Past Junior Geometry and Topology Seminar

9 February 2017
16:00
Giles Gardam
Abstract

Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.

  • Junior Geometry and Topology Seminar
26 January 2017
16:00
Gareth Wilkes
Abstract

The study of 3-manifolds is founded on the strong connection between algebra and topology in dimension three. In particular, the sine qua non of much of the theory is the Loop Theorem, stating that for any embedding of a surface into a 3-manifold, a failure to be injective on the fundamental group is realised by some genuine embedding of a disc. I will discuss this theorem and give a proof of it.

  • Junior Geometry and Topology Seminar
30 November 2016
16:00
Joshua Jackson
Abstract

A central tool in the construction of moduli spaces throughout algebraic geometry and beyond, geometric invariant theory (GIT) aims to sensibly answer the question, "How can we quotient an algebraic variety by a group action?" In this talk I will explain some basics of GIT and indicate how it can be used to build moduli spaces, before exploring one of its salient features: the non-canonicity of the quotient. I will show how the dependence on an additional parameter, a choice of so-called 'linearisation', leads to a rich 'wall crossing' picture, giving different interrelated models of the quotient. Time permitting, I will also speak about recent developments in non-reductive GIT, and joint work extending this dependence to the non-reductive setting.

  • Junior Geometry and Topology Seminar
24 November 2016
16:00
Daniel Bruegmann
Abstract

Spectra provide a way of understanding cohomology theories in terms of homotopy theory. Spectra are a bit like CW-complexes, they have homotopy groups which may be used to characterize homotopy equivalences. However, a spectrum has homotopy groups in negative degrees, too, and they are abelian groups in all degrees. We will discuss spectra representing ordinary cohomology, bordism, and K-theory.

  • Junior Geometry and Topology Seminar

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