IAS Seminar: Partitioning numbers into squares

IAS Guest Speaker Professor Don Zagier delivers a seminar on their research.

In the 18th century, Euler introduced the number p(n) of partitions of an integer n into positive parts (e.g. p(4)=5 because 4 can be partitioned in five ways: 1+1+1+1, 1+1+2, 1+3, 2+2, or 4) and its generating function p(0)+p(1)x+p(2)x^2+..., whose reciprocal later turned out to be the simplest example of a 'modular form'. A question arising in the theory of gravity and integrable systems required studying the numbers of partitions of a number into positive squares or higher powers. The corresponding generating functions are no longer modular, but satisfy certain surprising 'modular-type' transformation equations. There are also analogues of the famous Hardy-Ramanujan approximate partition formula, but far more subtle. Both results were discovered by numerical computations together with some nice tricks. The only prerequisites for the talk are basic mathematical background and some interest in numerical techniques.

Arrivals from 2.45pm for a 3pm start.