# Seminars & Events for 2010-2011

##### Restriction of sections for abelian schemes

I will describe work with Jason Starr in which we show that the group of sections of a family of abelian varieties over a higher dimensional base is determined by the restriction of the family to a "very general conic curve".

##### Modular representations of $p$-adic groups

The Langlands program relates complex representations of $GL_n(Q_p)$ to $n$-dimensional Galois representations. For $n=1$ this is explained by class field theory and for $n=2$ this is closely related to the theory of modular forms. For general $n$, this is now understood by the work of Harris-Taylor and Henniart.

##### Cops and robbers in random graphs

We will study the following game known as cops and robbers. There is a finite, connected, undirected graph $G$, and $m$ cops and one robber. At the start, each cop chooses one vertex, and then the robber makes his choice of a vertex. Then they move alternately (first the cops then the robber).

##### Geometry and combinatorics for revolute-jointed robot arms

We present a complete theoretical characterization and a method for calculating the reachable workspace boundary for all serial manipulators with revolute joints having any pair of consecutive joint axes coplanar. The number of joints is arbitrary.

##### Effective Limit Distribution of the Frobenius Numbers

The Frobenius number of a lattice point $\bf{a}$ with positive coprime coordinates, is the largest integer which can NOT be expressed as a non-negative integer linear combination of the coordinates of $\bf{a}$. Marklof showed in 2010 that the limit distribution of the Frobenius numbers is given by the distribution for the covering radius function of a random unimodular lattice.

##### Heuristics for lambda invariants

The $\lambda$-invariant is an invariant of an imaginary quadratic field that measures the growth of class numbers in cyclotomic towers over the field. It also measures the number of zeroes of an associated $p$-adic L-function. In this talk, I will discuss the following question: How often is the $p$-adic $\lambda$-invariant of an imaginary quadratic field equal to $m$? I'll explain how one

##### Compactness and Quantization phenomena in conformal geometry: Some recent results and open problems.

I will briefly introduce the GJMS operators, Q-curvature and their basic properties, then I will describe the possible behaviours of a given sequence of metrics (on a closed Riemannian manifold or on a domain in $R^n$ or similar) having prescribed Q-curvatures which converge to a given function. If the sequence is not-precompact various blow-up phenomena appear, only in part well understood.

##### Explicit formula for the solution of the cubic Szego equation on the real line and its applications

In this talk we consider the cubic Szego equation: $i u_t = Pi (|u|^2u)$ on the real line, where $Pi$ is the Szeg? projector on non-negative frequencies. This equation was introduced as a model of a non-dispersive Hamiltonian equation. Like 1-d cubic NLS and KdV, it is known to be completely integrable in the sense that it possesses a Lax pair structure.

##### Symplectic deformation invariance of rationally connected 3-folds

##### Color 6-critical graphs on surfaces

We give a simple proof of a theorem of Thomassen that for every surface S there are only finitely many 6-critical graphs that embed in S. With a little bit of additional effort we can bound the number of vertices of a 6-critical graph embedded in S by a function that is linear in the genus of S. This is joint work with Luke Postle.

##### On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a group $\pi$ to a Lie group $G$ denoted $Hom(\pi,G)$. The space of representations $Hom(\pi,G)/G$ obtained from the adjoint action of $G$ will be considered. In special cases, these spaces can be assembled into a single space analogous to the classifying space of the group $G$.

##### Right-angledness, flag complexes, asphericity

I will discuss three related constructions of spaces and manifolds and then give necessary and sufficient conditions for the resulting spaces to be aspherical. The first construction is the polyhedral product functor. The second construction involves applying the reflection group trick to a "corner of spaces". The third construction involves pulling back a corner of spaces via a coloring of

##### Whittaker Functions and Demazure Characters

It is well-known that there are connections between the representation theory of a reductive p-adic

##### On unique continuation for nonlinear elliptic equations

We will discuss the following issue: if two solutions of a nonlinear elliptic equation coincide in a small ball, do they necessarily coincide everywhere? The problem is fairly well understood in the linear setting, but it is open for most interesting nonlinear elliptic equations.

##### Invertibility of random matrices and applications

Joint Analysis Seminar and PACM Colloquium

##### Globally F-regular and Log Fano Varieties

Globally F-regular varieties are a class of projective varieties over a field of prime characteristic, closely related to the more well-known class of Frobenius split varieties, but more robust. Examples include Schubert and related varieties. In trying to understand their geometry, we discovered that they are very closely related to log Fano varieties.

##### New theory of hypergeometric functions

The lecture will be devoted to the new vintage in the theory of special functions, a unification of the Bessel, hypergeometric, spherical and Whittaker functions, their p-adic and difference counterparts, and of course the theta-functions (associated with root systems) in one definition.

##### Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions

We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions.

##### Rank bounds for design matrices with applications

We prove a lower bound on the rank of sparse matrices whose pattern of zeros and non zeros satisfies a certain 'design' like property. Namely, if the intersections of the supports of different columns are small compared to the size of the individual supports. This bound holds over fields of large characteristic or characteristic zero.