Claremont Center for the Mathematical Sciences

In this expository talk we look at look at two classes of metric spaces lying between the compact metric spaces and the complete metric spaces. The first is the well-known class of UC spaces, best known as the class of spaces on which each continuous function with values in a metric space is uniformly continuous. The second is the class of cofinally complete metric spaces, best known as the class of spaces in which
each “cofinally Cauchy” sequence has a cluster point. We show how these classes are in many ways parallel universes, and explain why.

This week's colloquium will be postponed due to the Poster Session held at the CMC Athenaeum. Details of the Poster Session event, forthcoming.

Alvin's life in mathematics is exemplary for two reasons: he insisted, and he persisted. He insisted that mathematics is something that people do, it is inseparable from people. And he persisted and persisted! In the spirit of Alvin's example, I will try to lead an informal conversation, linking two well-known controversies: The first controversial issue is about the many different approaches and styles to college math teaching. Noteworthy examples are from Emil Artin, Alonzo Church, William Feller, Robert Lee Moore, Richard Courant, George Polya, and Clarence Stephens. On the other hand, there is a controversy in the philosophy of mathematics about the nature of mathematical entities or \objects." Mathematical entities might actually be self-subsisting, external, outside of space and time (\Platonism"). Or they might just be arrangements of arbitrary symbols to form \patterns" (formalism, structuralism). Or they might be concepts, shared by thinking human beings (humanism). Is there an important linkage between these two controversies, about the teaching of mathematics and the nature of mathematics?

Much as our concept of `number' has evolved over time, what we mean by `knots' has recently undergone its own volutionary generalization. We will explore new types of generalized knots including virtual knots, singular knots, at virtual knots and more. These new knot types motivate related algebraic structures such as kei, quandles, racks and biquandles. This talk is based on an article scheduled to appear in Notices of the AMS in 2010.

TBA

TBA

TBA

We modify existing models of bacteriophage growth on an
exponentially growing bacterial population by including (1) density
dependent phage attack rates and (2) loss to phage due to adsorption
to both infected and uninfected bacteria. The effects of these
modifications on key pharmacokinetic parameters associated with
phage therapy are examined. More general phage growth models are
explored which account for infection-age of bacteria, bacteria-phage
complex formation, and decoupling phage progeny release from host
cell lysis.

A common theme of enumerative combinatorics are counting functions given
by polynomials that are evaluated at positive integers. For example, one
proves in any introductory graph theory course that the number of proper
k-colorings of a given graph G is a polynomial in k, the "chromatic
polynomial" of G. Combinatorial reciprocity theorems give interpretations
of these polynomials at negative integers. For example, when we evaluate
the chromatic polynomial of G at -1, we obtain (up to a sign) the number
of acyclic orientations of G, that is, those orientations of G that do not
contain a coherent cycle.

Combinatorics is abundant with polynomials that count something when
evaluated at positive integers, and many of these polynomials have a
completely different interpretation when evaluated at negative
integers. We follow a common thread of chromatic and flow polynomials of
graphs, the Euler characteristic of polyhedra, Ehrhart polynomials
enumerating integer points in polytopes, and characteristic polynomials of
hyperplane arrangements.

The great physicist James Clerk Maxwell was one of the first to realize that what we'd now call topological ideas were destined to play a key part in the future of physics. In an 1870 paper "On Hills and Dales" he devoted sustained attention to the apparently straightforward question of counting the "critical points" of the Earth's landscape: the mountain peaks, passes, and ocean deeps that stand out clearly on a contour map. In doing this he anticipated the key ideas of Morse theory, a central notion of 20th-century mathematics. Around 1980 Ed Witten reconnected Morse theory to physics by interpreting it in terms of the "tunneling" of certain (virtual) particles associated to the topology of a manifold. I'll explain this story in elementary terms and show how it relates to other key ideas in modern geometric analysis.