## UNIVERSITY OF PENNSYLVANIA

## Department of Mathematics

Fall 2003 -Hans Rademacher Lectures in Mathematics## Mathematics Department, University of California, Berkeley

Richard E. Borcherds

will deliver four lectures on

## Modular forms, Lie algebras & infinite products

Modular forms

Monday....September 22, 2003....4:30pm

Abstract:What are modular forms and why are they so important?

The Hardy-Ramanujan-Rademacher series

Tuesday....September 23, 2003....4:30pmAbstract:In a notoriously difficult paper, Hardy and Ramanujan gave a non-convergent but asymptotic series for the partition function. While preparing notes on their work, Rademacher found a simpler series that converged exactly.

Modular forms and infinite products

Wednesday....September 24, 2003....4:30pmAbstract:Some modular forms can be written as infinite products; a classical example is Euler's infinite product for the generating function of partitions. The zeros of these modular forms correspond to the terms of the Hardy-Ramanujan-Rademacher series.

Lie algebras and infinite products

Thursday....September 25, 2003....3:30pm

Abstract:The representations of semisimple Lie algebras are described by the Weyl character formula, whose denominator is given as a product. There is an analogue for some infinite dimensional Lie algebras, where the denominator turns out to be a sort of modular form given as an infinite product. In particular the root multiplicities of these Lie algebras can be calculated using the Hardy-Ramanujan-Rademacher series.

Previous Rademacher Lecturers

Lectures on Monday, Tuesday and Wednesday will be held in room A-6 of the David Rittenhouse Laboratory,

Thursday's lecture will be held in A-5 of the David Rittenhouse Laboratory

corner of 33rd and Walnut Streets, Philadelphia, PA.

Tea: 4E17 David Rittenhouse Laboratory, preceding the lectures at 4:00pm.For further information, please call the Department of Mathematics at the University of Pennsylvania - 215-898-8627.