Saturday, May 14, 2016
The Ohio State University
Eighteenth Avenue Building: EA 160
209 W 18th Ave
Columbus, OH 43210-1174
All refreshments and talks in EA 160
9:00 – 10:00 Coffee, bagels, fruit
10:00 – 11:00 Wouter van Limbeek (Michigan)
Strongly regularly self-covering manifolds and linear endomorphisms of tori
Abstract: Let M be a manifold that admits nontrivial cover diffeomorphic to itself. What can we then say about M? Examples are provided by tori, in which case the covering is a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that M admits the structure of a fiber bundle with torus fibers. We use this to give an application to holomorphic self-covers of Kaehler manifolds.
11:15 – 12:15 Emily Riehl (Johns Hopkins)
Model-independent ∞-category theory in the homotopy 2-category
Abstract: Quillen’s model category axioms provide a well-behaved homotopy category, spanned by the fibrant-cofibrant objects, in which the poorly behaved notion of weak equivalence is equated with a better behaved notion of homotopy equivalence. For many of the model categories presenting the homotopy theories of models of (∞, 1)-categories, their homotopy categories can be categorified, defining a homotopy 2-category with certain properties. We explain how the basic category theory of ∞-categories, objects of some homotopy 2-category, can be developed in a model independent and to a large extent model invariant fashion by working internally to the homotopy 2-category and its associated ∞-cosmos. This is joint work with Dominic Verity.
1:30 – 2:00 Coffee II
2:00 – 3:00 Ayelet Lindenstrauss (Indiana)
The Topological Hochschild Homology of Maximal Orders in Central Simple Algebras over the Rationals
Abstract: I will begin with the definition and the motivation for defining topological Hochschild homology, and survey calculations of this invariant for Eilenberg-Mac Lane spectra of rings. I will discuss methods which have proved useful for these calculations in the past, and explain how they are used in a recent calculation with H. Chan of the THH of maximal orders in central simple algebras over the rationals.
3:30 – 4:30 Michael Ching (Amherst)
Taylor towers and algebraic K-theory
Abstract: The Taylor tower of a functor (in the sense of Goodwillie's homotopy calculus) can be encoded via its symmetric sequence of derivatives together with the action of a certain comonad on that symmetric sequence. The form of the comonad depends on the source and target categories of the functor. In the case of functors from based spaces to spectra the relevant comonad can be explicitly stated in terms of the little disc operads. I will describe the resulting action on the derivatives of Waldhausen's algebraic K-theory of spaces functor. This is joint work with Greg Arone.
Algebraic K-theory can also be viewed as a functor from (associative) ring spectra to spectra and there is a comonad for this situation too. I will explain preliminary attempts to understand the action on the derivatives in this case. This is based on work of Lindenstrauss and McCarthy on the Taylor tower for the algebraic K-theory of tensor algebras.