L3

The classical associahedra are cell complexes, in fact polytopes,

introduced by Stasheff to parametrize the multivariate operations

naturally occurring on loop spaces of connected spaces.

They form a topological operad $ Ass_\infty $ (which provides a resolution

of the operad $ Ass $ governing spaces-with-associative-multiplication)

and the complexes of cellular chains on the associahedra form a dg

operad governing $A_\infty$-algebras (that is, a resolution of the

operad governing associative algebras).

In classical applications it was not necessary to consider units for

multiplication, or it was assumed units were strict. The introduction

of non-strict units into the picture was considerably harder:

Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in

their work on Lagrangian intersection Floer theory, and equivalent

descriptions of the dg operad for homotopy unital $A_\infty$-algebras

have now been given, for example, by Lyubashenko and by Milles-Hirsch.

In this talk we present the "missing link": a cellular topological

operad $uAss_\infty$ of "unital associahedra", providing a resolution

for the operad governing topological monoids, such that the cellular

chains on $uAss_\infty$ is precisely the dg operad of

Fukaya-Ono-Oh-Ohta.

(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)

I will discuss some aspects of the simplicial theory of

infinity-categories which originates with Boardman and Vogt, and has

recently been developed by Joyal, Lurie and others. The main purpose of

the talk will be to present an extension of this theory which covers

infinity-operads. It is based on a modification of the notion of

simplicial set, called 'dendroidal set'. One of the main results is that

the category of dendroidal sets carries a monoidal Quillen model

structure, in which the fibrant objects are precisely the infinity

operads,and which contains the Joyal model structure for

infinity-categories as a full subcategory.

(The lecture will be mainly based on joint work with Denis-Charles

Cisinski.)

Three short talks by the authors of essays on topics related to c3 Algebraic topology: Whitehead's theorem, Cohomology of fibre bundles, Division algebras

The WZW functional for a map from a surface to a Lie group has a role in the theory of harmonic maps, and it also arises as the determinant of a d-bar operator on the surface, as the action functional for a 2-dimensional quantum field theory, as the partition function of 3-dimensional Chern-Simons theory on a manifold with boundary, and as the norm-squared of a state-vector. It is intimately related to the quantization of the symplectic manifold of flat bundles on the surface, a fascinating test-case for different approaches to geometric quantization. It is also interesting as an example of interpolation between commutative and noncommutative geometry. I shall try to give an overview of the area, focussing on the aspects which are still not well understood.

I will give an overview of the description of imaginaries in algebraically closed (and some other) valued fields, and then discuss the related issue for valued fields with analytic structure (in the sense of Lipshitz-Robinson, and Denef – van Den Dries). In particular, I will describe joint work with Haskell and Hrushovski showing that in characteristic 0, elimination of imaginaries in the `geometric sorts’ of ACVF no longer holds if restricted exponentiation is definable.

This talk will be accessible to non-specialists and in particular details how model theory naturally leads to specific representations of abelian group rings as rings of global sections. The model-theoretic approach is motivated by algebraic results of Amitsur on the Semisimplicity Problem, on which a brief discussion will first be given.

To each additive definable category there is attached its category of pp-imaginaries. This is abelian and every small abelian category arises in this way. The connection may be expressed as an equivalence of 2-categories. We describe two associated spectra (Ziegler and Zariski) which have arisen in the model theory of modules.