C5

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.

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.

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.

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. 

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.

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.