# Past Junior Geometry and Topology Seminar

14 June 2012
12:00
Dawid Kielak
Abstract
This is intended as an introductory talk about one of the most important (and most geometric) aspects of Geometric Group Theory. No prior knowledge of any maths will be assumed.
• Junior Geometry and Topology Seminar
7 June 2012
12:00
Tom Hawes
Abstract
The aim of this talk is to give an introduction to Geometric Invariant Theory (GIT) for reductive groups over the complex numbers. Roughly speaking, GIT is concerned with constructing quotients of group actions in the category of algebraic varieties. We begin by discussing what properties we should like quotient varieties to possess, highlighting so-called good' and geometric' quotients, and then turn to search for these quotients in the case of affine and projective varieties. Here we shall see that the construction runs most smoothly when we assume our group to be reductive (meaning it can be described as the complexification of a maximal compact subgroup). Finally, we hope to say something about the Hilbert-Mumford criterion regarding semi-stability and stability of points, illustrating it by constructing the rough moduli space of elliptic curves.
• Junior Geometry and Topology Seminar
31 May 2012
12:00
Richard Manthorpe
Abstract
Given a manifold $M$ and a basepointed labelling space $X$ the space of unordered finite configurations in $M$ with labels in $X$, $C(M;X)$ is the space of finite unordered tuples of points in $M$, each point with an associated point in $X$. The space is topologised so that particles cannot collide. Given a compact submanifold $M_0\subset M$ we define $C(M,M_0;X)$ to be the space of unordered finite configuration in which points vanish' in $M_0$. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $\Sigma^nX$-bundle over $M$. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on little round $\varepsilon$-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.
• Junior Geometry and Topology Seminar
24 May 2012
12:00
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.
• Junior Geometry and Topology Seminar
17 May 2012
12:00
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.

• Junior Geometry and Topology Seminar
10 May 2012
12:00
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.
• Junior Geometry and Topology Seminar
3 May 2012
12:00
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.
• Junior Geometry and Topology Seminar
26 April 2012
12:00
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.
• Junior Geometry and Topology Seminar
8 March 2012
13:00
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.
• Junior Geometry and Topology Seminar
1 March 2012
13:00
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.
• Junior Geometry and Topology Seminar