Coassembly Maps, Gauge Groups, and K-Theory
Cary Malkiewich, Stanford University
Location: Stevenson 1432
Calculus of functors is a powerful technique from homotopy theory, which studies computationally difficult constructions by means of "linear approximations." We will describe a new variant of this calculus, based on the embedding calculus of Weiss and Goodwillie. This theory provides us with a sequence of approximations to the stable gauge group of a principal bundle, in which the linear approximation is the Cohen-Jones string topology spectrum. We will finish with some future applications to algebraic K-theory, a subject which provides a powerful (but difficult to compute) invariant for rings, algebras, and groups.
Error Bounds for Kernel-Based Numerical Differentiation
Oleg Davydov, Strathclyde University (Scotland)
Location: Stevenson 1307
The literature on meshless methods observed that kernel-based numerical differentiation formulae are highly accurate and robust. We present error bounds for such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds for kernel-based interpolation but are not applicable in this setting. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we show that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. The talk is based on joint research with Robert Schaback.
Counting Non-Simple Closed Geodesics on a Pair of Pants
Jenya Sapir, Stanford University
Location: Stevenson 1310
We give coarse bounds on the number of (non-simple) closed geodesics on a pair of pants with upper bounds on both length and intersection number. We acheive our bounds by transforming the problem of counting geodesics into a combinatorial problem of counting words with certain conditions. If time permits, we will give an idea of how to extend these results to a general surface, and give an application.