March 8, 2012
University of Virginia professor to present 45th William J. Spencer Lecture today
Nicholas Kuhn, professor of mathematics at the University of Virginia, will present the 45th William J. Spencer Lecture at 2:30 p.m. today in 1018 Throckmorton Hall. Kuhn’s lecture is titled “Polynomial Functors in Homotopy.”
The calculation of the homotopy groups of a topological space is notoriously difficult. By contrast, homology groups are much easier to calculate because they satisfy the nice Meyer-Vietoris decomposition rule.
Said differently, homology groups are the homotopy groups of a linear functor from spaces to spaces. Kuhn will discuss what this means and how this can be generalized to the notion of a polynomial functor. A general functor has a best approximation by a polynomial functor of degree n, with nth Taylor coefficient a geometric object with an action of the nth symmetric group.
When applied to the identity functor, and then specialized to spheres, many miracles happen, that give a conceptual explanation for a number of aspects of the homotopy groups of spheres.