The symplectic fiber sum construction is one of the few surgery technique available to construct new symplectic manifolds out of olds ones. The goal of the talk is to explain the construction and, if time permits, to give some applications .

Various model structures for the category of pro objects in simplicial sheaves will be displayed. These structures extend the standard homotopy theory for simplicial sheaves in the sense that their weak equivalences are locally defined. They also extend known results for simplicial sets, and thus, for example, give a common framework for modern simplicial sheaf homotopy theory and classical \'etale homotopy theory. One or more applications will be discussed.

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

I will try to cover the following topics: 1. Equivariant cohomology: Definition, motivation, 2. Two constructions of E, 3. Cartan Model and some examples.

We will describe basic facts about actions of finite groups on compact manifolds and then recall what is known about finite groups acting without fixed points on spheres. From there we will discuss more recent work on characterizing those groups acting freely on a product of two spheres. This will involve ingredients from representation theory, bundle theory and group cohomology.

This is the second part of a talk on stacks, and will continue from the first. This talk will particularly look at algebraic stacks.

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer $q>1$ and every topological group $G$, with realizations $B(q,G)$ that filter the classifying space $BG$. In particular for $q=2$ this yields a single space $B(2,G)$ assembled from all the $n$-tuples of commuting elements in $G$. Homotopy properties of the $B(q,G)$ will be described for finite groups, and cohomology calculations provided for compact Lie groups. Recent results on understanding both the number and stable homotopy type of the components of related spaces of representations will also be discussed.