Past 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
11 June 2015
16:00
Roland Grinis
Abstract

I plan to discuss finite time singularities for the harmonic map heat flow and describe a beautiful example of winding behaviour due to Peter Topping.

  • Junior Geometry and Topology Seminar
28 May 2015
16:00
Antonio De Capua
Abstract

A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.

  • Junior Geometry and Topology Seminar
21 May 2015
16:00
Matthias Wink
Abstract

In this talk we're going to discuss Hamilton's maximum principle for the Ricci flow. As an application, I would like to explain a technique due to Boehm and Wilking which provides a general tool to obtain new Ricci flow invariant curvature conditions from given ones. As we'll see, it plays a key role in Brendle and Schoen's proof of the differentiable sphere theorem.

  • Junior Geometry and Topology Seminar
14 May 2015
16:00
Carlos Alfonso Ruiz
Abstract
I will present a model theoretic point of view of algebraic geometry based on certain objects called Zariski Geometries. They were introduced by E. Hrushovski and B. Zilber and include classical objects like compact complex manifolds, algebraic varieties and rigid analytic varieties. Some connections with non commutative geometry have been found by B. Zilber too. I will concentrate on the relation between Zariski Geometries and schemes. 
  • Junior Geometry and Topology Seminar

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