# Seminars & Events for 2011-2012

##### Bocher Type Theorems for Degenerate Conformally Invariant Equations

A classical theorem of Bocher states that a positive harmonic function in a punctured ball can be written as the sum of a multiple of the fundamental solution and a regular harmonic function. I will describe generalizations for degenerate $\sigma_k$ equations. Joint work with Yanyan Li.

##### Hardy Spaces With Variable Exponents and Generalized Campanato Spaces

Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space H1, which substitute for L1, and the Hardy spaces $H^p$ with $0 < p < 1$, are different in that the latter contains non-regular distributions.

##### Topological Landscape of Networks

We will discuss how one can endow a network with a landscape in a very simple and natural way. Critical point analysis is introduced for functions defined on networks. The concept of local minima/maxima and saddle points of different indices are defined, by extending the notion of gradient flows and minimum energy path to the network setting.

##### Tensor Products On Triangulated and Abelian Categories

##### The Surface Subgroup Theorem and the Ehrenpreis Conjecture

We prove that there is a hyperbolic surface $S$ such that for any closed hy-perbolic 2 or 3-manifold M, and > 0, there is a nite cover ^ S of S, and a map f: ^ S ! M that is locally within of being an isometric immersion. When dimM = 3 this implies that 1(M) has a surface subgroup, and when dimM = 2 this is the Ehrenpreis conjecture.

##### Effective bisector estimate for PSL(2,C) with applications to circle packings

Let Gamma be a non-elementary discrete geometrically finite subgroup of PSL(2,C). Under the assumption that the critical exponent of Gamma is greater than 1 we prove an effective bisector counting theorem for Gamma. We then apply this Theorem to the Apollonian circle packing problem to get power savings and to compute the overall constant.

##### Pareto Optimal Solutions for Smoothed Analysts

Consider an optimization problem with n binary variables and $d+1$ linear objective functions. Each valid solution in ${0,1}^n$ gives rise to an objective vector in $R^{d+1}$, and one often wants to enumerate the Pareto optima among them.

##### Hermitian K-theory and Cobordism

I will discuss Z/2-equivariant motivic spectra. As an example, I will talk about a Z/2-equivariant motivic spectrum representing Karoubi's Hermitian K-theory, and my joint solution with Kriz and Ormsby of Thomason's homotopy limit problem. As another example, I will talk about motivic Hermitian cobordism, and its topological realization, topological Hermitian cobordism.

##### Counting Connections and the Ehrenpreis Conjecture

Let $S$ be a closed hyperbolic surface. We review the theory of counting connections on $S$---between points, between horocycles, and between geodesic segments---and we explain how this relates to the proof of the Ehrenpreis conjecture.

##### Mod $p$ Points on Shimura Varieties

A conjecture of Langlands-Rapoport predicts the structure of the mod $p$ points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions. I will report on progress towards the conjecture in the case of Shimura varieties attached to non-exceptional groups.

##### The Spacetime Positive Mass Theorem in Dimensions Less Than 8

After reviewing the proof of the Riemannian positive mass theorem in dimensions less than 8, I will briefly explain how to generalize the proof to slices of spacetime that are not time- symmetric. The basic idea is to replace minimal hypersurfaces by marginally outer-trapped hypersurfaces, and the main difficulty is to avoid using any minimization process.

##### Finite Point Configurations, Incidence Theory and Multi-linear Operators

A classical problem in geometric combinatorics is to determine how often a single distance may repeat among $n$ points in the plane. A related problem that has received much attention is how often a given triangle can repeat among $n$ points in the plane.

##### A Frobenius Variant of Seshadri Constants

I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle.

##### A Case Study for Critical Non-linear Dispersive quations: The Energy Critical Wave Equation

We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations.

##### Noncollision Singularities in Planar Two-center-two-body Problem

In this work we study a model called planar 2-center-2-body problem. In the plane, we have two fixed centers Q_1=(-\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses \mu. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers.

##### Tangles, Trees, and Flowers

Identifiable regions of high connectivity in a matroid or graph are captured by the notion of ``tangles''.

##### Hypergeometric Motives

The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide a fairly wide class which is numerically accessible. As an illustration we will consider the case of Artin L-functions.

##### A Khovanov Homotopy Type

We will start by describing Khovanov's categorification of the Jones polynomial from a cube of resolutions of a link diagram. We will then introduce the notion of a framed flow category, as defined by Cohen, Jones and Segal.

##### $L^p$ Bounds for Eigenfunctions on Locally Symmetric Spaces

There is a classical theorem of Sogge which provides bounds for the $L^p$ norms of a Laplace eigenfunction on a compact Riemannian manifold, which are sharp on the sphere and for spectral clusters.

##### University of North Carolina at Chapel Hill

It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. We consider a family of manifolds with trapped geodesics which are degenerately hyperbolic and prove a sharp local smoothing estimate with loss depending on the type of trapping.