Variability in the number of simultaneous flows present can have a substantial impact on the perceived performance of packet networks such as the Internet. While the packet level behaviour of a given set of flows is by now well understood, less is known about the stochastic behaviour of the number of flows in progress on different routes through the network. In this talk we describe recent work on Brownian models of networks in heavy traffic. Joint work with Ruth Williams.

Algebraic K-theory is a highly complicated invariant of
algebraic varieties that encodes arithmetic, geometric and algebraic
information.
In this talk, I will try to make this distinction somewhat less vague and
explain how to isolate some of the algebraic and geometric information
K-theory provides about singularities, leading to proofs of various
longstanding conjectures.

This talk describes how a symbolic computer algebra
tool (NCAlgebra) that handles symbolic matrix
(noncommutative) products can be used to assist the
numerical solution of semidefinite programs where the
variables are matrices. The idea is to keep matrix
variables aggregated at all steps of a primal-dual
interior-point algorithm in which symbolic expressions
are automatically generated and used iteratively.

For a given elliptic curve $E$ over a finite field $F_q$, we let $N_k =
\#E(F_{q^k})$, where $F_{q^k}$ is a $k$th degree extension of the finite
field $F_q$. Because the Zeta Function for $E$ only depends on $q$ and
$N_1$, the sequence $\{N_k\}$ only depends on those numbers as well.
More specifically, we observe that these bivariate expressions for $N_k$
are in fact polynomials with integer coefficients, which alternate in sign
with respect to the power of $N_1$.

This motivated a search for a combinatorial interpretation of these
coefficients, and one such interpretation involves spanning trees of a
certain family of graphs. In this talk, I will describe this
combinatorial interpretation, as well as applications and directions for
future research. This will include determinantal formulas for $N_k$,
factorizations of $N_k$, and the definition of a new sequence of
polynomials, which we call elliptic cyclotomic polynomials.

One of the important features of elliptic curves which makes them the
focus of contemporary research is that they admit a group structure.
During the remainder of this talk I will describe chip-firing games, how
they provide a group structure on the set of spanning trees, and numerous
ways that these groups are analogous to those of elliptic curves.

The existence of K\"ahler-Einstein metrics on a compact K\""ahler manifold

Many mathematical objects come in continuous families, prompting the
desire to define a ``universal family'' that contains each such object
exactly once up to isomorphism. When this isn't possible (because the
family would be too bad to be worthwhile -- I'll talk about this behavior),
we can try to come close, by including only ``stable'' objects.

Frequently the universal family is constructed by starting with a
bigger family that includes each object many times, then dividing
by a group action that implements the isomorphisms. There are two ways to
do this, one algebro-geometric (complex) and one symplecto-geometric
(real), and I'll give some idea of why they agree.

The main example will be the space of $N$ ordered points on the
Riemann sphere, modulo M\"obius transformations. These are unstable if

Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

Waldspurger established a connection between the vanishing of
certain L-values and the vanishing of period integrals over tori. Subsequent
work of Gross, Zhang and others has, in certain cases, made this connection
more precise. I will describe a different approach, via the relative trace
formula, to obtain such refinements in general. This is joint work with
Kimball Martin.

The study of minimal surfaces is a central problem in geometry and analysis that dates back to the 1700's when the catenoid and helicoid were discovered. I will survey recent advances, focusing on joint work with Bill Minicozzi that describes the structure of a general embedded minimal surface in terms of the catenoid and helicoid. I hope to give an overview of how these results have played a role in the solution of some old problems.

One of the most important choices a graduate student will make in their graduate career will be choosing a thesis advisor. It is never too early for students to begin thinking about choosing an area of specialty and choosing among the 55 UCSD math faculty who might supervise them.
How did other students find a thesis advisor? What are the key factors to consider when choosing an advisor? What do professors look for before they accept a student as their thesis student? How does finding a thesis advisor lead to find a problem for a thesis?
The goal of this seminar is to share information about the process of finding a thesis advisor. We will have three graduate students-Manda Riehl, Jonathan Armel and Kristin Jehring describe their experiences finding a thesis advisor and what happens after a thesis advisor is found.
We will also have two faculty, Lance Small and Michael Holst describe what they look for in a graduate student before they accept him or her as their thesis student. Jim Lin will serve as moderator. All students, especially first, second and third year students, are cordially invited to attend.