TBA
I will discuss groups generated by reflections on the spaces on constant curvature: the sphere, Euclidean space, or hyperbolic space. Then I will define the notion of a "Coxeter group" (or an "abstract reflection group"). I will show how to realize any given Coxeter group as a group generated by reflections on a particularly nice space. In the second talk I will give applications of this construction to some problems in geometric group theory and topology.
L-functions are very interesting, and equally mysterious, complex analytic invariants attached to certain arithmetic objects, geometric objects, and analytic objects. L-functions seem to have a wonderful ability to interpolate from the local to the global. The paradigm of an L-function is the Riemann zeta function and the arithmetic contained in it, such as the prime number theorem. Most of our understanding and expectations of L-functions come from examples.
In these two talks I hope to give a feeling for what L-functions are and some of the things we expect from them by discussing several examples, including the Riemann zeta function, the Dirichlet L-functions, the L-function of an elliptic curve, and perhaps Artin L-functions. I will also discuss the analytic side, so L-functions attached to modular or automorphic forms. If there is time, perhaps I may even be able to give a small introduction to the Langlands program.
Differential equations and numerical simulation have been widely used in modeling biological systems. In this talk, I will introduce two biological systems related to cell signaling:
Hurwitz numbers are certain combinatorial quantities that admit several different-looking definitions. One way to define Hurwitz numbers is to say that the are counts of covers of the Riemann sphere with specific ramifications. Hurwitz numbers are related to various different areas of mathematics, such as moduli theory and integrable systems. In this talk I will attempt to discuss some of the definitions of Hurwitz numbers, and outline the connections to other areas.
A significant difficulty in extending the theory of hyperbolic conservation laws beyond a single space dimension is identifying the singularities in solutions that may appear in multidimensional problems.
However, while searching for a simple prototype, I have come across a problem that is simple and natural. It appears to have something new to say even about the classical problem of steady transonic flow, and the solution uses the familiar tools named in the title.
The problem and its solution will be expounded in two talks, to an audience that is not assumed to have any prior acquaintance with the theory of conservation laws.