Topological methods in toric geometry, symplectic geometry and combinatorics (10w5026)


Tony Bahri (Rider University)

Fred Cohen (University of Rochester)

(University of Western Ontario)

Samuel Gitler (Cinvestav, Sam Pedro Zacatenco)

Megumi Harada (McMaster University)


The Banff International Research Station will host the "Topological methods in toric geometry, symplectic geometry and combinatorics" workshop from November 7 to 12, 2010.

A recent workshop at the Banff International Research Station brought together leading experts in toric topology and related mathematical areas. Topology is a branch of geometry which studies
the properties of geometrical figures which persist even when the figures are twisted, turned, and otherwise drastically deformed.
(For example, if a typical coffee mug with one handle were made out of Play-Dough rather than porcelain or glass, then we could, without
tearing the PlayDough, reshape and "deform" this mug into the shape of a doughnut with one hole.) The group of mathematicians gathered at BIRS for the 5-day workshop, however, were concerned with not just the geometrical properties of their figures, but also in their symmetries; that's because it often happens in science and in mathematics that symmetries help us to understand the observed phenomena of the world around us. For example, Newton formulated his universal law of gravity based on Kepler's realization that the
paths of the planets around the sun were orderly and symmetric ellipses; Watson and Crick's fundamental discoveries about DNA were partially based on their understanding of the symmetric double-helix structure of the DNA's geometry. The BIRS workshop concerned a particular kind of symmetry called toric symmetries; a simple
example is the circular symmetry of our planet in its familiar 24-hour rotation around the north-south axis.

The researchers at BIRS were concerned with further developing the general theory of geometric objects with such toric symmetries. This theory has far-reaching connections to many other areas of mathematics, even those which seem at first glance unrelated: examples include the algebraic study of equations, as well as combinatorics (which is the mathematical study of finite collections of discrete objects, a field important in modern computer science). Recently, toric topology has also made contact with engineering and robotics, in the study of the motions of robot legs as encoded by ``gait states''. Many young and promising mathematicians are working in this exciting area which has diverse applications and great potential. The workshop was explicitly designed to especially encourage these younger participants, as well as to foster among all
participants (both junior and senior) the development of interdisciplinary interaction between experts in different 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 Tecnologia (CONACYT).