Math 583GC
Sara Billey and James Morrow
Spring 2005, Friday 2:30-5:30
A matrix is totally positive if every minor is positive. Similarly, a
matrix is totally nonnegative if every minor is nonnegative. Such matrices
appear in a wide range of mathematical subjects including combinatorics,
probability, stochastic processes, representation theory, and inverse
problems. For example, totally positive matrices characterize certain
invertible electrical networks studied by Curtis, Morrow and several of
their REU students. Other examples come from the recent work of Fomin and
Zelevinsky on stratified spaces, double Bruhat cells, and cluster algebras.
Total positivity is a very active area of research. This course will
survey some of the basic theorems in total positivity and a variety of
applications particularly related to graphs and electrical networks with an
eye toward current research. We will read several recent papers related to
totally positive matrices. The lectures will be 3 hours long in order to
cover each application throughly. Students will be expected to contribute
to the lectures through presentations and discussion. This class would be
appropriate for first year graduate students or advanced undergraduates
interested in REU opportunities for the following summer.
Two good references for more information on applications of total
positivity that will be relevant to this course are:
- Edward B. Curtis and James A. Morrow, Inverse Problems for
Electrical Networks World Scientific Pub Co; (May 2000).
- S.Fomin and A.Zelevinsky, Total positivity: tests and
parametrizations, Math. Intelligencer 22 (2000), 23-33.