# Mathematics Colloquium

#### Thursday, April 18, 1996, 4:00pm

#### 102 Bradley Hall

## Professor Andrei Zelevinsky

#### Northeastern University

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.