In this talk we shall discuss rigidity aspects of infinite discrete groups, which arise naturally in Geometry (as fundamental groups of manifolds), in Algebraic groups (as lattices) and, more generally, as symmetries of various mathematical objects.

Starting from classical by now rigidity results of Mostow, Margulis, Zimmer, we shall turn to the recently active area of Measurable Group Theory, which is closely related to Ergodic Theory, von Neumann algebras, and has applications to such fields as Descriptive Set Theory.

We will establish a bijective correspondence between finite type associative cones in $\R^7$ and their spectral data, which consists of a hexagonal algebraic curve and a planar flow of line bundles in its Jacobian. We characterize the spectral data by identifying various symmetries on them. We prove generic smoothness of these spectral curves, compute their genus, and compute the dimension of the moduli of such curves. Then we identify a Prym-Tjurin subtorus of the Jacobian, in which the direction of the flow must lie, and compute its dimension. Finally we characterize finite type special Lagrangian cones in $\C^3$ as a subclass of such associative cones in terms of the spectral data. These computations are mainly motivated by Hitchin's recent work on G_2 spectral curves and Langland duality.

A smooth plane projective cubic curve (also known as an elliptic curve or a curve of genus 1) carries a natural structure of a commutative group: the addition is defined geometrically by the "chord and tangent method". An attempt "to add" points on a curve of arbitrary positive genus g leads to the notion of the jacobian of the curve. This jacobian is a g-dimensional commutative algebraic group that is a projective algebraic variety; in particular, it cannot be realized as a matrix group. Geometric properties of jacobians play a crucial role in the study of arithmetic and geometric properties of curves involved. One of the most important geometric invariants of a jacobian is its endomorphism ring.

We discuss how to compute explicitly endomorphism rings of jacobians for certain interesting classes of curves that may be viewed as natural (and useful) generalizations of elliptic curves.

Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process.
Under mild assumptions, we prove existence and uniqueness of fluid model solutions.
Furthermore, we are able to characterize the left edge of fluid model
solutions as the right continuous inverse of a simple functional of the initial condition,
arrival rate, and service time distribution. When applied to various examples, this
characterization reveals the dependence on service time distribution of the rate at which the
left edge of the fluid model increases.

The totally asymmetric simple exclusion process (TASEP) is one of the
simplest models of interacting particle systems on the one-dimensional
lattice. It is equivalent to a random growth model from the
Kardar-Parisi-Zhang universality class. We focus on fluctuations of the
particle positions for a nonequilibrium TASEP that starts from certain
deterministic initial conditions. We (rigorously) derive the scaling
exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian
Orthogonal and Unitary ensembles of the random matrix theory.