A matrix is stable if its eigenvalues are in the left half of the complex
plane. More practical stability measures include the pseudospectral abscissa
(maximum real part of the pseudospectrum) and the distance to instability
(minimum norm perturbation required to make a stable matrix unstable).
Likewise, the classical definition of controllability is not as useful
as a measure of the distance to uncontrollability.

Matrices often arise in applications as parameter dependent.
Optimization of stability or controllability measures over parameters is
challenging because the objective functions are nonsmooth and nonconvex.
We solve such optimization problems, locally at least, via a novel method
based on gradient sampling. One of our stability optimization examples is a
difficult problem from the control literature: finding stable low-order
controllers for a model of a Boeing 767 at a flutter condition. We also give
a controllability optimization example and explain its connection with an
interesting open question: how many connected components are possible for
pseudospectra of rectangular matrices?

joint work with
James V. Burke, University of Washington, Seattle, WA
Adrian S. Lewis, Simon Fraser University, Burnaby, BC, Canada

Have you ever seen someone juggle and wonder how he or she does
it? Or, are you able to juggle but have wondered how the process might be
described mathematically? In this talk, I will introduce the concept of a
juggling sequence and explain how juggling sequences can be used to
describe simple juggling patterns and will address some of the
mathematical questions related to juggling sequences. I will also
illustrate some juggling patterns by juggling them (when I\'m not picking
up the balls off the floor) and by using a juggling animator program to
juggle patterns that are too difficult for me.

Refreshments will be served!

In this talk, we will explore the connections between
special values of zeta functions, invariants of toric
varieties, and generalized Dedekind sums. We use invariants
arising in formulas for the Todd class of a toric variety
to give formulas for the zeta function of a real quadratic
number field at nonpositive integers.

We study the evolutionary game dynamics of a two-strategy game. In infinite populations, the well-known replicator equations describe the deterministic evolutionary dynamics. There are three generic selection scenarios. The dynamics of a finite group of players has received little attention. We provide a framework for studying stochastic evolutionary game dynamics in finite populations. We define a Moran process with frequency dependent fitness. We find that there are eight selection scenarios. And for a given payoff matrix, a number of these sceanrios can occur for different population size. Our results have interesting applications in biology and economics. In particular,
we obtain new results on the evolution of cooperation in the classic repeated Prisoner\'s Dilemma game. This is joint work with Drew Fudenberg and Martin Nowak.

A Topological Quantum Field Theory (TQFT) is a functorial extension of invariants of 3-manifolds to manifolds with boundaries. They are thus highly structured and imply, for example, nontrivial representations of the mapping class groups. A large family of such TQFT\'s is given by the Witten-Reshetikhin-Turaev TQFT\'s. Assuming a mild modification of the TQFT axioms it is possible to define them over the cyclotomic integers (rather than just the complex numbers). The rich ideal structure of this ring combined with the modified functoriality yields a new and quite subtle tool to investigate various properties of the mapping class groups, specific 3-manifolds, and some of their classical invariants. In the talk I will give several examples of such applications.