| DATE: | Tuesday, February 1, 2000 |
| TIME: | 4:30 - 5:30 PM |
| PLACE: | SW 323 |
| SPEAKER: | Matthias Beck |
| TITLE: | Dedekind sums: a geometric viewpoint |
We define a generalized Dedekind sum as an expression of
the form
Here the sum is taken over all a-th roots of unity for which the summand
is not singular. Sums of this type have intrigued mathematicians from
various areas such as Number Theory, Topology, and Combinatorial Geometry
since their introduction by Dedekind in 1892. Our definition, which is due
to Gessel, includes as special cases the classical Dedekind sum
(essentially the case n=2, t=0) and its generalizations due to
Rademacher (n=2, arbitrary t), and Zagier (t=0,
arbitrary n). Our
interest in these sums stems from the appearance of Dedekind's and
Zagier's sums in lattice point count formulas for polytopes. Using
generating functions, we show that generalized Dedekind sums are natural
ingredients for such formulas. By applying our approach to a certain
class of polytopes, we obtain reciprocity laws of Dedekind, Zagier, and
Gessel as immediate 'geometric' corollaries to our formulas.