# Seminars & Events for 2009-2010

##### Torsion in the homology of arithmetic groups

##### The Yamabe problem on manifolds with boundary

##### Some variants on the flows of suspensions: Diffusion, dispersion, and biofilms

In this talk I will present several fluid mechanics problems that concern the flow of particles and suspensions. This topic has many variants, which I will introduce to provide breadth and perspective for the listener (most of you) who has not studied the topic.

##### Cocompact imbeddings and critical nonlinearity revisited

We introduce a notion of cocompact imbeddings relative to a group of linear isometries.We discuss the notion of critical Sobolev nonlinearity in connection with the usual dilation actions that make the (non-compact) limit Sobolev imbedding co-compact and yield solutions of Talenti type for semilinear elliptic equations with self-similar autonomous nonlinearities of critical growth.

##### The stability of the irrotational Euler-Einstein system with a positive cosmological constant

The irrotational Euler-Einstein system models the evolution of a dynamic spacetime containing a perfect fluid with vanishing vorticity.

##### BGG correspondence and the cohomology of compact Kaehler manifolds

The cohomology algebra of the sheaf of holomorphic functions on a compact Kaehler manifold can be naturally viewed as a module over the exterior algebra of a vector space. A well-known result of Bernstein-Gelfand-Gelfand gives a correspondence between such "exterior" modules and linear complexes of modules over the symmetric algebra, i. e. the polynomial ring.

##### The Congruence Subgroup Problem for $SL(n,Z)$

The group $SL(n, k)$ ($k$ a field) is simple modulo its center, but for a ring $A$, $SL(n, A)$ is not: the kernels of the maps $SL(n,A)\rightarrow SL(n,A/I)$ ($I<A$ an ideal) give many normal subgroups. The congruence subgroup problem asks whether all finite index normal subgroups of $SL(n,A)$ are obtained in this way.

##### The minimum number of monochromatic 4-term progressions

It is not difficult to see that whenever you 2-color the elements of $Z/pZ$, the number of monochromatic 3-term arithmetic progressions depends only on the density of the color classes. The analogous statement for 4-term progressions is false.

##### On the KO-theory of toric spaces

Central in toric geometry and topology are several important spaces which include moment-angle complexes, the Davis-Januszkiewicz space and toric manifolds. In any complex-oriented cohomology theory, the cohomology rings of many of these spaces have elegant descriptions in terms of the underlying combinatorics. For KO-theory however the situation is more complex.

##### Link surgery, monopole Floer homology, and odd Khovanov homology

I'll describe new bigraded invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction relates the topology of link surgeries to the combinatorics of polytopes called graph associahedra.

##### Generalizations of the Sato-Tate conjecture

I will discuss a recent joint work with Thomas Barnet-Lamb and Toby Gee in which we prove the Sato-Tate conjecture for non-CM regular algebraic cuspidal automorphic representations of GL_2 over a totally real field.

##### Center of mass and constant mean curvature foliations in general relativity

We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. We will first introduce the background and then discuss how the foliation relates to the concept of center of mass in general relativity.

##### Modularity lifting for n-dimensional ordinary Galois representations

I will discuss a generalization of the modularity lifting theorems of Clozel, Harris and Taylor to the case of ordinary Galois representations. The result is obtained by applying the Taylor-Wiles method (with innovations due to Kisin and Taylor) over a Hida family. A key step is to construct an appropriate ordinary lifting ring and determine its irreducible components.

##### Spaces of knots and configuration spaces

The structure of the space of 'long knots in \R3' was developed in a series of papers by R. Budney with applications in joint work with Budney. An introduction to this space together with homological properties, and their connections to configuration spaces will be developed. For example, the singular homology of the space of knots will be considered.

##### Mean values with $GL(2)\times GL(3)$ functions

##### Geometry and Analysis of point sets in high dimensions

The analysis of high dimensional data sets is useful in a large variety of applications, from machine learning to dynamical systems: data sets are often modeled as low-dimensional, noisy data sets embedded in high-dimensional spaces; dynamical systems often have very high-dimensional state spaces but sometimes interesting dynamics occurs on low-dimensional sets.

##### A variational model for crystals with defects

This talk will be devoted to the reduced Hartree-Fock model for crystals with defects.

The main idea is to describe at the same time the electrons bound by the defect and the (nonlinear) behavior of the infinite crystal. This leads to a bounded-below nonlinear functional whose variable is however an operator of infinite-rank.

##### Rational Simple Connectedness

Rational simple connectedness is an analog of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order.

##### Strongly Focused Gravitational Waves

Christodoulou (2008) proved that trapped spheres can form in evolution, through the focusing of gravitational waves. Recently, with E. Trubowitz, we considered the same physical problem, using very different mathematical methods. Our approach is based on a systematic use of formal expansions, scaling symmetries and energy estimates.

##### A model-theoretic approach to certain diophantine problems

I will describe some results and problems about the distribution of rational points on certain non-algebraic sets in real space. They find their natural setting in the model-theoretic notion of an 'o-minimal structure over the real numbers.' I will describe a result joint with Alex Wilkie in this setting.