Past 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
Henry Bradford
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
23 February 2012
13:00
Christian Paleani
Abstract
After giving a brief physical motivation I will define the notion of generalized pseudo-holomorphic curves, as well as tamed and compatible generalized complex structures. The latter can be used to give a generalization of an energy identity. Moreover, I will explain some aspects of the local and global theory of generalized pseudo-holomorphic curves.
  • Junior Geometry and Topology Seminar
16 February 2012
13:00
Roberto Rubio
Abstract
Basic and mild introduction to Generalized Geometry from the very beginning: the generalized tangent space, generalized metrics, generalized complex structures... All topped with some Lie type B flavour. Suitable for vegans. May contain traces of spinors.
  • Junior Geometry and Topology Seminar
9 February 2012
13:00
Hemanth Saratchandran
Abstract
I will give a brief introduction into how Elliptic curves can be used to define complex oriented cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.
  • Junior Geometry and Topology Seminar
2 February 2012
13:00
Chris Hopper
Abstract
I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].
  • Junior Geometry and Topology Seminar
26 January 2012
13:00
Jakob Blaavand
Abstract
In this talk we will discuss geometric quantization. First of all we will discuss what it is, but shall also see that it has relations to many other parts of mathematics. Especially shall we see how the Hitchin connection in geometric quantization can give us representations of a certain group associated to a surface, the mapping class group. If time permits we will discuss some recent results about these groups and their representations, results that are essentially obtained from geometrically quantizing a moduli space of flat connections on a surface."
  • Junior Geometry and Topology Seminar

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