SR1

The derived category of a variety has (relatively) recently come into play as an invariant of the variety, useful as a tool for classification. As the derived category contains cohomological information about the variety, it is perhaps a natural question to ask how close the derived category is to the motive of a variety.

We will begin by briefly recalling Grothendieck's category of Chow motives of smooth projective varieties, recall the definition of Fourier-Mukai transforms, and state some theorems and examples. We will then discuss some conjectures of Orlov http://arxiv.org/abs/math/0512620, the most general of which is: does an equivalence of derived categories imply an isomorphism of motives?

In this talk I shall present some ongoing work on principal G-Higgs bundles, for G a simple Lie group. In particular, we will consider two non-compact real forms of GL(p+q,C) and SL(p+q,C), namely U(p,q) and SU(p,q). By means of the spectral data that principal Higgs bundles carry for these non-compact real forms, we shall give a new description of the moduli space of principal U(p,q) and SU(p,q)-Higgs bundles. As an application of our method, we will count the connected components of these moduli spaces.

In spirit with John's talk we will discuss how topological invariants can be defined within a purely algebraic framework. After having introduced étale fundamental groups, we will discuss conjectures of Gieseker, relating those to certain "flat bundles" in finite characteristic. If time remains we will comment on the recent proof of Esnault-Sun.

I'll start by defining the zeta function and stating the Weil conjectures (which have actually been theorems for some time now). I'll then go on by saying things like "Weil cohomology", "standard conjectures" and "Betti numbers of the Grassmannian". Hopefully by the end we'll all have learned something, including me.

Mikhail Borovoi's theorem states that any simply connected compact semisimple Lie group can be understood (as a group) as an amalgam of its rank 1 and rank 2 subgroups. Here we present a recent extension of this, which allows us to understand the same objects as a colimit of their rank 1 and rank 2 subgroups under a final group topology in the category of Lie groups. Loosely speaking, we obtain not only the group structure uniquely by understanding all rank 1 and rank 2 subgroups, but also the topology.

The talk will race through the elements of Lie theory, buildings and category theory needed for this proof, to leave the audience with the underlying structure of the proof. Little prior knowledge will be assumed, but many details will be left out.

We will give a quick and dirty introduction to Gromov-Witten theory and discuss some integrality properties of GW invariants. We will start by briefly recalling some basic properties of the Deligne Mumford moduli space of curves. We will then try to define GW invariants using both algebraic and symplectic geometry (both definitions will be rather sloppy, but hopefully the basic idea will become visible), talk a bit about the axiomatic definition due to to Kontsevich and Manin, and discuss some applications like quantum cohomology. Finally, we will talk a bit about integrality and the Gopakumar-Vafa conjecture. Just as a word of warning: this talk is intended as an introduction to the

subject and should give an overview, so we will perhaps be a bit sloppy here and there...

A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.

I will explain how Chen's iterated integrals can be used to construct an $A_\infty$-version of de Rham's theorem (originally due to Gugenheim). I will then explain how to use this result to construct generalized holonomies and integrate homotopy representations in Lie theory.