According to a classical theorem of Bieberbach, if G is a discrete subgroup of the group of isometries of R^n then G contains a subgroup of finite index consisting of parallel translations. An analogous statement for discrete groups of affine transformations is not true. Nevertheless there is Auslander's conjecture which atates that if G is a subgroup of the group of affine transformations of R^n which properly discontinuously and cocompactly on R^n then G contains a solvable subgroup of finite index. I will describe some partial results related to this conjecture and will also talk about what happens if the assumption of cocompactness of R^n/G is dropped.

Singular symplectic varieties and their resolutions of singularities lie at the crossroads of algebraic and symplectic geometry, representation theory, and integrable systems. Central examples include the nilpotent cone of a complex semisimple Lie algebra and its resolution by the cotangent bundle of the flag variety (the Springer resolution); the nth symmetric product of the affine plane and its resolution by the Hilbert scheme of points; and a Kleinian surface singularity and its minimal resolution. A singular variety and its resolution never have equivalent geometry (as encoded, for example, in their derived categories). Replacing a symplectic variety by a quantization, however---an algebro-geometric analog of passing to a Fukaya-type category---one miraculously finds that such equivalences are common. I'll discuss singular symplectic varieties and their resolutions, examples, quantization, and a general criterion for such geometric equivalences that extends classical results (for example, Beilinson-Bernstein localization). Time permitting, I'll also discuss some additional features of these quantizations that parallel emerging structures in the (much more complicated) world of Fukaya categories. This is based on joint work with K. McGerty.

The goal of this talk is to familiarize the audience with some basic properties of quadratic forms on fields of characteristic different from 2. I will discuss results such as Witt's theorem and Hasse Minkowski, and briefly recall some classical theorems on sums of squares.

Just like the Poincare group, (a cover of) the isometry group of affine Minkowski space, is the symmetry group for classical theories, the Super Poincare group is the symmetry group for super spacetime, and thus plays an important role in super-symmetric quantum mechanics. We will discuss the construction of the super Poincare group by first describing Clifford algebras and spinors in signature metrics and then construct the super Poincare group and its group law via its algebra.

I will finish describing the "easy" part of the AGT conjecture, which gives a rather unexpected relation between moduli spaces of instantons in four dimensions and the Virasoro algebra.

Using reverse phase protein arrays (RPPA) we measure time-course protein expression for a set of selected markers that are known to co-regulate biological functions in a pathway structure. To accommodate the complex dependent nature of the data, including temporal correlation and pathway dependence for the protein markers, we propose a mixed effects model with temporal and protein-specific components. We develop a sequence of random probability measures (RPM) to account for the dependence in time of the protein expression measurements.

The second and third lectures will be the continuation, on a more technical level, of the first lecture. Topics to be (very briefly) covered include: (1) recurrence to compact sets and Diophantine approximation on manifolds; (2) orbit closures, the Oppenheim conjecture and the Littlewood conjecture; (3) classification of ergodic invariant measures and equidistribution; (4) quantitative Oppenheim conjecture and counting of integral points on homogenous varieties; (5) effective results in homogeneous dynamics.

We shall discuss the number theoretic ingredients that go into the proof of equidistribution of mass for classical Hecke eigenforms. This is joint work with Kannan Soundararajan. We shall also present a revision to the argument by Henryk Iwaniec.

We are developing a two-scale mortar iterative method for problems in porous media. We iterate the fine scale mortar by PCG and use coarse mortar to damp out the smooth error. The numerical examples show the condition number is O( log(H/h) ).

This will be a presentation of our recent results joint with J. Friedlander, B. Mazur and K. Rubin, which concern a new symbol associated to ideals in Galois number fields, the spin. The spin takes two values +1, -1 and the question is, how often does it change over prime ideals. The answer is (in case of cubic fields) it assumes each value half of the time. The result has applications for asymptotic distribution of the rank of 2-Selmer group of families of elliptic curves.

In the classical Obstacle Problem, one has the added luxury of having an obstacle that is apriori very smooth. There are situations where this assumption has to be dropped and the most one can hope for is some weaker notion of concavity or convexity. In this talk I hope to present some tools to understand obstacle problems with implicit constraints, when the obstacle depends on the solution you are trying to find! This will lead us into the realm of subdifferentials and nonlinear elliptic PDE theory. I hope to show how some of these tools are used, and why one should care about implicit constraint obstacle problems.