Forthcoming events in this series


Thu, 12 Nov 2015

16:00 - 17:00
C5

Finite covers of 3-manifold groups

Gareth Wilkes
(Oxford)
Abstract

I will introduce the profinite completion as a way of aggregating information about the finite-sheeted covers of a 3-manifold, and discuss the state of the homeomorphism problem for 3-manifolds in this context; in particular, for geometrizable 3-manifolds.

Thu, 05 Nov 2015

16:00 - 17:00
C5

Deformation K-theory

Simon Gritschacher
(Oxford)
Abstract

Deformation K-theory was introduced by G. Carlsson and gives an interesting invariant of a group G encoding higher homotopy information about its representation spaces. Lawson proved a relation between this object and a homotopy theoretic analogue of the representation ring. This talk will not contain many details, instead I will outline some basic constructions and hopefully communicate the main ideas.
 

Thu, 29 Oct 2015

16:00 - 17:00
C5
Thu, 22 Oct 2015

16:00 - 17:00
C5

Einstein metrics on 4-manifolds

Alejandro Betancourt
(Oxford)
Abstract


Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Thu, 11 Jun 2015

16:00 - 17:00
C2

What is bubbling?

Roland Grinis
(Oxford)
Abstract

I plan to discuss finite time singularities for the harmonic map heat flow and describe a beautiful example of winding behaviour due to Peter Topping.

Thu, 28 May 2015

16:00 - 17:00
C2

Hyperbolic volume of links, via pants graph and train tracks

Antonio De Capua
(Oxford)
Abstract

A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.

Thu, 21 May 2015

16:00 - 17:00
C2

Ricci flow invariant curvature conditions

Matthias Wink
(Oxford)
Abstract

In this talk we're going to discuss Hamilton's maximum principle for the Ricci flow. As an application, I would like to explain a technique due to Boehm and Wilking which provides a general tool to obtain new Ricci flow invariant curvature conditions from given ones. As we'll see, it plays a key role in Brendle and Schoen's proof of the differentiable sphere theorem.

Thu, 14 May 2015

16:00 - 17:00
C2

Zariski Geometries

Carlos Alfonso Ruiz
(Oxford)
Abstract
I will present a model theoretic point of view of algebraic geometry based on certain objects called Zariski Geometries. They were introduced by E. Hrushovski and B. Zilber and include classical objects like compact complex manifolds, algebraic varieties and rigid analytic varieties. Some connections with non commutative geometry have been found by B. Zilber too. I will concentrate on the relation between Zariski Geometries and schemes. 
Thu, 07 May 2015

16:00 - 17:00
C2

The geometry of the Ising model

Bruce Bartlett
(Oxford)
Abstract

The Ising model is a well-known statistical physics model, defined on a two-dimensional lattice. It is interesting because it exhibits a "phase transition" at a certain critical temperature. Recent mathematical research has revealed an intriguing geometry in the model, involving discrete holomorphic functions, spinors, spin structures, and the Dirac equation. I will try to outline some of these ideas.

Thu, 12 Mar 2015

16:00 - 17:00
C2

Multiplicative quiver varieties and their quantizations

Iordan Ganev
(University of Texas at Austin)
Abstract

Quiver varieties and their quantizations feature prominently in
geometric representation theory. Multiplicative quiver varieties are
group-like versions of ordinary quiver varieties whose quantizations
involve quantum groups and $q$-difference operators. In this talk, we will
define and give examples of representations of quivers, ordinary quiver
varieties, and multiplicative quiver varieties. No previous knowledge of
quivers will be assumed. If time permits, we will describe some phenomena
that occur when quantizing multiplicative quiver varieties at a root of
unity, and work-in-progress with Nicholas Cooney.

Thu, 05 Mar 2015

16:00 - 17:00
C2

Introduction to deformation quantization

Pavel Safronov
(Oxford)
Abstract

I will explain the basics of deformation quantization of Poisson
algebras (an important tool in mathematical physics). Roughly, it is a
family of associative algebras deforming the original commutative
algebra. Following Fedosov, I will describe a classification of
quantizations of (algebraic) symplectic manifolds.
 

Thu, 26 Feb 2015

16:00 - 17:00
C2

On Weyl's Problem of Isometric Embedding

Siran Li
(Oxford)
Abstract

In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$. 

    The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture. 

    Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.

Thu, 12 Feb 2015

16:00 - 17:00
C2

Introduction to conformal symmetry

Agnese Bissi
(Oxford)
Abstract

 In this talk I will present a basic introduction to conformal symmetry from a physicist perspective. I will talk about infinitesimal and finite conformal transformations and the conformal group in diverse dimensions. 

Thu, 05 Feb 2015

16:00 - 17:00
C2

G-Higgs bundles, mirror symmetry and Langlands duality

Lucas Branco
(Oxford)
Abstract

The moduli space of G-Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.

We shall begin the talk by first introducing G-Higgs bundles for reductive Lie groups and the associated Hitchin fibration, and sketching how to realize Langlands duality through spectral data. We shall then look at particular types of branes (BAA-branes) which correspond to very interesting geometric objects, hyperholomorphic bundles (BBB-branes). 

The presentation will be introductory and my goal is simply to sketch some of the ideas relating these very interesting areas. 

Thu, 29 Jan 2015

16:00 - 17:00
C2

Simple Homotopy Theory and the Poincaré Conjecture

Robert Kropholler
(Oxford)
Abstract

I will introduce simple homotopy theory and then discuss relations between some conjectures in 2 dimensional simple homotopy theory and the 3 and 4 dimensional Poincaré conjectures.

Thu, 04 Dec 2014

16:00 - 17:00
C2

Introduction to Concepts of General Relativity

Felix Tennie
(Oxford University)
Abstract

Since its genesis in 1915, General Relativity has proven to be one of the most successful physical theories ever invented. Providing a description of the large scale structure of the universe it continues to be in agreement with all experimental tests to high accuracy. By merging Classical Mechanics and Electrodynamics to a consistent geometrical theory of space-time it has become one of the two pillars of modern theoretical physics alongside Quantum Mechanics. This talk aims to give an introduction to the ideas and concepts of General Relativity. After briefly reviewing Classical (Newtonian) Mechanics and experiments in contradiction with it the framework and axioms of General Relativity will be introduced. This will be followed by a survey on major implications of the (new) geometrical description of gravity. Finally an outlook on physics beyond General Relativity will be provided. 

Thu, 27 Nov 2014

16:00 - 17:00
C2

Lagrangian Floer theory

Lino Campos
(Oxford University)
Abstract

Lagrangian Floer cohomology categorifies the intersection number of (half-dimensional) Lagrangian submanifolds of a symplectic manifold. In this talk I will describe how and when can we define Lagrangian Floer cohomology. In the case when Floer cohomology cannot be defined I will describe an alternative invariant known as the Fukaya (A-infinity) algebra.

Thu, 20 Nov 2014

16:00 - 17:00
C2

Cancelled

Felix Tennie
(Oxford University)
Thu, 13 Nov 2014

16:00 - 17:00
C2

Non-commutative topology and K-theory for applications to topological insulators

Guo Chuan Thiang
(Oxford University)
Abstract

I will recall basic notions of operator K-theory as a non-commutative (C*-algebra) generalisation of topological K-theory. Twisted crossed products will be introduced as generalisations of group C*-algebras, and a model of Karoubi's K-theory, which makes sense for super-algebras, will be sketched. The motivation comes from physics, through the study of quantum mechanical symmetries, charged free quantum fields, and topological insulators. The relevant theorems, which are interesting in their own right but scattered in the literature, will be consolidated.

Thu, 30 Oct 2014

16:00 - 17:00
C2

Finiteness properties of Kähler groups

Claudio Llosa
(Oxford University)
Abstract

In this talk we want to discuss results by Dimca, Papadima, and Suciu about the finiteness properties of Kähler groups. Namely, we will sketch their proof that for every $2\leq n\leq \infty$ there is a Kähler group with finiteness property $\mathcal{F}_n$, but not $FP_{n+1}$. Their proof is by explicit construction of examples. These examples all arise as subgroups of finite products of surface groups and they are the first known examples of Kähler groups with arbitrary finiteness properties. The talk does not require any prior knowledge of finiteness properties or of Kähler groups.

Thu, 23 Oct 2014

16:00 - 17:00
C2

Manifolds of positive curvature

Alejandro Betancourt
(Oxford University)
Abstract

Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.

Thu, 16 Oct 2014

16:00 - 17:00
C2

Yau's Proof of the Calabi Conjecture

Roland Grinis
(Oxford University)
Abstract

The Calabi conjecture, posed in 1954 and proved by Yau in 1976, guaranties the existence of Ricci-flat Kahler metrics on compact Kahler manifolds with vanishing first Chern class, providing examples of the so called Calabi-Yau manifolds. The latter are of great importance to the fields of Riemannian Holonomy Groups, having Hol0 as a subgroup of SU; Calibrated Geometry, more precisely Special Lagrangian Geometry; and to String theory with the discovery of the phenomenon of Mirror Symmetry (to mention a few!). In the talk, we will discuss the necessary background to formulate the Calabi conjecture and explain some of the main ideas behind its proof by Yau, which itself is a jewel from the point of view of non-linear PDEs.

Thu, 19 Jun 2014

16:00 - 17:00
C6

Introduction to Lie algebroids

Brent Pym
(Oxford University)
Abstract

Lie algebroids are geometric structures that interpolate between finite-dimensional Lie algebras and tangent bundles of manifolds. They give a useful language for describing geometric situations that have local symmetries. I will give an introduction to the basic theory of Lie algebroids, with examples drawn from foliations, principal bundles, group actions, Poisson brackets, and singular hypersurfaces.

Thu, 12 Jun 2014

16:00 - 17:00
C6

Spectral Networks and Abelianization

Omar Kidwai
(Oxford University)
Abstract

Spectral networks are certain collections of paths on a Riemann surface, introduced by Gaiotto, Moore, and Neitzke to study BPS states in certain N=2 supersymmetric gauge theories. They are interesting geometric objects in their own right, with a number of mathematical applications. In this talk I will give an introduction to what a spectral network is, and describe the "abelianization map" which, given a spectral network, produces nice "spectral coordinates" on the appropriate moduli space of flat connections. I will show that coordinates obtained in this way include a variety of previously known special cases (Fock-Goncharov coordinates and Fenchel-Nielsen coordinates), and mention at least one reason why generalising them in this way is of interest.

Thu, 05 Jun 2014

16:00 - 17:00
C6

Kitaev's Lattice Model and 123-TQFTs

Gerrit Goosen
Abstract

We give an overview of Kitaev's lattice model in the setting of an arbitrary finite group G (where $G = Z_{2}$ is the famous Toric Code). We also exhibit the connection this model has with so-called 123-TQFTs (topological quantum field theories), making use of ideas coming from higher gauge theory and Hopf algebra representations.