Past Junior Geometry and Topology Seminar

22 November 2012
15:00
Shehryar Sikander
Abstract
In this talk we show how Teichmüller curves can be used to compute quantum invariants of certain Pseudo-Anasov mapping tori. This involves computing monodromy of the Hitchin connection along closed geodesics of the Teichmüller curve using iterated integrals. We will mainly focus on the well known Teichmüller curve generated by a pair of regular pentagons. This is joint work with J. E. Andersen.
  • Junior Geometry and Topology Seminar
15 November 2012
12:00
Søren Fuglede Jørgensen
Abstract
Quantum representations are finite-dimensional projective representations of the mapping class group of a compact oriented surface that arise from the study of Chern--Simons theory; a 3-dimensional quantum field theory. The input to Chern--Simons theory is a compact, connected and simply connected Lie group $G$ (and in my talks, the relevant groups are $G = SU(N)$) and a natural number $k$ called the level. In these talks, I will discuss the representations from two very different and disjoint viewpoints. <b>Part I: Quantum representations and their asymptotics</b> The characters of the representations are directly related to the so-called quantum SU(N)-invariants of 3-manifolds that physically correspond to the Chern--Simons partition function of the 3-manifold under scrutiny. In this talk I will give a definition of the quantum representation using the geometric quantization of the moduli space of flat $SU(N)$-manifolds, where Hitchin's projectively flat connection over Teichmüller space plays a key role. I will give examples of the large level asymptotic behaviour of the characters of the representations and discuss a general conjecture, known as the Asymptotic Expansion Conjecture, for the asymptotics. Whereas I will likely be somewhat hand-wavy about the details of the construction, I hope to introduce the main objects going into it -- some prior knowledge of the geometry of moduli spaces of flat connections will be an advantage but not necessarily necessary. <b>Part II: Quantum representations and their algebraic properties</b> In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial. Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.
  • Junior Geometry and Topology Seminar
8 November 2012
15:00
Martin Palmer
Abstract
Fix a connected manifold-with-boundary M and a closed, connected submanifold P of its boundary. The set of all possible submanifolds of M whose components are pairwise unlinked and each isotopic to P can be given a natural topology, and splits into a disjoint union depending on the number of components of the submanifold. When P is a point this is just the usual (unordered) configuration space on M. It is a classical result, going back to Segal and McDuff, that for these spaces their homology in any fixed degree is eventually independent of the number of points of the configuration (as the number of points goes to infinity). I will talk about some very recent work on extending this result to higher-dimensional submanifolds: in the above setup, as long as P is of sufficiently large codimension in M, the homology in any fixed degree is eventually independent of the number of components. In particular I will try to give an idea of how the codimension restriction arises, and how it can be improved in some special cases.
  • Junior Geometry and Topology Seminar
18 October 2012
15:00
Alberto Cazzaniga
Abstract
We will go through the GIT construction of the moduli space of quiver representations. Concentrating on examples (probably the cases of Hilbert schemes of points of $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$) we will try to give an idea of why this methods became relevant in modern (algebraic) geometry. No prerequisites required, experts would probably get bored.
  • Junior Geometry and Topology Seminar
11 October 2012
12:00
Jakob Blaavand
Abstract
This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.
  • 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

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