# Mathematics Colloquia and Seminars

### Universality and Scaling for Zeros of Random Polynomials

Colloquium

 Speaker: Pavel Bleher, Indiana University-Purdue University Indianapolis & MSRI Location: 693 Kerr Start time: Mon, May 24 1999, 4:10PM

A random polynomial is a polynomial whose coefficients are (independent) random variables, which can be real or complex. The basic questions about real random polynomials are:
1. How many zeros are real?
2. What is the probability distribution of real zeros?
3. What is the probability distribution of complex zeros?
4. What are the correlations between zeros? etc., asymptotically as the degree N of the random polynomial goes to infinity.
Remarkably, in the limit when N goes to infinity the joint distribution of zeros approaches, after an appropriate rescaling, some universal limit, which is characterized by repulsion between zeros at small distances and fast decay of correlations at large distances. We will discuss all these questions in the talk along with their natural extension to zeros of complex random polynomials and to random algebraic varieties.

Coffee and Tea before the talks at 3:30 in the Fifth Floor Commons Room, Kerr Hall.