Past Junior Geometry and Topology Seminar

28 February 2013
15:00
Henry Bradford
Abstract
In this first of two talks, I shall introduce the Virtual Haken Conjecture and the major players involved in the proof announced by Ian Agol last year. These are the special cube complexes studied by Dani Wise and his collaborators, with a large supporting cast including the not-inconsiderable presence of Perelman’s Geometrization Theorem and the Surface Subgroup Theorem of Kahn and Markovic. I shall sketch how the VHC follows from Agol’s result that, in spite of the name, specialness is entirely generic among non-positively curved cube complexes.
  • Junior Geometry and Topology Seminar
21 February 2013
15:00
Thomas Wasserman
Abstract
<p><span>Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.</span><br /><span>One can do better and compute the cohomology in terms of this function using the so-called Witten complex.</span><br /><span>Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.</span><br /><span>I will explain Witten's (heuristic) arguments and describe the Witten complex.</span></p>
  • Junior Geometry and Topology Seminar
7 February 2013
15:00
Jakob Blaavand
Abstract
This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.
  • Junior Geometry and Topology Seminar
31 January 2013
15:00
Tom Hawes
Abstract
The aim of this talk is to introduce the notion of a stack, by considering in some detail the example of the the stack of vector bundles on a curve. One of the key areas of modern geometry is the study of moduli problems and associated moduli spaces, if they exist. For example, can we find a `fine moduli space' which parameterises isomorphism classes of vector bundles on a smooth curve and contains information about how such vector bundles vary in families? Quite often such a space doesn't exist in the category where we posed the original moduli problem, but we can enlarge our category and construct a `stack' which in a reasonable sense gives us the key properties of a fine moduli space we were looking for. This talk will be quite sketchy and won't even properly define a stack, but we hope to at least give some feel of how these objects are defined and why one might want to consider them.
  • Junior Geometry and Topology Seminar
17 January 2013
15:00
Jan Vonk
Abstract
Algebraic geometry has become the standard language for many number theorists in recent decades. In this talk, we will define modular forms and related objects in the language of modern geometers, thereby giving a geometric motivation for their study. We will ask some naive questions from a purely geometric point of view about these objects, and try to answer them using standard geometric techniques. If time permits, we will discuss some rather deep consequences in number theory of our geometric excursion, and mention open problems in geometry whose solution would have profound consequences in number theory.
  • 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

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