Forthcoming events in this series


Thu, 03 May 2018
16:00
C5

TBA

Joshua Jackson
(Oxford University)
Wed, 25 Apr 2018
16:00
C5

Symplectic cohomology and its (non)vanishing

Filip Zivanovic
(Oxford University)
Abstract

Symplectic cohomology is a Floer cohomology invariant of compact symplectic manifolds 
with contact type boundary, or of open symplectic manifolds with a certain geometry 
at the infinity. It is a graded unital K-algebra related to quantum cohomology, 
and for cotangent bundle, it recovers the homology of a loop space. During the talk 
I will define symplectic cohomology and show some of the results on its (non) vanishing. 
Time permitting, I will also mention natural TQFT algebraic structure on it.

Thu, 08 Mar 2018
16:00
C5

TBA

Lawrence Barrott
(University of Cambridge)
Thu, 01 Mar 2018
16:00
C5

TBA

Emily Maw
(UCL London)
Thu, 22 Feb 2018
16:00
C5

Thick triangles and a theorem of Gromov

Matthias Wink
(Oxford University)
Abstract

A theorem of Gromov states that the number of generators of the fundamental group of a manifold with nonnegative 
curvature is bounded by a constant which only depends on the dimension of the manifold. The main ingredient 
in the proof is Toponogov’s theorem, which roughly speaking says that the triangles on spaces with positive 
curvature, such as spheres, are thick compared to triangles in the Euclidean plane. In the talk I shall explain 
this more carefully and deduce Gromov’s result.

Thu, 08 Feb 2018
16:00
C5

Symplectic reduction and geometric invariant theory

Maxence Mayrand
(Oxford University)
Abstract

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

Thu, 01 Feb 2018
16:00
C5

The Reidemeister graphs (Joint work with Daniele Celoria)

Agnese Barbensi
(Oxford University)
Abstract

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

Wed, 29 Nov 2017
16:00
C5

Classifying Higgs bundles, stable and unstable

Eloise Hamilton
(Oxford University)
Abstract

 The aim of this talk is to describe the classification problem for Higgs bundles and to explain how a combination of classical and Non-Reductive Geometric Invariant Theory might be used to solve this classification problem.
 
I will start by defining Higgs bundles and their physical origins. Then, I will present the classification problem for Higgs bundles. This will involve introducing the "stack" of Higgs bundles, a purely formal object which allows us to consider all isomorphism classes of Higgs bundles at once. Finally, I will explain how the stack of Higgs bundles can be described geometrically. As we will see, the stack of Higgs bundles can be decomposed into disjoint strata, each consisting of Higgs bundles of a given "instability type". Both classical and Non-Reductive GIT can then be applied to obtain moduli spaces for each of the strata.

Thu, 23 Nov 2017
16:00
C5

Operads with homological stability and infinite loop space structures

Tom Zeman
(Oxford University)
Abstract

In a recent preprint, Basterra, Bobkova, Ponto, Tillmann and Yeakel
defined operads with homological stability (OHS) and showed that after
group-completion, algebras over an OHS group-complete to infinite loop
spaces. This can in particular be used to put a new infinite loop space
structure on stable moduli spaces of high-dimensional manifolds in the
sense of Galatius and Randal-Williams, which are known to be infinite
loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of
the OHS condition and construct yet another infinite loop space
structure on these stable moduli spaces. This structure turns out to be
equivalent to that constructed by Basterra et al. It is believed that
the infinite loop space structure due to Galatius--Randal-Williams is
also equivalent to these two structures.

Thu, 16 Nov 2017
16:00
C5

The Einstein Equation on Manifolds with Large Symmetry Groups

Timothy Buttsworth
(The University of Queensland)
Abstract

In this talk I will discuss the problem of finding Einstein metrics in the homogeneous and cohomogeneity one setting. 
In particular, I will describe a recent result concerning existence of solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.

Thu, 09 Nov 2017
16:00
C5

The Quantum Steenrod Square and its Properties

Nicholas Wilkins
(Oxford University)
Abstract

Topologists have the Steenrod squares, a collection of additive homomorphisms on the Z/2 cohomology of a space M. They can be defined axiomatically and are often be regarded as algebraic operations on cohomology groups (for many purposes). However, Betz and Cohen showed that they could be viewed geometrically. 

Symplectic geometers have quantum cohomology, which on a symplectic manifold M is a deformation of singular cohomology using holomorphic spheres.

The geometric definition of the Steenrod square extends to quantum cohomology. This talk will describe the Steenrod square and quantum cohomology in terms of the intersection product, and then give a description of this quantum Steenrod square by putting these both together. We will describe some properties of the quantum squares, such as the quantum Cartan formula, and perform calculations in certain cases.

Thu, 02 Nov 2017
16:00
C5

C^infinity Algebraic Geometry (with corners)

Kelli Francis-Staite
(Oxford University)
Abstract

Manifolds, the main objects of study in Differential Geometry, do not have nice categorical properties. For example, the category of manifolds with smooth maps does not contain all fibre products.
The algebraic counterparts to this (varieties and schemes) do have nice categorical properties. 

A method to ‘fix’ these categorical issues is to consider C^infinity schemes, which generalise the category of manifolds using algebraic geometry techniques. I will explain these concepts, and how to translate to manifolds with corners, which is joint work with my supervisor Professor Dominic Joyce.

Thu, 26 Oct 2017
16:00
C5

Quiver varieties revisited

Filip Zivanovic
(Oxford University)
Abstract

Quiver varieties are an attractive research topic of many branches of contemporary mathematics - (geometric) representation theory, (hyper)Kähler differential geometry, (symplectic) algebraic geometry and quantum algebra.

In the talk, I will define different types of quiver varieties, along with some interesting examples. Afterwards, I will focus on Nakajima quiver varieties (hyperkähler moduli spaces obtained from framed-double-quiver representations), stating main results on their topology and geometry. If the time permits, I will say a bit about the symplectic topology of them.

Thu, 19 Oct 2017
16:00
C5

The Drinfeld Centre of a Symmetric Fusion Category

Thomas Wasserman
(Oxford University)
Abstract


This talk will be a gentle introduction to braided fusion categories, with the eventual aim to explain a result from my thesis about symmetric fusion categories. 


Fusion categories are certain kinds of monoidal categories. They can be viewed as a categorification of the finite dimensional algebras, and appear in low-dimensional topological quantum field theories, as well as being studied in their own right. A braided fusion category is additionally commutative up to a natural isomorphism, symmetry is an additional condition on this natural isomorphism. Computations in these categories can be done pictorially, using so-called string diagrams (also known as ``those cool pictures''). 


In this talk I will introduce fusion categories using these string diagrams. I will then discuss the Drinfeld centre construction that takes a fusion category and returns a braided fusion category. We then show, if the input is a symmetric fusion category, that this Drinfeld centre carries an additional tensor product. All of this also serves as a good excuse to draw lots of pictures.
 

Fri, 11 Aug 2017

13:00 - 14:00
C1

Invertible Topological Field Theories

Benedict Morrissey
(UPenn)
Abstract

Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime.  This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra.  Constructions of field theories corresponding to specific maps of spectra will be described.
 

Thu, 15 Jun 2017
16:00
C5

A discussion of Lurie's proof of the cobordism hypothesis

Adrian Clough
(Univerity of Texas at Austin)
Abstract

Despite its fame there appears to be little literature outlining Lurie's proof sketched in his expository article "On the classification of topological field theories." I shall embark on the quixotic quest to explain how the cobordism hypothesis is formalised and give an overview of Lurie's proof in one hour. I will not be able to go into any of the motivation, but I promise to try to make the talk as accessible as possible. 

Wed, 31 May 2017

16:00 - 17:00
C1

Moduli spaces of singular curves

Joshua Jackson
((Oxford University))
Abstract

Moduli spaces attempt to classify all mathematical objects of a particular type, for example algebraic curves or vector bundles, and record how they 'vary in families'. Often they are constructed using Geometric Invariant Theory (GIT) as a quotient of a parameter space by a group action. A common theme is that in order to have a nice (eg Hausdorff) space one must restrict one's attention to a suitable subclass of 'stable' objects, in effect leaving certain badly behaved objects out of the classification. Assuming no prior familiarity, I will elucidate the structure of instability in GIT, and explain how recent progress in non-reductive GIT allows one to construct moduli spaces for these so-called 'unstable' objects. The particular focus will be on the application of this principle to the GIT construction of the moduli space of stable curves, leading to moduli spaces of curves of fixed singularity type.
 

Thu, 25 May 2017
16:00
C5

Manifolds with a-corners & moduli space of Morse flows

Yixuan Wang
((Oxford University))
Abstract

Manifolds with ordinary boundary/corners have found their presence in differential geometry and PDEs: they form Man^b or Man^c category; and for boundary value problems, they are nice objects to work on. Manifolds with analytical corners -- a-corners for short -- form a larger category Man^{ac} which contains Man^c, and they can in some sense be viewed as manifolds with boundary at infinity.
In this talk I'll walk you through the definition of manifolds with corners and a-corners, and give some examples to illustrate how the new definition will help.

Thu, 18 May 2017

16:00 - 17:00
C5

Symplectic Cohomology for Quiver Varieties

Filip Zivanovic
((Oxford University))
Abstract

Floer (co)homology, invariant which recovers periodic orbits of a Hamiltonian system, is the central topic of symplecic topology at present. Its analogue for open symplecic manifolds is called symplectic (co)homology. Our goal is to compute this invariant for big family of spaces called Nakajima's Quiver Varieties, spaces obtained as hyperkahler quotients of representation spaces of quivers.
 

Wed, 03 May 2017

16:00 - 17:00
C1

Integrating without integrating: weights of Kontsevich graphs

Ricardo Buring
(University of Groningen)
Abstract

Abstract: The Kontsevich graph weights are period integrals whose
values make Kontsevich's star-product associative for any Poisson
structure. We illustrate, by using software, to what extent these
weights are determined by their properties: the associativity
constraint for the star-product (for all Poisson structures), the
multiplicativity (decomposition into prime graphs), the cyclic
relations, and some relations due to skew-symmetry. Up to the order 4
in ℏ we express all the weights in terms of 10 parameters (6
parameters modulo gauge-equivalence), and we verify pictorially that
the star-product expansion is associative modulo ō(ℏ⁴) for every value
of the 10 parameters. This is joint work with Arthemy Kiselev.
 

Wed, 08 Mar 2017

16:00 - 17:00
C1

C^infinity Rings and Manifolds with Corners

Kelli Staite
((Oxford University))
Abstract

Manifolds with corners are similar to manifolds, yet are locally modelled on subsets $[0,\infty)^k \times R^{n-k}$. I will discuss some of the theory of these objects, as well as introducing $C^\infty$-rings. This will explain the background to my current research in $C^\infty$-Algebraic Geometry. Time permitting, I will briefly discuss my current research on $C^\infty$-schemes with corners and motivation of this research.