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# Abstracts

### Andy Baker

TAQ and all that

I will survey the basic notions of Topological Andr\'e Quillen (co)homology for commutative $S$-algebras and in particular discuss some sample calculations related to Thom spectra.

### Eugenia Cheng

The universal loop space operad and generalisations
Grothendieck hypothesised that "\omega-groupoids'' could be used to model
homotopy types, and many different ways of approaching this have been
proposed in recent years.  I will discuss a theory proposed by
Batanin/Leinster using a generalised form of operad called "globular
operads''.  There are various analogies with the use of operads to study
loop spaces; I will recall the loop space operad E called universal'' by
Salvatore, and present an analogous globular operad G which can be used to recognise the fundamental \omega-groupoid of a space.  I will give a
categorical proof of the universal property of E, and generalise it to
prove a universal property of G.

### Jeffrey Giansiracusa

Formality of the framed little discs operad

Tamarkin and Kontsevich proved that the little n-discs operad is formal in the following sense: the operad formed by taking its real chains is quasi-isomorphic as an operad to its real homology. This theorem had important applications in deformation quantization and also in computing the homology of spaces of knots. The framed little discs operad is an interesting relative - it actually forms a cyclic operad, meaning that inputs and outputs can be interchanged. Paolo Salvatore and I prove that the framed little 2-discs is formal as a *cyclic* operad. The proof introduces a new and interesting graph complex in which the differential is a combination of edge contractions and deletions.

### Ian Grojnowski

Almost local differential operators

One of the most fundamental theorems in representation theory
is the Beilinson-Bernstein theorem, which describes the representations
of a Lie algebra such as $sl_n$ in much simpler terms---as
sheaves of modules for differential operators on a smooth algebraic variety.

I will describe this theorem, and some work in progress to generalise
it to describe  non-commutative deformations of enveloping algebras
in terms of 'almost local differential operators'.

The goal is to compute the algebraic K-theory of these algebras
by reducing it to the work of Hesselholt and Madsen.

### Jelena Grbic

The homotopy theory of moment angle complexes

In recent years, toric topology has been recognised as an area of topology which has deep connections with combinatorics, algebraic and symplectic geometry, and homological algebra.

The moment-angle complex, an object initially studied in algebraic geometry, has become one of the central objects of toric topology. Thus its homotopy type, stable and unstable, is of great interest. In this talk I'll explain a relation between simplicial complexes and moment-angle complexes and determine the unstable homotopy type of moment-angle complexes related to shifted complexes.

In addition, I'll describe how higher Whitehead products arise in the homotopy theory of moment-angle complexes and their relations with the loop homology of Davis-Januszkiewicz spaces.

Moduli spaces and topology
The lecture will survey a number of results about the stable homological structure of the mapping class group,the moduli space of surfaces in a background space and the moduli space of manifolds embedded in high dimensional euclidian space.Applications will include the stable structure of the Atiyah-Bott moduli space of flat connections on principal bundles over Riemann surfaces.I will also discuss the relationship between the moduli space of manifolds and Waldhausen's algebraic K-theory of spaces.

### Martin Markl

Homotopy invariant structures in algebra and transfers

We discuss the concept of homotopy invariance in algebra and
recall some basic examples of structures having this property. We also
explain that each "reasonable" algebraic structure has its homotopy
invariant version.

We prove that homotopy invariant structures can be transferred over
maps having a one-sided homotopy inverse. We will then recall the
notion of strong homotopy equivalence and show how it explains the
relation between the above transfer property and the classical
perturbation lemmas.

### Constanze Roitzheim

Hochschild cohomology of A-infinity algebras.

In the 1960s, A-infinity algebras were introduced to study the cohomology of topological spaces with products and are now known to arise widely in various areas of mathematics. Roughly speaking, A-infinity algebras are generalisations of associative algebras. We are going to explain how to extend the definition of Hochschild cohomology from associative algebras to A-infinity algebras and how this will help solving realizability problems in topology and present some new results concerning derived A-infinity algebras.

### Shoham Samir

A noncommutative analog of a local-cohomology spectral sequence for chains on loops of certain spaces.

I will present a spectral sequence which is analogous to the local cohomology spectral sequence of Dwyer, Greenlees and Iyengar for the cochains of a space. This spectral sequence can be applied, in some cases, to the dga which is the chains on the loops of a space (with coefficients in a field). As with the local cohomology spectral sequence, when this dga is Gorenstein the spectral sequence converges to the cochains of the loop space.

### Ron Umble

On the Transfer of Multiplicative Structure.

Kadeishvili, Markl, and others have studied the transfer of multiplicative structure from an $A_\infty$-algebra $A$ to a DG module $B$ along a map $f : B \to A$. Kadeishvili assumes that $A$ and $B$ are free modules and $f$ is an inclusion; Markl assumes that $f$ has a left-homotopy inverse. To set the stage for our more general result, we discuss an example in which $A$ and $B$ have torsion and $f$ has a left but no right-homotopy inverse. Theorem. Let $\bar{f} : Hom(B^{\otimes n},B) \to Hom(B^{\otimes n},A)$ be the induced map. The $A_\infty$-algebra structure on $A$ pulls back along $f$ to an $A_\infty$-algebra structure on $B$ whenever $\bar{f}$ is a quasi-isomorphism. In fact, if $\bar{f}$ is a quasi-isomorphism, so is $f$, but not conversely. This result extends immediately to $A_\infty$-bialgebras and gives the Corollary: If $F$ is a field, $H_*(\Omega(X);F)$ is an $A_\infty$-bialgebra. We conclude with a discussion of the $A_\infty$-bialgebra operations on $H_*(\Omega(X);F)$ and the structure relations among them.

Contact details

Department of Mathematics
University of Leicester
Leicester LE1 7RH
United Kingdom

Tel.: +44 (0)116 229 7407

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