

APRIL 2009 


Automorphic Forms and Galois Representations Seminars 
Topic: 
Phimodules 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: 
The hypoelliptic Dirac operator 
Presenter: 
Jean Michel Bismut, Universite ParisSud 
Date: 
Wednesday, April 22, 2009, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
See http://www.math.princeton.edu/colloq/bismut.pdf 


Graduate Student Seminar 
Topic: 
Local entropy and projections of dynamically defined fractals 
Presenter: 
Kevin Hughes, Princeton University 
Date: 
Thursday, April 23, 2009, Time: 12:30 p.m., Location: Fine Hall 314 
Abstract: 
What's the smallest area a subset of the plane containing a unit line segment in every direction can have? Besicovitch showed that you can get 0 measure and Fefferman used the existence of this set to provide a solution to the ball multiplier problem. Kakeya sets have been important in understanding other phenomena like the BochnerRiesz conjecture as the Hausdorff/Minkowski dimension of Kakeya sets relate to the two. Building on Fefferman's work, Bourgain improved our understanding introducing an ingenious bush argument . I will use the analytic story to motivate the corresponding problems for varieties over finite fields and recent work done there. 


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 ***Please note special time and location 
Topic: 
Packing seagulls in graphs with no stable set of size three 
Presenter: 
Maria Chudnovsky, Columbia University 
Date: 
Thursday, April 23, 2009, Time: 3:30 p.m., Location: Fine Hall 801 
Absbtract: 
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 nonadjacent 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 threevertex 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: 
KaiWen Lan, Princeton University 
Date: 
Thursday, April 23, 2009, Time: 4:30 p.m., Location: Fine Hall 214 
Abstract: 
We will explain how toroidal compactifications of certain Kuga families of abelian varieties over integral models of PELtype 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 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. 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
Ancient solutions to the curve shortening flow and the Ricci flow in 2 dimensions 
Presenter: 
Natasa Sesum, Columbia University 
Date: 
Friday, April 24, 2009, Time: 3:00 p.m., Location: Fine Hall 314 
Abstract: 
I will give a classification of ancient convex closed embedded ancient solutions to the curve shortening flow and the ancient solutions to the Ricci flow on surfaces. This is a joint work with Daskalopoulos and Hamilton. 


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. 


Department Colloquium *** Please note special date, time, and location 
Topic: 
Mapping class groups, relative hyperbolicity and rigidity 
Presenter: 
Yair Minsky , Yale University 
Date: 
Tuesday, April 28, 2009, Time: 3:30 p.m., Location: Fine Hall 214 


Algebraic Geometry Seminar 
Topic: 
CalabiYau 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: 
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. 


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 
Abstract: 
We associate padic 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 localglobal 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 vcomponent of the cusp form. Moreover, the Swan conductor is essentially the depth. 


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 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
The KahlerRicci flow and canonical measures 
Presenter: 
Jian Song, Rutgers University 
Date: 
Friday, May 1, 2009, Time: 3:00 p.m., Location: Fine Hall 314 
Abstract: 
We define and prove the existence of canonical measures of Einstein type on algebraic manifolds of nonnegative Kodaira dimension. We also show that the KahlerRicci flow can be uniquely defined on algebraic varieties with log terminal singularities. It reveals the deep connection between the Ricci flow and the classification of algebraic varieties. 


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 


Differential Geometry and Geometric Analysis Seminar 
Topic: 
TBA 
Presenter: 
Justin Corvino, Lafayette College 
Date: 
Friday, May 8, 2009, Time: 3:00 p.m., Location: Fine Hall 314 



