While a symplectic quotient coming from a Hamiltonian action of a compact Lie group is generally not a manifold (it is a stratified space), one can still define a notion of differential form on it. Indeed, one can obtain a de Rham Theorem, Poincaré Lemma, and a version of Stokes' Theorem using this de Rham complex of forms. I will show how these forms are defined, and then explore the question of intrinsicality of the complex. This question leads into a discussion of different definitions of a smooth structure on the quotient, and the pros and cons of each

In this talk, I will highlight some differences between the moduli space of Higgs bundles (in the sense of Hitchin) on a curve of positive genus and the the moduli space of "twisted" Higgs bundles at genus 0. The Betti numbers of both spaces can be determined by a localization calculation, with respect to an $S^1$ action. This was exactly Hitchin's method for obtaining the Betti numbers of the rank-2 instance of the usual Higgs moduli space. The $S^1$ fixed points are what are called "holomorphic chains": these are similar to complexes of vector bundles, but the differential (the Higgs field itself) is nilpotent with order equal to the length of the complex. I will show how the localization calculation can be made very combinatorial in the genus 0 case. The appropriate language for organizing this data is that of quivers, which we use to represent (and construct) families of chains.

In this introductory talk I will start with basic definitions and examples. After defining natural transformations I will prove the Yoneda lemma which is a fundamental result in category theory. It will indeed be a good opportunity for the beginners to befriend the theory and refresher for the others.

this will be the first talk in a short series looking over some basics of tropical geometry.

Let f be a function. The two most basic ways to approximate f with a polynomial function are, probably, the interpolation polynomial and the Taylor polynomial. The interpolation polynomial (of degree n) is determined by the n+1 numbers f(0), f(1), ..., f(n). The Taylor polynomial is determined by a different set of n+1 numbers - the first n+1 derivatives of f (at 0 say).

In the talk we will explore the analogues of these two constructions for functors that arise in topology. It turns out that while a polynomial function is determined by a sequence of numbers, a polynomial functor is determined by a (truncated) symmetric sequence with an extra structure. The extra structure can be expressed in terms of operads and their modules. The relationship between the interpolation and the Taylor polynomial can be understood in terms of (a version of) Koszul duality between operads.

A good example to test the theory on is the mapping bi-functor that sends a pair of topological spaces (X, Y) to the space of maps F(X, Y). An equally interesting example is the functor that sends a pair ofsmooth manifolds (M, N) to the space of smooth embeddings Emb(M, N). We will use these functors, and others related to them, to illustrate the general theory.

Recently Positselski proved that an unbounded derived category of quasi-coherent D-modules on a smooth algebraic variety X is equivalent to a so-called coderived category of quasi-coherent DG-modules over the de Rham algebra of X. I will explain how to work with this coderived category.