# Seminars & Events for 2010-2011

##### An integral Eisenstein-Sczech cocycle on $GL_n(Z)$ and $p$-adic L-functions of totally real fields

In 1993, Sczech defined an $n-1$ cocycle on $GL_n(Z)$ valued in a certain space of distributions. He showed that specializations of this cocyle yield the values of the partial zeta functions of totally real fields of degree $n$ at nonpositive integers. In this talk, I will describe an integral refinement of Sczech's cocycle.

##### Regularity of absolutely minimizing Lipschitz extensions

I will present joint work with Lawrence C. Evans on the everywhere differentiability of absolutely minimizing Lipschitz extensions.

##### Extrapolation Models

We discuss the role of linear models for two extrapolation problems. The rst is the ex-trapolation to the limit of innite series, i.e. convergence acceleration.

##### A limiting interaction energy for Ginzburg-Landau vortices

This is a joint work with Etienne Sandier where we study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields $H_{c1}$ and $H_{c2}$. In that regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices.

##### Charge screening in quantum crystals

Density Functional Theory (DFT) has become a major tool in chemistry, materials science, molecular biology and nanotechnology. It is also an inexhaustible source of exciting mathematical and numerical issues. In this talk, I will present some of the variational models derived from DFT, and discuss their mathematical properties.

##### Extended States in a Lifshitz Tail Regime for Random Operators on Trees

We will discuss the spectral properties of random operators on regular tree graphs. The models have have been among the earliest studied for Anderson localization, and they continue to attract attention because of analogies with localization issues for many particles. The talk will focus on the location of the mobility edge.

##### Equivariant birational maps and resolutions of categorical quotients

If $X^{ss}$ is the set of semi-stable points for a linearized action of a reductive group on a smooth projective variety $X$ then there two procedures (Kirwan's procedure or change of linearization) for constructing a partial resolution of singularities of the categorical quotient $X^{ss}/G$.

##### Microscopic Models of Macroscopic Transport: A Selective Overview

I will describe various attempts to derive, heuristically or rigorously, diffusive behavior of energy (particle) transport, i.e. Fourier's law (Fick's law) from classical microscopic models (mostly deterministic). Computer simulations showing presence or absence of such behavior in ordered and disordered systems will be described.

##### Incompressible Fluids: Simple Models, Complex Fluids

Complex fluids are fluids with particles suspended in them. The particles are carried by the fluid, interact among themselves, and influence the fluid's behavior. I will describe some of the basic questions of existence, uniqueness, regularity and stability of solutions of models of complex fluids, in the broader context of incompressible hydrodynamic PDE.

##### Coherent-constructible correspondence and its applications

I will describe a coherent-constructible correspondence, which is a monoidal dg functor between the category of equivariant perfect sheaves of a toric variety and the category of some constructible sheaves over a real vector space (dual Lie algebra of the torus).

##### The Spectrum of an Hermitian Matrix With Dependent Entries Constructed from Random Independent Images

In this talk we will present a preliminary analysis and numerical results for the distribution of eigenvalues of a certain random N by N Hermitian matrix, whose construction is motivated by a problem in structural biology. The matrix is built from N images, where each image is an array of P pixels, and the pixels are i.i.d standard Gaussians.

##### Nearly Tight Low Stretch Spanning Trees

We prove that any graph $G$ on $n$ vertices has a distribution over its spanning trees such that for any edge $(u,v)$ the expected stretch $E_T[d_T(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem. Joint work with Ittai Abraham and Yair Bartal

##### Holomorphic traingle maps in sutured Floer homology

Honda, Kazez and Matic defined maps on sutured Floer homology induced by a contact structure. I'll explain how to compute these maps using holomorphic triangle counts and give some applications to computing sutured Floer homologies and sutured contact invariants.

##### Even Galois Representations and the Fontaine-Mazur conjecture

Fontaine and Mazur have a remarkable conjecture that predicts which (p-adic) Galois representations arise from geometry. In the special case of two dimensional representations with distinct Hodge-Tate weights, they further conjecture that these "geometric" representations are also modular.

##### Counterexamples to Min-Oo's Conjecture

Consider a compact Riemannian manifold $M$ of dimension $n$ whose boundary $\partial M$ is totally geodesic and is isometric to the standard sphere $S^{n-1}$. A natural conjecture of Min-Oo asserts that if the scalar curvature of $M$ is at least $n(n-1)$, then $M$ is isometric to the hemisphere $S_+^n$ equipped with its standard metric.

##### Information Aggregation in Complex Networks

Over the past few years there has been a rapidly growing interest in analysis, design and optimization of various types of collective behaviors in networked dynamic systems.

##### Asymptotic Behavior of Spacetimes Approaching a Schwarzschild solution

Consider a spacetime which approaches a Schwarzschild solution. We will discuss the following problem: Assuming decay of appropriate norms of the Ricci rotation coefficients and their derivatives, can one prove boundedness/ decay for the curvature components and their derivatives?

##### Global wellposedness and scattering for the inhomogeneous fourth-order Schrodinger equation

Fourth-order Schrödinger equations were proposed as a correction to the standard model for propagation of laser in nonlinear media and have since appeared in different contexts. In this talk, I will consider the inhomogeneous mass-critical fourth-order Schrödinger equation $iu_t+D^2u-Du+|u|^{8/n}u=0$ and prove global existence and scattering in $L^2$ in high dimensions.

##### A vanishing theorem in characteristic p

While the classical Kodaira vanishing theorem is false in general in characteristic $p>0$, Deligne, Illusie and Raynaud proved that it remains true under some mild (liftability and dimension) conditions.

##### From microscopic hamiltonian dynamics to heat equation

One of the main problems in non-equilibrium statistical mechanics is to derive, by space-time rescaling, macroscopic irreversible diffusive evolution for the co nserved quantities of an (large) hamiltonian system. I will describe the mathematical setup of the problem, and some recent progress when the hamiltonian dynamics is perturbed by energy conserving stochastic collisions.