## UNIVERSITY OF PENNSYLVANIA

## Department of Mathematics

Spring 2003 -Hans Rademacher Lectures in Mathematics## Mathematics Department, Pennsylvania State University

George E. Andrews

will deliver four lectures on

## Rademacher, Ramanujan, Rogers and Partitions

Rademacher and Ramanujan

Monday....March 31, 2003....4:30pm

Abstract:Rademacher was one of the great analytic number theorists of the 20th century. He perfected the Hardy-Ramanujan formula for p(n), the number of partitions of n, and with his students extended his methods to many aspects of additive number theory and modular forms. We shall survey some of his work which leads up to still open conjectures that either Rademacher made himself or that stemmed from his ideas. In addition to this, we shall briefly recount how recent work of Augustine Munagi provides support for one of the most tantalizing of these problems.

The Lost Notebook of Ramanujan

Tuesday....April 1, 2003....4:30pmAbstract:In 1976 quite by accident, I stumbled across a collection of about 100 sheets of mathematics in Ramanujan's handwriting; they were stored in a box in the Trinity College Library in Cambridge. I titled this collection "Ramanujan's Lost Notebook" to distinguish it from the famous notebooks that he had prepared earlier in his life. On and off for the past 27 years, I have studied these wild and confusing pages. Some of the weirder results have yielded entirely new lines of research. I will try to provide an account of where these efforts have led. The result that most frightened me (I tried to ignore it for 26 years) will conclude the presentation.

Rogers and Ramanujan

Wednessday....April 2, 2003....4:30pmAbstract:In Hardy's encomium for Ramanujan he asserts: "I am inclined to think that it was in the theory of partitions, and the allied parts of the theories of elliptic functions and continued fractions, that Ramanujan shews at his very best work...It would be difficult to find more beautiful formulae than the 'Rogers-Ramanujan' identities...; but here Ramanujan must take second place to Prof. Rogers." This pair of famous formulas has a fascinating history. Their discovery resembles detective fiction, and their aid in the description of the behaviour of liquid helium on a graphite plate came as a complete surprise. We shall try to give an account of both the history of the Rogers-Ramanujan identities as well as current research.

Partition Computations and Ramanujan

Thursday....April 3, 2003....4:00pm

Abstract:In this talk I will describe my work with mathematicians and computer scientists at the University of Linz (Paule, Riese and Zimmermann) to implement algorithms in computer algebra that are effective and fruitful in the study of partitions. We shall concentrate on two projects: The Omega package and The Generalized Engel Transformation. The first of these arises from the work of P.A. MacMahon, the computational and combinatorial mathematician whose work was invaluable to Ramanujan and Hardy. The second arises from a study by A. and J. Knopfmacher extending ideas of Engel, a student of Perron. When this algorithm is applied to the product side of either Rogers-Ramanujan identity, it produces the q-series side as output. The object of this talk will be to introduce gently the methods of each algorithm and to describe some of the applications.

Previous Rademacher Lecturers

All lectures will be held in room A-6 of the David Rittenhouse Laboratory,

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

Tea: 4E17 David Rittenhouse Laboratory, preceding the lectures at 4:00pm. Tea on Thursday will be at 3:30pmFor further information, please call the Department of Mathematics at the University of Pennsylvania - 215-898-8627.