Past Junior Geometry and Topology Seminar

23 June 2010
11:00
Ben Davison
Abstract
In this talk I will discuss "motivic" Donaldson-Thomas invariants, following the now not-so-recent paper of Kontsevich and Soibelman on this subject. I will, in particular, present some understanding of the mysterious notion of "orientation data," and present some recent work. I will of course do my best to make this talk "accessible," though if you don't know what a scheme or a category is it will probably make you cry.
  • Junior Geometry and Topology Seminar
22 June 2010
11:00
Robert Clancy
Abstract
This talk will begin with an introduction to calibrations and calibrated submanifolds. Calibrated geometry generalizes Wirtinger's inequality in Kahler geometry by considering k-forms which are analogous to the Kahler form. A famous one-line proof shows that calibrated submanifolds are volume minimizing in their homology class. Our examples of manifolds with a calibration will come from complex geometry and from manifolds with special holonomy. We will then discuss the deformation theory of the calibrated submanifolds in each of our examples and see how they differ from the theory of complex submanifolds of Kahler manifolds.
  • Junior Geometry and Topology Seminar
10 June 2010
12:00
Thomas Bruun Madsen
Abstract
On any Hermitian manifold there is a unique Hermitian connection, called the Bismut connection, which has torsion a three-form. One says that the triplet consisting of the Hermitian structure together with the Bismut connection specifies a Kähler-with-torsion structure, or briefly a KT structure. If the torsion three-form is closed, we have a strong KT structure. The first part of this talk will discuss these notions and also address the problem of classifying strong KT structures. \paragraph{} Despite their name, KT manifolds are generally not Kähler. In particular the fundamental two-form is not closed. If the KT structure is strong, we have instead a closed three-form. Motivated by the usefulness of moment maps in geometries involving symplectic forms, one may ask whether it is possible to construct a similar type of map, when we replace the symplectic form by a closed three-form. The second part of the talk will explain the construction of such maps, which are called multi-moment maps.
  • Junior Geometry and Topology Seminar
3 June 2010
12:00
Oscar Randal-Williams
Abstract
I will discuss what is known about the cohomology of several moduli spaces coming from algebraic and differential geometry. These are: moduli spaces of non-singular curves (= Riemann surfaces) $M_g$, moduli spaces of nodal curves $\overline{M}_g$, moduli spaces of holomorphic line bundles on curves $Hol_g^k \to M_g$, and the universal Picard varieties $Pic^k_g \to M_g$. I will construct characteristic classes on these spaces, talk about their homological stability, and try to explain why the constructed classes are the only stable ones. If there is time I will also talk about the Picard groups of these moduli spaces. Much of this work is due to other people, but some is joint with J. Ebert.
  • Junior Geometry and Topology Seminar
27 May 2010
12:00
Frank Gounelas
Abstract
This talk will largely be a survey and so will gloss over technicalities. After introducing the basics of the theory of the étale fundamental group I will state the theorems and conjectures related to Grothendieck's famous "anabelian" letter to Faltings. The idea is that the geometry and arithmetic of certain varieties is in some sense governed by their non-abelian (anabelian) fundamental group. Time permitting I will discuss current work in this area, particularly the work of Minhyong Kim relating spaces of (Hodge, étale) path torsors to finiteness theorems for rational points on curves leading to a conjectural proof of Faltings' theorem which has been much discussed in recent years.
  • Junior Geometry and Topology Seminar
20 May 2010
12:00
Flavio Cordeiro
Abstract
\paragraph{} Poisson quasi-Nijenhuis structures with background (PqNb structures) were recently defined and are one of the most general structures within Poisson geometry. On one hand they generalize the structures of Poisson-Nijenhuis type, which in particular contain the Poisson structures themselves. On the other hand they generalize the (twisted) generalized complex structures defined some years ago by Hitchin and Gualtieri. Moreover, PqNb manifolds were found to be appropriate target manifolds for sigma models if one wishes to incorporate certain physical features in the model. All these three reasons put the PqNb structures as a new and general object that deserves to be studied in its own right. \paragraph{} I will start the talk by introducing all the concepts necessary for defining PqNb structures, making this talk completely self-contained. After a brief recall on Poisson structures, I will define Poisson-Nijenhuis and Poisson quasi-Nijenhuis manifolds and then move on to a brief presentation on the basics of generalized complex geometry. The PqNb structures then arise as the general structure which incorporates all the structures referred above. In the second part of the talk, I will define gauge transformations of PqNb structures and show how one can use this concept to construct examples of such structures. This material corresponds to part of the article arXiv:0912.0688v1 [math.DG].\\ \paragraph{} Also, if time permits, I will shortly discuss the appearing of PqNb manifolds as target manifolds of sigma models.
  • Junior Geometry and Topology Seminar
13 May 2010
12:00
Vicky Hoskins
Abstract
A moduli problem in algebraic geometry is essentially a classification problem, I will introduce this notion and define what it means for a scheme to be a fine (or coarse) moduli space. Then as an example I will discuss the classification of coherent sheaves on a complex projective scheme up to isomorphism using a method due to Alvarez-Consul and King. The key idea is to 'embed' the moduli problem of sheaves into the moduli problem of quiver representations in the category of vector spaces and then use King's moduli spaces for quiver representations. Finally if time permits I will discuss recent work of Alvarez-Consul on moduli of quiver sheaves; that is, representations of quivers in the category of coherent sheaves.
  • Junior Geometry and Topology Seminar
6 May 2010
12:00
Markus Roeser
Abstract
A Hyperkähler manifold is a riemannian manifold carrying three complex structures which behave like quaternions such that the metric is Kähler with respect to each of them. This means in particular that the manifold is a symplectic manifold in many different ways. In analogy to the Marsden-Weinstein reduction on a symplectic manifold, there is also a quotient construction for group actions that preserve the Hyperkähler structure and admit a moment map. In fact most known (non-compact) examples of hyperkähler manifolds arise in this way from an appropriate group action on a quaternionic vector space. In the first half of the talk I will give the definition of a hyperkähler manifold and explain the hyperkähler quotient construction. As an important application I will discuss the moduli space of solutions to the gauge-theoretic "Self-duality equations on a Riemann surface", the space of Higgs bundles, and explain how it can be viewed as a hyperkähler quotient in an infinite-dimensional setting.
  • Junior Geometry and Topology Seminar
29 April 2010
12:00
Maria Buzano
Abstract
The aim of this talk is to get a feel for the Ricci flow. The Ricci flow was introduced by Hamilton in 1982 and was later used by Perelman to prove the Poincaré conjecture. We will introduce the notions of Ricci flow and Ricci soliton, giving simple examples in low dimension. We will also discuss briefly other types of geometric flows one can consider.
  • Junior Geometry and Topology Seminar

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