## Saturday, May 14, 2016

The Ohio State University

## Eighteenth Avenue Building: EA 160

209 W 18th Ave

Columbus, OH 43210-1174

## Schedule

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.