Math.Koll. am 18.10.2006/Prof. Rosengrens

[Institut für Mathematik]

Mathematisches Kolloquium

EINLADUNG

zu einem

VORTRAG

von

Prof. Hjalmar Rosengrens
(Chalmers University of Technology and Göteborg University)

mit dem Thema:

''Elliptic pfaffians and sums of squares''

Abstract: A classical problem of number theory is to count the
representations of an integer n as a sum of k squares. For instance, Gauss
proved that the number of integer solutions to n=x^2+y^2 equals
4(d_1(n)-d_3(n)), where d_i(n) denotes the number of divisors to n that are
congruent to i modulo 4. Jacobi obtained similar results for sums of four,
six, and eight squares.
In 1996, Stephen Milne announced a far-reaching generalization of Jacobi's
four and eight squares formulas, giving exact formulas for the number of
representations of an integer as a sum of 4m^2 squares and 4m(m+1) squares,
for any m. Milne's proof is based on continued fractions. Ono, and Long and
Yang, have given alternative proofs based on modular forms.
In this talk we will present a third approach to Milne's formulas, using
explicit evaluation of pfaffians with elliptic function entries. This puts
Milne's results in a wider context, by relating them to correlation
functions for random hermitian matrices, classical orthogonal polynomials
and symmetric functions.

Zeit: Mittwoch, 18. Oktober 2006, 15.00 Uhr (Kaffeejause), anschlieszend
15.15 Uhr Vortrag

Ort: Fakultaet fuer Mathematik der Universitaet Wien, Nordbergstr. 15,
Seminarraum C 2.09

Harald Rindler
Christian Krattenthaler