Mechanical Engineering Building Room 102
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 2 hours long with a break in the middle in order to cover each application throughly and have group discussions. 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 with background in algebra and combinatorics. Anyone interested in research opportunities for the following summer should contact the professors at the beginning of the course to discuss the options.
Assignments: In class assignments will be an important part of the course. During the break for each lecture, several easy questions will be posed. These are meant to insure everyone is understanding the material and give the students a chance to ask for clarifications as necessary. Harder problems will also be given. Everyone is expected to consider the harder problems and turn in any significant progress.
Grades: Grades are entirely based on participation. Students are expected to ask questions, solve assigned problems, and present material. Never hesitate to ask a question in this forum.
Last modified: Tue Mar 22 12:17:12 PST 2005