Wed, 22 Jun 2011

11:30 - 12:30
ChCh, Tom Gate, Room 2

Things I haven't managed to do

David Craven
(University of Oxford)
Abstract

This talk will summarize some of the problems and conjectures that I haven't managed to solve (although I have tried to) while spending my three years in this job. It will cover the areas of group theory, representation theory, both of general finite groups and of symmetric groups, and fusion systems.

Wed, 11 May 2011

16:00 - 17:00
SR1

3-manifolds and their fundamental groups

Alessandro Sisto
(University of Oxford)
Abstract

We'll discuss 2 ways to decompose a 3-manifold, namely the Heegaard

splitting and the celebrated geometric decomposition. We'll then see

that being hyperbolic, and more in general having (relatively)

hyperbolic fundamental group, is a very common feature for a 3-manifold.

Wed, 04 May 2011

11:30 - 12:30
ChCh, Tom Gate, Room 2

Unbounding Ext

David Stewart
(University of Oxford)
Wed, 01 Jun 2011

11:30 - 12:30
ChCh, Tom Gate, Room 2

Sophic groups

Elisabeth Fink
(University of Oxford)
Abstract

The talk will start with the definition of amenable groups. I will discuss various properties and interesting facts about them. Those will be underlined with representative examples. Based on this I will give the definition and some basic properties of sofic groups, which only emerged quite recently. Those groups are particularly interesting as it is not know whether every group is sofic.

Fri, 20 May 2011

12:00 - 13:00
SR1

Spectral data for principal Higgs bundles

Laura Schaposnik
(University of Oxford)
Abstract

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.

Fri, 10 Jun 2011

12:00 - 13:00
SR1

Fundamental groups and positive characteristic

Michael Groechenig
(University of Oxford)
Abstract

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.

Fri, 03 Jun 2011

12:00 - 13:00
SR1

Some random facts about the Weil conjectures

John Calabrese
(University of Oxford)
Abstract

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.

Fri, 24 Jun 2011

12:00 - 13:00
L3

Betti numbers of twisted Higgs bundles on P^1

Steven Rayan
(University of Oxford)
Abstract

As with conventional Higgs bundles, calculating Betti numbers of twisted Higgs bundle moduli spaces through Morse theory requires us to

study holomorphic chains. For the case when the base is P^1, we present a recursive method for constructing all the possible stable chains of a given type and degree by representing a family of chains by a quiver. We present the Betti numbers when the twists are O(1) and O(2), the latter of which coincides with the co-Higgs bundles on P^1. We offer some open questions. In doing so, we mention how these numbers have appeared elsewhere recently, namely in calculations of Mozgovoy related to conjectures coming from the physics literature (Chuang-Diaconescu-Pan).

Thu, 12 May 2011

13:00 - 14:00
SR1

Topological Borovoi's theorem

David Hume
(University of Oxford)
Abstract

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.

Fri, 17 Jun 2011

12:00 - 13:00
SR1

Gromov-Witten Invariants and Integrality

Benjamin Volk
(University of Oxford)
Abstract

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...

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