Forthcoming events in this series


Thu, 24 May 2012

12:00 - 13:00
L3

Unoriented cobordism categories and Klein TQFTs

Rosalinda Juer
Abstract

The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $\mathscr{K}$, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $\mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}}$). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.

Thu, 17 May 2012

12:00 - 13:00
L3

Hyperkähler Metrics in Lie Theory

Markus Röser
Abstract

In this talk our aim is to explain why there exist hyperkähler metrics on the cotangent bundles and on coadjoint orbits of complex Lie groups. The key observation is that both the cotangent bundle of $G^\mathbb C$ and complex coadjoint orbits can be constructed as hyperkähler quotients in an infinite-dimensional setting: They may be identified with certain moduli spaces of solutions to Nahm's equations, which is a system of non-linear ODEs arising in gauge theory. 

In the first half we will describe the hyperkähler quotient construction, which can be viewed as a version of the Marsden-Weinstein symplectic quotient for complex symplectic manifolds. We will then introduce Nahm's equations and explain how their moduli spaces of solutions may be related to the above Lie theoretic objects.

Thu, 10 May 2012

12:00 - 13:00
L3

Spectral data for the Hitchin fibration

Laura Schaposnik
Abstract

We shall dedicate the first half of the talk to introduce

classical Higgs bundles and describe the fibres of the corresponding

Hitchin fibration in terms of spectral data. Then, we shall define

principal Higgs bundles and look at some examples. Finally, we

consider the particular case of $SL(2,R)$, $U(p,p)$ and $Sp(2p,2p)$ Higgs

bundles and study their spectral data. Time permitting, we shall look

at different applications of our new methods.

Thu, 03 May 2012

12:00 - 13:00
L3

Expander Graphs and Property $\tau$

Henry Bradford
Abstract

Expander graphs are sparse finite graphs with strong connectivity properties, on account of which they are much sought after in the construction of networks and in coding theory. Surprisingly, the first examples of large expander graphs came not from combinatorics, but from the representation theory of semisimple Lie groups. In this introductory talk, I will outline some of the history of the emergence of such examples from group theory, and give several applications of expander graphs to group theoretic problems.

Thu, 26 Apr 2012

12:00 - 13:00
SR1

Teichmüller space: complex vs hyperbolic geometry

Alessandro Sisto
Abstract

Complex structures on a closed surface of genus at least 2 are in

one-to-one correspondence with hyperbolic metrics, so that there is a

single space, Teichmüller space, parametrising all possible complex

and hyperbolic structures on a given surface (up to isotopy). We will

explore how complex and hyperbolic geometry interact in Teichmüller

space.

Thu, 08 Mar 2012

13:00 - 14:00
L3

Twistor Geometry

Markus Röser
Abstract

Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $\mathbb R^4$ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $\mathbb R^4$ , or $S^4$, as the space of certain "real" lines in the (projective) Twistor space $\mathbb{CP}^3$. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $S^4$ as certain holomorphic vector bundles on $\mathbb{CP}^3$ due to Ward.

Thu, 01 Mar 2012

13:00 - 14:00
L3

Applications of non-linear analysis to geometry

Robert Clancy
Abstract

I will claim (and maybe show) that a lot of problems in differential geometry can be reformulated in terms of non-linear elliptic differential operators. After reviewing the theory of linear elliptic operators, I will show what can be said about the non-linear setting.

Thu, 23 Feb 2012

13:00 - 14:00
SR2

Pseudo-Holomorphic Curves in Generalized Geometry

Christian Paleani
Abstract

After giving a brief physical motivation I will define the notion of generalized pseudo-holomorphic curves, as well as tamed and compatible generalized complex structures. The latter can be used to give a generalization of an energy identity. Moreover, I will explain some aspects of the local and global theory of generalized pseudo-holomorphic curves.

Thu, 16 Feb 2012

13:00 - 14:00
SR2

Generalized Geometry - a starter course.

Roberto Rubio
Abstract

Basic and mild introduction to Generalized Geometry from the very beginning: the generalized tangent space, generalized metrics, generalized complex structures... All topped with some Lie type B flavour. Suitable for vegans. May contain traces of spinors.

Thu, 09 Feb 2012

13:00 - 14:00
L3

Elliptic Curves and Cohomology Theories

Hemanth Saratchandran
Abstract

I will give a brief introduction into how Elliptic curves can be used to define complex oriented

cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto

formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated

to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.

Thu, 02 Feb 2012

13:00 - 14:00
SR2

Monotonicity, variational methods and the Ricci flow

Chris Hopper
Abstract

I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].

Thu, 26 Jan 2012

13:00 - 14:00
SR2

Geometric Quantization - an Introduction

Jakob Blaavand
Abstract

In this talk we will discuss geometric quantization. First of all we will discuss what it is, but shall also see that it has relations to many other parts of mathematics. Especially shall we see how the Hitchin connection in geometric quantization can give us representations of a certain group associated to a surface, the mapping class group. If time permits we will discuss some recent results about these groups and their representations, results that are essentially obtained from geometrically quantizing a moduli space of flat connections on a surface."

Fri, 20 Jan 2012

12:00 - 13:45
L3

Derived Algebraic Geometry: a global picture II

Vittoria Bussi
Abstract

This is the second of two talks about Derived Algebraic Geometry. We will go through the various geometries one can develop from the Homotopical Algebraic Geometry setting. We will review stack theory in the sense of Laumon and Moret-Bailly and higher stack theory by Simpson from a new and more general point of view, and this will culminate in Derived Algebraic Geometry. We will try to point out how some classical objects are actually secretly already in the realm of Derived Algebraic Geometry, and, once we acknowledge this new point of view, this makes us able to reinterpret, reformulate and generalize some classical aspects. Finally, we will describe more exotic geometries. In the last part of this talk, we will focus on two main examples, one addressed more to algebraic geometers and representation theorists and the second one to symplectic geometers.

Thu, 19 Jan 2012

12:00 - 13:45
L3

Derived Algebraic Geometry: a global picture I

Vittoria Bussi
Abstract

This is the first of two talks about Derived Algebraic Geometry. Due to the vastity of the theory, the talks are conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as it might reveal interesting new insights also on classical objects, rather than a detailed and precise exposition. We will start with an introduction on the very basic idea of this theory, and we will expose some motivations for introducing it. After a brief review on the existing literature and a speculation about homotopy theories and higher categorical structures, we will review the theory of dg-categories, model categories, S-categories and Segal categories. This is the technical part of the seminar and it will give us the tools to understand the basic setting of Topos theory and Homotopical Algebraic Geometry, whose applications will be exploited in the next talk.

Thu, 01 Dec 2011

12:00 - 13:00

Thom spectra and cobordism rings

Martin Palmer
Abstract

After recalling some definitions and facts about spectra from the previous two "respectra" talks, I will explain what Thom spectra are, and give many examples. The cohomology theories associated to various different Thom spectra include complex cobordism, stable homotopy groups, ordinary mod-2 homology.......

I will then talk about Thom's theorem: the ring of homotopy groups of a Thom spectrum is isomorphic to the corresponding cobordism ring. This allows one to use homotopy-theoretic methods (calculating the homotopy groups of a spectrum) to answer a geometric question (determining cobordism groups of manifolds with some specified structure). If time permits, I'll also describe the structure of some cobordism rings obtained in this way.

Thu, 17 Nov 2011

12:00 - 13:00
SR2

Perspectives on Spectra

Michael Gröchenig
Abstract

This is the first in a series of $\geq 2$ talks about Stable Homotopy Theory. We will motivate the definition of spectra by the Brown Representability Theorem, which allows us to interpret a spectrum as a generalized cohomology theory. Along the way we recall basic notions from homotopy theory, such as suspension, loop spaces and smash products.

Thu, 10 Nov 2011

12:00 - 13:00
SR2

Holomorphic analogues of Chern-Simons gauge theory and Wilson operators

Tim Adamo
Abstract

Chern-Simons theory is topological gauge theory in three dimensions that contains an interesting class of operators called Wilson lines/loops, which have connections with both physics and pure mathematics. In particular, it has been shown that computations with Wilson operators in Chern-Simons theory reproduce knot invariants, and are also related to Gauss linking invariants. We will discuss the complex generalizations of these ideas, which are known as holomorphic Chern-Simons theory, Wilson operators, and linking, in the setting of Calabi-Yau three-folds. This will (hopefully) include a definition of all three of these holomorphic analogues as well as an investigation into how these ideas can be translated into simple homological algebra, allowing us to propose the existence of "homological Feynman rules" for computing things like Wilson operators in a holomorphic Chern-Simons theory. If time permits I may say something about physics too.

Thu, 03 Nov 2011

12:00 - 13:00
SR2

Some Remarks on d-manifolds and d-bordism

Benjamin Volk
Abstract

We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $C^\infty$-rings, $C^\infty$-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.

Thu, 27 Oct 2011

12:00 - 13:00
SR2

Stability conditions on K3 surfaces

Heinrich Hartmann
Abstract

We will explain Bridgelands results on the stabiltiy manifold of a K3 surface. As an application we will define the stringy Kaehler moduli space of a K3 surface and comment on the mirror symmetry picture.

Thu, 20 Oct 2011

12:00 - 13:00
SR2

Stability conditions, rational elliptic surfaces and Painleve equations

Tom Sutherland
Abstract

We will describe the space of Bridgeland stability conditions

of the derived category of some CY3 algebras of quivers drawn on the

Riemann sphere. We give a biholomorphic map from the upper-half plane to

the space of stability conditions lifting the period map of a meromorphic

differential on a 1-dimensional family of elliptic curves. The map is

equivariant with respect to the actions of a subgroup of $\mathrm{PSL}(2,\mathbb Z)$ on the

left by monodromy of the rational elliptic surface and on the right by

autoequivalences of the derived category.

The complement of a divisor in the rational elliptic surface can be

identified with Hitchin's moduli space of connections on the projective

line with prescribed poles of a certain order at marked points. This is

the space of initial conditions of one of the Painleve equations whose

solutions describe isomonodromic deformations of these connections.

Thu, 13 Oct 2011

12:00 - 13:00
L3

Type I singularities and ancient solutions of homogeneous Ricci flow

Maria Buzano
Abstract

We will present a class of compact and connected homogeneous

spaces such that the Ricci flow of invariant Riemannian metrics develops

type I singularities in finite time. We will describe the singular

behaviours that we can get, as we approach the singular time, and the Ricci

soliton that we obtain by blowing up the solution near the singularity.

Finally, we will investigate the existence of ancient solutions when the

isotropy representation decomposes into two inequivalent irreducible

summands.

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).

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...

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.