Dedekind sums in geometry, topology, and arithmetic (09w5102)


(San Francisco State University)

(University of Massachusetts)

Adam Sikora (SUNY Buffalo)


The Banff International Research Station will host the "Dedekind sums in geometry, topology, and arithmetic" workshop next week, October 11 - October 16, 2009.

This workshop plans to focus on intriguing relationships between several different fields of mathematics: number theory, convex geometry, and topology. Number theory is one of the oldest fields in mathematics. It studies properties of the whole numbers and fractions as well as more exotic constructions, and today fines many applications in computer science and cryptography. Convex geometry studies the properties of rigid shapes in space built from points, lines, and polygons, such as the pyramid and the cube. Topology is also known as "rubber-sheet" geometry: its practitioners investigate
properties of geometric objects that remain unchanged under continuous

Although these branches of mathematics sound wildly different, it turns out there are fascinating relationships between them. One relationship is captured by Dedekind sums. These sums often arise in the different subjects in the course of computing other quantities, and usually reflect some kind of geometric complexity in the quantities under study. The goals of the workshop are to explore connections between these subjects using Dedekind sums as a bridge, to try to understand the significance of Dedekind sums in these fields,
and to try to unearth new relationships between these areas.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).