# Seminars & Events for 2014-2015

##### Zarhin's trick and geometric boundedness results for K3 surfaces

**Please note special time. **Tate's conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin's trick.

##### The structure of flow in Hydrodynamics, Thermodynamics and General Relativity, from Navier Stokes to Tolman

Problems with the formulation of Relativistic Astrophysics lead to the need for a variational formulation of thermodynamics. This is easy and immensely rewarding in the non relativistic context, so long as the motion is irrotational. The main topic of this talk is to overcome this limitation. The solution is amazingly simple; one has to combine two familiar forms of hydrodynamics.

##### Universal Chow group of zero-cycles on cubic hypersurfaces

**Please note special day and time. ** We discuss the universal triviality of the CH0-group of cubic hypersurfaces, or equivalently the existence of a Chow-theoretic decomposition of their diagonal. The motivation is the study of stable irrationality for these varieties. Our main result is that this decomposition exists if and only if it exists on the cohomological level.

##### Infinite Dimensional Stochastic Differential Equations for Dyson's Brownian Motion

Dyson's Brownian Motion (DBM) describes the evolution of the spectra of certain random matrices, and is governed by a system of stochastic differential equations (SDEs) with a singular, long-range interaction. In this talk I will outline a construction of the strong solution of the infinite dimensional SDE that corresponds to the bulk limit of DBM.

##### Computing Fukaya categories

The Fukaya category has become a central tool in symplectic topology. In this talk, I will begin with some motivating examples, and explain methods for making computations of the Fukaya categories sufficiently explicit in order to extract information about Lagrangian submanifolds.

##### Nonabelian Hodge theory and uniformization

Classical Hodge theory provides a link between the topology and the analytic geometry of a compact Kaehler manifold X via harmonic forms. Similarly, in nonabelian Hodge theory (developed by Simpson based on works of Hitchin, Corlette, Donaldson, Uhlenbeck-Yau, and others), harmonic metrics on vector bundles are used to study the fundamental group of X, a nonabelian topological invariant.

##### Piecewise isometric dynamics on the square pillowcase

I will begin by describing a method to renormalize a dynamical system associated with a class of tilings in the plane related to corner percolation studied by Gábor Pete. I will explain how these ideas give rise to a renormalization scheme for a 2-parameter family of piecewise isometries of the square pillowcase. I'll describe some results about the dynamics of these maps.

##### Saturation in the hypercube

A subgraph of the d-dimensional cube Qd is (Qd,Qm)-saturated if it does not contain a copy of Qm, but adding any missing edge creates a copy of Qm. The subgraph is weakly (Qd,Qm)-saturated if we can add the missing edges one at a time, creating a new copy of Qm at each step.

##### 3-manifolds, Lipschitz geometry, and equisingularity

The local topology of isolated complex surface singularites is long understood, as cones on closed 3-manifolds obtained by negative definite plumbing. On the other hand a full understanding of the analytic types is out of reach, motivating Zariski's efforts into the 1980's to give a good concept of "equisingularity" for families of singularities.

##### Fourier--Jacobi periods on unitary groups

We formulate a conjectural identity relating the Fourier--Jacobi periods on unitary groups and the central value of certain Rankin--Selberg L-functions. This refines the Gan--Gross--Prasad conjecture. We give some examples supporting this conjecture.

##### Infinite volume limit for the Nonlinear Schrodinger Equation and Weak turbulence

The theory of weak turbulence has been put forward by applied mathematicians to describe the asymptotic behavior of NLS set on a compact domain - as well as many other infinite dimensional Hamiltonian systems. It is believed to be valid in a statistical sense, in the weakly nonlinear, infinite volume limit.

##### A free boundary problem in kinetic theory

We consider a rigid body colliding with a continuum of particles. We assume that the body is moving at a velocity close to an equilibrium velocity V_{infty} and that the particles colliding with the body reflect probabilistically with some probablility distribution K.

##### Conservation of knottiness in real and idalized fluids

**This is a joint talk with PACM/MAE. ** To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core.

##### TBA - Friedman

##### Superfluid behavior of a Bose-Einstein condensate in a random potential

We investigate the relation between Bose-Einstein condensation (BEC) and superfluidity in the ground state of a one-dimensional model of interacting Bosons in a strong random potential. We prove rigorously that in a certain parameter regime the superfluid fraction can be arbitrarily small while complete BEC prevails.

##### Equivalence of decay of correlations, the log-Sobolev inequality, and of the spectral gap

In this talk we consider a lattice system of unbounded real-valued spins, which is described by its Gibbs measure mu. We discuss how a known result for finite-range interaction is generalized to infinite-range.

##### Playing with the waves outside a black hole

In the recent years, a lot of research has been devoted to the study of the behaviour of scalar waves in the exterior region of black hole spacetimes. This is intimately connected to the problem of non linear stability of such black hole spacetimes as solutions to the Einstein equations. In this talk, we will explore some interesting theorems and results in this area.

##### A constructive induction for interval exchanges and applications

We explain the induction process initiated by L. Zamboni and myself, which was designed to understand the word combinatorics of the natural codings, but is now better described through a geometrical model introduced by Delecroix and Ulcigrai, with a natural extension where convex polygons (parallelograms in the hyperelliptic case) replace the rectangles of the Rauzy-Veech induction. This induction is used to build families of examples of interval exchange transformations, with weak mixing or with eigenvalues, with Veech's simplicity property, or satisfying a criterion due to Bourgain which in turn implies Sarnak's conjecture on the orthogonality of the trajectories with the Moebius function.

##### Topological Witt groups and Reality

This is joint work with Karoubi. Given an algebraic variety over the real numbers, we compare its Witt groups with the topological Witt groups of the underlying manifold with involution using Atiyah's Real K-theory, Brumfiel's Theorem and a conjecture of Bruce Williams.

##### Asymptotics of representations of classical Lie groups

**Please note special time.** We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition.