Selected Mathematical Topics of Interest to the Faculty

Algebraic Combinatorics and Its Applications

Combinatorics could be defined as the study of arrangements of objects according to specified rules. It is an area of mathematics that has experienced tremendous growth during the last few decades. Algebraic combinatorics is concerned with using combinatorial structures for modeling complex algebraic objects and for performing complex computations. Applications have been found to the representation theory of groups and algebras, commutative algebra, algebraic geometry, algebraic topology etc. Professor Cristian Lenart is interested in developing combinatorial models for the representation theory of semisimple Lie algebras (and, more generally, symmetrizable Kac-Moody algebras), as well as for the geometry of generalized flag varieties; he is also interested in the relation of combinatorics to various formal group laws in algebraic topology.

The Borel and Novikov Conjectures

Both conjectures address the basic question of distinguishing between homotopy equivalent topological or differential manifolds. Most of the techniques of algebraic topology are designed to study the homotopy type of spaces, so new and finer algebraic and topological techniques need to be developed. This often leads back to problems and ideas from algebraic topology and their often surprising relations with geometry. This topic is of great interest to Boris Goldfarb.

The Theory of Bergman Spaces

The theory of Bergman spaces is a modern research area in Complex Analysis and Operator Theory. This department offers an excellent research environment for anyone interested in Bergman spaces; several faculty members (Professors Korenblum, Range, Stessin, and Zhu) of the department are very much involved in the the recent developments in this area.

Hopf Algebras and Local Galois Module Theory

A research area in algebraic number theory that began with the 1984 Albany Ph. D. dissertation of Susan Hurley under the direction of Professor Lindsay Childs, who in February 2000 completed the manuscript for a monograph Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory.

The Non-Linear Similarity Problem

Two matrices A and B are linearly similar if there is an invertible matrix C with CA=BC. Professor Mark Steinberger studies the relationship between linear similarity and the following topological analogue: the n by n real matrices A and B are topologically similar if there is a continuous bijection with continuous inverse h:R^n -> R^n of Euclidean n-space with hA=Bh, where A and B are now regarded as linear functions from R^n to itself. This problem is fundamental in studying group actions on manifolds. It is the first step in asking whether two smooth actions of a finite group on a manifold can be G-homeomorphic.

Wiles, Ribet, Shimura-Taniyama-Weil and Fermat's Last Theorem

A partially affirmative answer by Andrew Wiles of Princeton University in 1993 to a question about elliptic curves, that had lingered possibly since the 1930's and at least since the time of a 1955 mathematical meeting in Japan, generated a great deal of interest due to its connection with the unproved proposition known as "Fermat's Last Theorem" (1637). Basically, thanks to Ken Ribet (1986) and others, we knew that FLT was a consequence of knowing that every elliptic curve defined by a cubic with rational coefficients is "modular". Wiles showed that every semi-stable elliptic curve is modular, and that is enough for FLT. In 1999 Breuil, Conrad, Diamond, and Taylor showed that every elliptic curve is modular.

Professors Antun Milas and Anupam Srivastav have active research interests related to the area of elliptic curves.

Professor William Hammond recently refreshed the write-up of his 1993 survey talk on the background of the excitement in that year over the work of Andrew Wiles. In refreshing that write-up he had, in particular, the purpose of demonstrating a new system called GELLMU of XML-based, TeX-related infra-structure to facilitate the simultaneous generation of mathematical articles for both print and online presentation.

Mathematics and physics

The commentaries in this area posted by John Baez in the Usenet newsgroup sci.math.research and archived at http://math.ucr.edu/home/baez/TWF.html give an excellent illustration of how most of the mathematics created during the “axiomatic era” (1920-1960), based solely on its intrinsic interest to the discipline by itself, is turning out to be extremely useful as the study of physics evolves.

The URL recipe for individual weeks appears to be:
http://math.ucr.edu/home/baez/weekN.html
where N is the week number.