Special Structures in Riemannian Geometry (08w5039)

Mathematicians who work in the area of modern geometry study the kinds of shapes that can exist, usually of higher dimension, and the properties that these shapes possess. Consider, for example, a two-dimensional shape. This means that at every point on the shape we have two perpendicular directions in which we can move. It turns out that the way that the shape can curve in these two different directions is related to the number of holes that the shape has. On a sphere, which has no holes, at each point the sphere curves toward its center in every direction. Meanwhile on a donut, there are some points at which the shape always curves toward the center (on the outer edge, for example), and there are some points (like the inner edge) at which the shape curves inward in one direction and outward in the perpendicular direction.

For higher dimensional shapes, many more surprising and complicated relationships exist between the way a shape can curve and its other geometric properties. The dimension of shape is just the number of independent quantities that are needed to describe it. Such a shape need not be visualizable, and does not have to represent physical space. A remarkable fact is that modern theories of physics, which attempt to understand the relationship between Einstein's theory of general relativity and quantum mechanics, require for their description the use of such higher dimensional shapes that possess some special properties. Mathematicians have been working together with these physicists for the last thirty years trying to understand these mysterious and yet beautiful relationships. The more they learn, the richer and more complex this marriage of physics and geometry appears to be.

This workshop, held at the Banff International Research Station on Feb 17-22, 2008, brings together these researchers to continue the investigations into these fundamental scientific questions.

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).