# Seminars & Events for 2008-2009

##### Bounding sup-norms of cusp forms

Given an $L^2$-normalized cusp form $f$ on a modular curve $X_0(N)$, what can be said about pointwise bounds for $f$? For Hecke eigenforms, we will prove the first non-trivial bound in terms of the level $N$ as well as hybrid bounds in terms of the level and the Laplacian eigenvalue. Similar techniques work for functions on other spaces, e.g. quotients of quaternion algebras.

##### Ricci flow on ALE spaces

##### Trouble with a chain of stochastic oscillators

I will discuss some recent (but modest) results showing the existence and slow mixing of a stationary chain of Hamiltonian oscillators subject to a heat bath. Such systems are used as simple models of heat conduction or energy transfer. Though the unlimite goal might be seen to under stand the "fourier" like law in this setting, I will be less ambitious.

##### What makes the ergodic theory of Markov chains in infinite dimensions different (and difficult)?

I will discuss how Markov chains in infinite dimensions generically have typically have properties which make their ergodic theory difficult. Such properties are very pathological in finite dimensions, but in some sense generic in infinite dimensions. I will draw examples from stochastically forced PDEs and stochastic delay equations.

##### Algebraic surfaces and hyperbolic geometry

The intersection form on the group of line bundles on a complex algebraic surface always has signature $(1,n)$ for some $n$. So the automorphism group of an algebraic surface always acts on hyperbolic $n$-space.

##### Internal aggregation Models: From Diaconis-Fulton addition to a free boundary problem

Start with $n$ particles at each of $k$ points in the $d$-dimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton.

##### Lie Groups: Decomposition and Exponentiation

A manifold with a smooth group structure is called a Lie group. Most of the information about Lie groups is captured by the tangent space at the identity and its Poisson bracket. The map relating these two structures is the exponential map (which in the compact case is the same as the geodesic exponential map).

##### Hénon Renormalization

The geometry of strongly dissipative infinite renormalizable Hénon maps of period doubling type is surprisingly different from its one-dimensional counterpart. There are universal geometrical properties. However, the Cantor attractor is not geometrically rigid. Typically, it doesn't have bounded geometry. The average Jacobian is a topological invariant of the global attractor.

##### New bounds on the size of Kakeya sets in finite fields

A Kakeya set is a set in $(F_q)^n$ (the $n$ dimensional vector space over a field of $q$ elements) which contains a line in every direction. In this talk I will present a recent result which gives a lower bound of $(q/2)^n$ on the size of such sets. This bound is tight to within a multiplicative factor of two from the known upper bounds.

##### Subgroup classification in $Out(F_n)$

We prove that for every subgroup $G$ of $Out(F_n)$, one of two alternatives holds: either there is a finite index subgroup $H<G$ and a nontrivial proper free factor $A$ of $F_n$ such that each element of $H$ fixes the conjugacy class of $A$; or there is an element $g\in G$ such that no nontrivial power of $g$ fixes the conjugacy class of any nontrivial proper free factor of $F_n$.

##### A "Relative" Langlands Program and Periods of Automorhic Forms

Motivated by the relative trace formula of Jacquet and experience on period integrals of automorphic forms, we take the first steps towards formulating a "relative" Langlands program, i.e. a set of conjectures on H-distinguished representations of a reductive group G (both locally and globally), where H is a spherical subgroup of G. We prove several results in this direction.

##### A generalization of compact operators and its application to the existence of local minima without convexity

We will introduce a certain property for a continuous (non-linear) operator that allows for the existence of local minima for functionals when the derivative complies with such a condition, without the need to check either weak lower semicontinuity or convexity. It turns out that this property is a generalization of the standard compactness for a continuous, non-linear operator.

##### Greatest lower bounds on the Ricci curvature of Fano manifolds

On Fano manifolds we study the supremum of the possible t such that there exists a metric in the first Chern class with Ricci curvature bounded below by t. For the projective plane blown up in one point we show that this supremum is 6/7.

##### Compressive Optical Imaging

Recent work in the emerging field of compressed sensing indicates that, when feasible, judicious selection of the type of image transformation induced by imaging systems may dramatically improve our ability to perform reconstruction, even when the number of measurements is small relative to the size and resolution of the final image.

##### Compactified Jacobians and Abel maps for singular curves

We will discuss the problem of extending the construction of the classical Abel maps for smooth curves to the case of singular curves. The construction of degree-1 Abel maps will be shown, together with an approach for constructing higher degree Abel maps.

##### Quantum Unique Ergodicity and Number Theory

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds.

##### Morse theory

Morse theory gives the cell structure of a manifold in terms of the critical points of a 'random' real-valued function on this manifold. Besides being clever and pleasing to the eye, it has given us Bott periodicity, counts of geodesics, periodic orbits of dynamical systems, Heegard Floer homology, the foundations of Mirror Symmetry and many many more riches.

##### Random walks with memory and statistical mechanics

This talk will review some results and conjectures about history dependent random walks. For example, edge reinforced random walk (ERRW) is a random walk which prefers to visit edges it has visited in the past. Diaconis showed that ERRW can be expressed as a random walk in a random environment. This environment is highly correlated and is described in terms of statistical mechanics.

##### Inverse Littlewood-Offord theory, Smooth Analysis and the Circular Law

A corner stone of the theory of random matrices is Wigner's semi-circle law, obtained in the 1950s, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random hermitian matrix with iid (upper diagonal) entries follows the semi-circle law.

##### Congruence subgroup problem for mapping class groups

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.