# Seminars & Events for 2009-2010

##### An arithmetic fundamental lemma for unitary group of three variable

In this talk I'll present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad conjecture) for unitary group. In particular, an arithmetic fundamental lemma (AFL) is proposed. Some results proved recently will be presented, including the AFL for unitary group of three variable.

##### Pseudo-Riemannian Calibrated Geometry and Optimal Transportation

Given a manifold $M$, there is a naturally occurring pseudo-Riemannian metric and Kähler form on the product $M\times M$. The graph of the solution to the optimal transportation problem for given smooth densities on $M$ is then a calibrated maximal Lagrangian submanifold in $M\times M$, with respect to a conformal metric on $M\times M$.

##### Harmonic Analysis and Geometries of Digital Data Bases

Given a matrix (of Data) we describe methodologies to build two multiscale (inference) Geometries/Harmonic Analysis one on the rows , the other on the columns . The geometries are designed to simplify the representation of the data base . We will provide a number of examples including; matrices of operators , psychological questionnaires, vector valued images, scientific articles, etc.

##### Ground states of the $L^2$-critical boson star equation

The boson star equation $\sqrt{-\Delta} u - (|x|^{-1} * |u|^2) u = -u$ in $R3$ involves both a non-local kinetic and potential energy and is $L^2$-critical. We establish uniqueness, radial symmetry (up to translations) and analyticity of non-negative solutions. We also prove the nondegeneracy of the linearization.

##### Ergodicity of Markov Processes: A marriage of topology and measure theory

One very widely used criterion in the theory of Markov chains states that if a Markov operator has the strong Feller property and is topologically irreducible, then it can have at most one invariant measure. While this criterion is very useful in finite-dimensional situations, it fails for many infinite-dimensional problems.

##### Entropy in Measurable Dynamics

In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation. Entropy has since been central to the classification theory of measurable dynamics. In the 70s and 80s researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss).

##### Harmonic measure

Harmonic measure is a measure on the boundary of domains in the complex plane whose associated integral generates the solution of the Dirichlet problem. I will explain Nevanlinna's "principle of harmonic measure," and give a number of examples of its use.

##### On the Schrijver-Seymour Conjecture

Two interesting classical results in combinatorial number theory are the Erdos-Ginzburg-Ziv Theorem and the Furedi-Kleitman Theorem.

##### A priori bounds for bounded-primitive renormalization

We say that an infinitely renormalizable quadratic polynomial has bounded-primitive type if we can find an infinite sequence of primitive renormalization times, such that the ratio between consecutive terms of the sequence is bounded. We prove that any such polynomial has the a priori bounds: there is a lower bound on the modulus of all renormalizations.

##### Immersed surfaces in closed hyperbolic 3-manifolds

Given any closed hyperbolic 3-manifold $M$ and $\epsilon > 0$, we find a closed hyperbolic surface $S$ and a map $f\to S\to M$ such that $f$ lifts to a $1+\epsilon$-quasi-isometry from the universal cover of $S$ to the universal cover of $M$. This is joint work with Vladimir Markovic.

##### Volume estimates in analytic and adelic geometry

The solution to many classical counting asymptotics problems in number theory goes by comparison with an analogous volume asumptotics.

##### More on immersed surfaces in closed hyperbolic 3-manifolds

##### The Static Extension Problem in General Relativity

There are several competing definitions of quasi-local mass in General Relativity. A very promising and natural candidate, proposed by Bartnik, seeks to localize the ADM or total mass. Fundamental to understanding Bartnik's construction, is the question of existence for a canonical geometric boundary value problem associated with the static vacuum Einstein equations.

##### Singularities, test configurations and constant scalar curvature Kahler metrics

##### Counterexamples to the Strichartz estimates for the wave equation in domains

We prove that the Strichartz estimates for the wave equation inside a strictly convex domain $\Omega$ of dimension $2$ suffer losses when compared to the usual case $\mathbb{R}2$, (at least for a subset of the usual range of indices) and this is due to microlocal phenomena such as caustics generated in arbitrariIly small time near the boundary.

##### Algebraic curves with CM

Is every Abelian variety isogenous with the Jacobian of an algebraic curve? We will study also several other questions in arithmetic geometry and show various implications. We will mention some solutions to these problems.

##### Symplectic Embeddings and Continued Fractions

It has been known since the time of Gromov that questions about symplectic embeddings lie at the heart of symplectic geometry. This talk will mostly be about some recent work with Schlenk in which we work out precisely when a four dimensional ellipsoid embeds symplectically in a ball.

##### Applying a Local Lemma to Thue games

In this talk, I will discuss probabilistic proofs for the existence of winning strategies in sequence games where the goal is nonrepetitiveness. The technique involves a 'one-sided' generalization of the Local Lemma, which allows us to ignore the dependencies on `future' events which would normally prevent this kind of proof from working.

##### A pair correlation bound implies the Central Limit Theorem for Sinai Billiards

It is an open problem in the study of dynamical systems whether fast decay of correlations alone is sufficient for the Central Limit Theorem (CLT) to hold. On the one hand, there are no examples of dynamical systems for which correlations decay quickly but the CLT fails. On the other, existing CLT proofs rely on statistical properties much stronger than correlation decay.

##### Cusp volume of fibered 3-manifolds

Consider a 3-manifold $M$ that fibers over the circle, with fiber a punctured surface $F$. I will explain how the volume of a maximal cusp of $M$ (in the hyperbolic metric) is determined up to a bounded constant by combinatorial properties of the arc complex of the fiber surface $F$. This is joint work with Saul Schleimer.