Past 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
9 June 2016
16:00
Aurelio Carlucci
Abstract

Donaldson-Thomas theory was born as a mean to attach to Calabi-Yau 3-manifolds integers, invariant under small deformation of the complex structure. Subsequent evolutions have replaced integers with cohomological invariants, more flexible and with a broader range of applicable cases.

This talk is meant to be a gentle induction to the topic. We start with an introduction on virtual fundamental classes, and how they relate to deformation and obstruction spaces of a moduli space; then we pass on to the Calabi-Yau 3-dimensional case, stressing how some homological conditions are essential and can lead to generalisation. First we describe the global construction using virtual fundamental classes, then the local approach via the Behrend function and the virtual Euler characteristic.
We introduce quivers with potential, which provide a profitable framework in which to build DT-theory, as they are a source of moduli spaces locally presented as degeneracy loci. Finally, we overview the problem of categorification, introducing the DT-sheaf and showing how it relates to the numerical invariants.

  • Junior Geometry and Topology Seminar
2 June 2016
16:00
Maxence Mayrand
Abstract

Abstract: A hyperkähler manifold is a Riemannian manifold $(M,g)$ with three complex structures $I,J,K$ satisfying the quaternion relations, i.e. $I^2=J^2=K^2=IJK=-1$, and such that $(M,g)$ is Kähler with respect to each of them. I will describe a construction due to Kronheimer which gives such a structure on the cotangent bundle of any complex reductive group.
 

  • Junior Geometry and Topology Seminar
26 May 2016
16:00
Alejandro Betancourt
Abstract

Abstract: Ricci solitons are genralizations of Einstein metrics which have become subject of much interest over the last decade. In this talk I will give a basic introduction to these metrics and discuss how to reformulate the Ricci soliton equation as a Hamiltonian system assuming some symmetry conditions. Using this approach we will construct explicit solutions to the soliton equation for manifolds of dimension 5.

  • Junior Geometry and Topology Seminar

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