Past Junior Geometry and Topology Seminar

18 February 2016
16:00
Thomas Wasserman
Abstract

Topological Quantum Field Theories are functors from a category of bordisms of manifolds to (usually) some categorification of the notion of vector spaces. In this talk we will first discuss why mathematicians are interested in these in general and an overview of the relevant notions. After this we will have a closer look at the example of functors from the bordism category of 1-, 2- and 3-dimensional manifolds equipped with principal G-bundles, for G a finite group, to nice categorifications of vector spaces.

  • Junior Geometry and Topology Seminar
4 February 2016
16:00
Lucas Branco
Abstract

Higgs bundles have a rich structure and play a role in many different areas including gauge theory, hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, mirror symmetry and Langlands duality. In this introductory talk I will explain some basic notions of G-Higgs – including the Hitchin fibration and spectral data - and illustrate how this relates to mirror symmetry.

  • Junior Geometry and Topology Seminar
3 December 2015
16:00
Renee Hoekzema
Abstract

In 1954 Thom showed that there is an isomorphism between the cobordism groups of manifolds and the homotopy groups of the Thom spectrum. I will define what these words mean and present the explicit, geometric construction of the isomorphism.

  • Junior Geometry and Topology Seminar
26 November 2015
16:00
Matthias Wink
Abstract

A basic result in Morse theory due to Reeb states that a compact manifold which admits a smooth function with only two, non-degenerate critical points is homeomorphic to the sphere. We shall apply this idea to distance function associated to a Riemannian metric to prove the diameter-sphere theorem of Grove-Shiohama: A complete Riemannian manifold with sectional curvature $\geq 1$ and diameter $> \pi / 2$ is homeomorphic to a sphere. I shall not assume any knowledge about curvature for the talk.

  • Junior Geometry and Topology Seminar
19 November 2015
16:00
Robert Kropholler
Abstract

I will discuss the theory of branched covers of cube complexes as a method of hyperbolisation. I will show recent results using this technique. Time permitting I will discuss a form of Morse theory on simplicial complexes and show how these methods combined with the earlier methods allow one to create groups with interesting finiteness properties. 

  • Junior Geometry and Topology Seminar
12 November 2015
16:00
Gareth Wilkes
Abstract

I will introduce the profinite completion as a way of aggregating information about the finite-sheeted covers of a 3-manifold, and discuss the state of the homeomorphism problem for 3-manifolds in this context; in particular, for geometrizable 3-manifolds.

  • Junior Geometry and Topology Seminar
5 November 2015
16:00
Simon Gritschacher
Abstract

Deformation K-theory was introduced by G. Carlsson and gives an interesting invariant of a group G encoding higher homotopy information about its representation spaces. Lawson proved a relation between this object and a homotopy theoretic analogue of the representation ring. This talk will not contain many details, instead I will outline some basic constructions and hopefully communicate the main ideas.
 

  • Junior Geometry and Topology Seminar
22 October 2015
16:00
Alejandro Betancourt
Abstract


Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

  • Junior Geometry and Topology Seminar

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