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.

Consider a particle in a finite dimensional Euclidean space performing a Brownian motion with an instantaneous drift vector at every time point determined by the order in which the coordinates of its location can be arranged as a decreasing sequence. These processes appear naturally in a variety of areas from queueing theory, statistical physics, and economic modeling. One is generally interested in the spacings between the ordered coordinates under such a motion.

For finite n, the invariant distribution of the vector of spacings can be completely described and is a function of the drift. We show, as n grows to infinity, a curious phenomenon occurs. We look at a transformation of the original process by exponentiating the location coordinates and dividing them by their total sum. Irrespective of the drifts, under the invariant distribution, only one of three things can happen to the transformed values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to some member of a two parameter family of random point processes. This family known as the Poisson-Dirichlet's appears in genetics and renewal theory and is well studied. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. We also consider another alternative starting with a countable collection of Brownian motions. This countable model is related to the Harris model of elastic collisions and the discrete Ruzmaikina-Aizenmann model for competing particles.

This is based on separate joint works with Sourav Chatterjee and Jim Pitman.