# 2007 Chern Lectures

*The 2007 Chern Lectures will be delivered by Vladimir Igorevich Arnold on April 2, 4 and 6, 2007.*

Department of Mathematics, University of California, Berkeley, presents

The 2007 Chern Lectures

Vladimir Igorevich Arnold

Steklov Mathematical Institute

Moscow, Russia

**Talk 1: Statistics and topology of Morse functions**

*April 2, 10 Evans, 4:00-5:00pm*

On the 2-sphere, there is exactly 17746 topologically different Morse
functions with 4 saddles. This result, based on the combinatorics of
random graphs, was obtained only a couple of years ago (in the process
of study of Hilbert's 16th problem in real algebraic geometry on the
topological classification of polynomials).

Last year L. Nicolaescu (University of Notre Dame) proved (using methods
going back to conformal field theory and the physics theory of mirror
symmentry) a conjecture of Arnold that the number of Morse functions
with T saddles on the 2-sphere grows as T to the power 2T.

The talk will be about this research and about its analogues in the
theory of smooth fucntions on other manifolds, e.g. for functions on the
torus, and for trigonometric polynomials with a fixed Newton polyhedron
of a given affine Coxeter group.

**Talk 2: Complexity of finite sequences of zeros and ones**

*April 4, 50 Birge, 4:00-5:00pm*

**Talk 3: Statistics of Young diagrams of random permutations, and of periodic orbits of dynamical systems**

*April 6, 10 Evans, 4:00-5:00pm*

**Reception in 1015 Evans Hall following the April 4 lecture.**