Abstracts of plenary talks
Abstract: Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some kind of "higher gauge theory" that describes the parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, it seems we must "categorify" concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. We give an overview of higher gauge theory, with an emphasis on the concept of "2-connection" for a principal 2-bundle. ( Slides are available )
Abstract: This talk will discuss some of the unexpected relations between geometric modeling and commutative algebra. I will discuss moving lines and moving planes. These give interesting ways of representing parametric curves and surfaces in the plane and 3-dimensional space, respectively, and are closely related to syzygy modules in commutative algebra. The story for curves will involve free resolutions and the Hilbert-Burch theorem, while the surface case has relations to the Boeing 777, the Guggenheim Bilbao, resultants, local complete intersections, and the Serre conjecture.
Abstract: To a group one can associate its classifying space. This allows one to apply the toolkit of algebraic topology to groups. One can for instance localize a group at a prime, or ask if it can be suitably glued together out of smaller groups. One can also ask if groups can exist at just one prime, and to which extend such p-local objects glue together to give global groups? My talk will be an introduction to this kind of "homotopical group theory."