar 
Topic: 
An Asymptotic Expansion for the Dimer Lambda_d 
Presenter: 
Paul Federbush, University of Michigan 
Date: 
Tuesday, April 14, 2009, Time: 4:30 p.m., Location: Jadwin 343 
Abstract: 
The dimer problem is to count the number of ways a ddimensional "chessboard" can be completely covered by nonoverlapping dimers (dominoes), each dimer covering two nearest neighbor boxes. The number is ~exp(Lambda_d*V) as the volume V goes to infinity. It has been long known Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)1) We derive an asymptotic expansion whose first few terms are Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)1) +(1/8)(1/d) + (5/96)(1/d2) + (5/64)(1/d3) The last term here was calculated by computer, and we conjecture the next term will never be explicitly computed ( just by reason of required computer time ). The expansion is not yet rigorously established. 


Department Colloqium 
Topic: 
TBA 
Presenter: 
Giovanni Forni, University of Maryland 
Date: 
Wednesday, April 15, 2009, Time: 4:30 p.m., Location: Fine Hall 314 


Ergodic Theory and Statistical Mechanics Seminar 
Topic: 
TBA. 
Presenter: 
Giovanni Forni, University of Maryland 
Date: 
Thursday, April 16, 2009, Time: 2:00 p.m., Location: Fine Hall 401 


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 
Abstract: 
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 epsilonnet problem). I will present a new class of constructions that yield the first nontrivial lower bound on the weak epsilonnet 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 
Abstract: 
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 CohenLenstra 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 CohenLenstra "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 


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 NavierStokesFourier system with longrange interactions 
Presenter: 
Diogo Arsenio, Courant Institute 
Date: 
Monday, April 20, 2009, Time: 4:00 p.m., Location: Fine Hall 110 
Abstract: 
Boltzmann's equation is known to converge, under a certain hydrodynamic regime, to an incompressible NavierStokesFourier 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 nonintegrable 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 misregulation. 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 
Topic: 
An explicit approach to the control of Lyapunov exponents 
Presenter: 
Ilya Goldsheid, Queen Mary, University of London 
Date: 
Tuesday, April 21, 2009, Time: 2:00 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 infinitedimensional unitary representations. The method I am going to discuss uses finitedimensional 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. 


Department Colloqium 
Topic: 
TBA 
Presenter: 
J. M. Bismut 
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 
Abstract: 
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 selfsimilar 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: 
TBA 
Presenter: 
Maria Chudnovsky, Columbia University 
Date: 
Thursday, April 23, 2009, Time: 2:15 p.m., Location: Fine Hall 224 


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 nullhomologous 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 selflinking 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 selflinking 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: 
Stateoftheart 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 
Abstract: 
To simulate supernova explosions, one must solve simultaneously the nonlinear, 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 stateoftheart 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 highdensity, hightemperature, highspeed 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 exascale 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, multiphysics capabilities that promise credibly to crack open this obdurate astrophysical nut. 


Algebraic Geometry Seminar 
Topic: 
TBA 
Presenter: 
Chad Schoen, Duke University 
Date: 
Tuesday, April 28, 2009, Time: 4:30 p.m., Location: Fine Hall 322 


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: 
LeeYang 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 LeeYang 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. 


Topology Seminar 
Topic: 
On the geometry of spacetime 
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 3dimensional 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 spacetimes with
constant curvature (i.e locally modeled on Minkowski, de Sitter or antide
Sitter space). In the 2+1 dimensional case, G. Mess gave a very nice
description of these spacetimes and a close connection with Teichmüller space.
In the higher dimensional case, they give rise to a proof of the following
theorem:
Theorem:
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 



