# Mathematics Colloquium

## Professor Andrei Zelevinsky

speaks on

### Total positivity: old and new

Abstract. A square matrix is totally positive if all its minors (of all sizes, including matrix entries) are positive real numbers. The study of these matrices has a long history, they play an important role in different ares of mathematics, from differential equations to combinatorics. Recently, G. Lusztig discovered a remarkable parallelism between the cone of totally positive matrices, and the canonical basis of the corresponding quantum group. We will give a survey of recent results triggered by this discovery. Some of these results (obtained jointly with A. Berenstein and S. Fomin) are of fairly classical nature. In particular, we obtain a family of criteria for total positivity: each of them says that to check that a given $n \times n$ matrix $x$ is totally positive, it is enough to check that certain $n^2$ minors of $x$ are $> 0$.

Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at the Emmy's after the talk.
Host. Dan Rockmore will be the host. Anybody who is interested in having dinner with the speaker should contact Dan at 646-3260.