Abstracts for Plenary Speakers

s for Plenary Speakers John Baez (University of California, Riverside) “Higher Gauge Theory: 2-Connections” 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 at http://math.ucr.edu/home/baez/union/) David Cox (Amherst College) “Two stories about implicitization and commutative algebra” 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. Jesper Grodal (University of Chicago) “Homotopical group theory” 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. Abstracts for Parallel Sessionss for Parallel Sessions John Armstrong (Yale University) “The Extension of Knot Groups to Tangles” Abstract: The extension of the knot group π1(S \K) to the category of tangles is introduced via a new category-theoretic construction. Through this presentation, a new avenue of proof for results about knot groups is opened. Mark Behrens (MIT) “Hypercohomology of categories” Abstract: How do you combine sheaf cohomology with the cohomology of a category? I will discuss enriched categories where the objects, as well as the morphisms, are enriched. What form does a coefficient system (enriched functor) take for this set-up? This talk is largely expository, combining a definition in old paper of Morava’s with some modern terminology.