Past Seminars › Seminar Series

Junior Geometry and Topology Seminar (past)

Mon, 06/12/2010
12:00 		Heinrich Hartmann (Oxford University) Junior Geometry and Topology Seminar Add to calendar SR1
Cusps of the Kaehler moduli space and stability conditions on K3 surfaces

We will state a theorem of Shouhei Ma (2008) relating the Cusps of the Kaehler moduli space to the set of Fourier--Mukai partners of a K3 surface. Then we explain the relationship to the Bridgeland stability manifold and comment on our work relating stability conditions "near" to a cusp to the associated Fourier--Mukai partner.


Thu, 02/12/2010
13:00 		Arman Taghavi-Chabert (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
A Lie-theoretic approach to prolongations of differential systems
I will sketch a method to prolong certain classes of differential equations on manifolds using Lie algebra cohomology. The talk will be based on articles by Branson, Cap, Eastwood and Gover (arXiv:math/0402100 and ESI preprint 1483).
Thu, 25/11/2010
13:00 		Robert Clancy (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Constructing manifolds with special holonomy by resolving orbifolds
All of Joyce's constructions of compact manifolds with special holonomy are in some sense generalisations of the Kummer construction of a K3 surface. We will begin by reviewing manifolds with special holonomy and the Kummer construction. We will then describe Joyce's constructions of compact manifolds with holonomy G_2 and Spin(7).
Thu, 18/11/2010
13:00 		Stuart J Hall ((Imperial College, London)) Junior Geometry and Topology Seminar Add to calendar SR1
Algebraic approximations to special Kahler metrics
I will begin by defining the space of algebraic metrics in a particular Kahler class and recalling the Tian-Ruan-Zelditch result saying that they are dense in the space of all Kahler metrics in this class.  I will then discuss the relationship between some special algebraic metrics called 'balanced metrics' and distinguished Kahler metrics (Extremal metrics, cscK, Kahler-Ricci solitons...). Finally I will talk about some numerical algorithms due to Simon Donaldson for finding explicit examples of these balanced metrics (possibly with some pictures).
Thu, 11/11/2010
13:00 		Christopher Hopper (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Maximum principle for tensors with applications to the Ricci flow
The maximum principle is one of the main tools use to understand the behaviour of solutions to the Ricci flow. It is a very powerful tool that can be used to show that pointwise inequalities on the initial data of parabolic PDE are preserved by the evolution. A particular weak maximum principle for vector bundles will be discussed with references to Hamilton's seminal work [J. Differential Geom. 17 (1982), no. 2, 255–306; MR664497] on 3-manifolds with positive Ricci curvature and his follow up paper [J. Differential Geom. 24 (1986), no. 2, 153–179; MR0862046] that extends to 4-manifolds with various curvature assumptions.
Thu, 04/11/2010
13:00 		Markus Röser (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Hypersymplectic Manifolds and Harmonic Maps
In the first part of this talk we introduce hypersymplectic manifolds and compare various aspects of their geometry with related notions in hyperkähler geometry. In particular, we explain the hypersymplectic quotient construction. Since many examples of hyperkähler structures arise from Yang-Mills moduli spaces via the hyperkähler quotient construction, we discuss the gauge theoretic equations for a (twisted) harmonic map from a Riemann surface into a compact Lie group. They can be viewed as the zero condition for a hypersymplectic moment map in an infinite-dimensional setup.
Thu, 28/10/2010
13:00 		Maria Buzano (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Homogeneous Riemannian manifolds, Einstein metrics and the Ricci flow
We will recall basic definitions and facts about homogeneous Riemannian manifolds and we will discuss the Einstein condition on this kind of spaces. In particular, we will talk about non existence results of invariant Einstein metrics. Finally, we will talk briefly about the Ricci flow equation in the homogeneous setting.
Thu, 21/10/2010
13:00 		Alan Thompson (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Models for threefolds fibred by K3 surfaces of degree two
A K3 surface of degree two can be seen as a double cover of the complex projective plane, ramified over a nonsingular sextic curve. In this talk we explore two different methods for constructing explicit projective models of threefolds admitting a fibration by such surfaces, and discuss their relative merits.
Thu, 14/10/2010
12:00 		Michael Groechenig (Oxford University Mathematical Institute) Junior Geometry and Topology Seminar Add to calendar SR1
Homotopy theory for C*-algebras
The theory of C*-algebras provides a good realisation of noncommutative topology. There is a dictionary relating commutative C*-algebras with locally compact spaces, which can be used to import topological concepts into the C*-world. This philosophy fails in the case of homotopy, where a more sophisticated definition has to be given, leading to the notion of asymptotic morphisms. As a by-product one obtains a generalisation of Borsuk's shape theory and a universal boundary map for cohomology theories of C*-algebras.
Wed, 23/06/2010
11:00 		Ben Davison (Oxford) Junior Geometry and Topology Seminar Add to calendar L3
Orientation Data and motivic DT invariants
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.
Tue, 22/06/2010
11:00 		Robert Clancy (Oxford) Junior Geometry and Topology Seminar Add to calendar L3
Deformations of Calibrated Submanifolds
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.
Thu, 10/06/2010
12:00 		Thomas Bruun Madsen (Odense) Junior Geometry and Topology Seminar Add to calendar SR1
Geometry with torsion and multi-moment maps
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.

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.
Thu, 03/06/2010
12:00 		Oscar Randal-Williams (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Cohomology of moduli spaces
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.
Thu, 27/05/2010
12:00 		Frank Gounelas (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Anabelian Geometry
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.
Thu, 20/05/2010
12:00 		Flavio Cordeiro (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Poisson quasi-Nijenhuis manifolds with background

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.

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].

Also, if time permits, I will shortly discuss the appearing of PqNb manifolds as target manifolds of sigma models.
Thu, 13/05/2010
12:00 		Vicky Hoskins (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Moduli of sheaves and quiver sheaves
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.
Thu, 06/05/2010
12:00 		Markus Roeser (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Hyperkähler Quotients and Metrics on Moduli Spaces
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.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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