# Past 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
24 January 2013
15:00
David Hume
Abstract
The Borel conjecture is one of the most important (and difficult) conjectures in Topology. We explain how some weaker but highly related conjectures are being tackled through the coarse geometry of finitely generated groups.
• 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
29 November 2012
15:00
Abstract
This talk surveys the well known relationship between half-flat SU(3) structures on 6-manifolds M and metrics with holonomy in G_2 on Mx(a,b), focusing on the case in which M=S3xS3 with solutions invariant by SO(4).
• 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
25 October 2012
15:00
Eirik Svanes
Abstract
I will give an introduction to how SU(3)-structures appear in heterotic string theory and string compactifications. I will start by considering the zeroth order SU(3)-holonomy Calabi-Yau scenario, and then see how this generalizes when higher order effects are considered. If time, I will discuss some of my own work.
• 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