Past Seminars › Seminar Series

Junior Geometry and Topology Seminar (past)

Thu, 25/06/2009
12:00 		Ben Davison (Oxford) Junior Geometry and Topology Seminar Add to calendar L3
Perfect obstruction theories and virtual fundamental classes
I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.
Thu, 18/06/2009
12:15 		Magnus Lauridsen (Aarhus) Junior Geometry and Topology Seminar Add to calendar SR1
The AJ conjecture from a gauge-theoretical viewpoint
The AJ conjecture relates two different knot invariants, namely the coloured Jones polynomial and the A-polynomial. The approach we will use will be that of 2+1 dimensional Topological Quantum Field Theory. Indeed, the coloured Jones polynomial is constructed in Reshetikhin and Turaev's formulation of a TQFT using quantum groups. The A-polynomial is defined by a subvariety of the moduli space of flat SL(2,C) connections of a torus.  Geometric quantization on this moduli space also gives a TQFT, and the correspondence between these provides a framework where the knot invariants can be compared. In the talk I will sketch the above constructions and show how we can do explicit calculations for simple knots. This is work in progress joint with J. E. Andersen.
Thu, 11/06/2009
12:15 		Frank Gounelas (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Grothendieck's Brauer group and the Manin obstruction
In this talk I will outline the two constructions of the Brauer group Br($ X $) of a scheme $ X $, namely via etale cohomology and Azumaya algebras and briefly describe how one may compute this group using the Hochschild-Serre spectral sequence. In the early '70s Manin observed that one can use the Brauer group of a projective variety $ X/k $ to define an obstruction to the existence of rational points on $ X $. I will discuss this arithmetic application and time permitting, outline an example for $ X $ a K3 surface.
Thu, 04/06/2009
12:15 		Arman Taghavi-Chabert (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Pure spinors and twistors
Thu, 28/05/2009
12:15 		Dawid Kielak (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Helly's Theorem through dimension theory
Thu, 21/05/2009
12:15 		Dirk Schlueter (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Universal moduli of parabolic bundles on stable curves
A parabolic bundle on a marked curve is a vector bundle with extra structure (a flag) in each of the fibres over the marked points, together with data corresponding to a choice of stability condition Parabolic bundles are natural generalisations of vector bundles when the base comes with a marking (for example, they partially generalise the Narasimhan-Seshadri correspondence between representations of the fundamental group and semistable vector bundles), but they also play an important role in the study of pure sheaves on nodal curves (which are needed to compactify moduli of vector bundles on stable curves). Consider the following moduli problem: pairs $ (C,E) $ of smooth marked curves $ C $ and semistable parabolic bundles $ E\rightarrow C $. I will sketch a construction of projective moduli spaces which compactify the above moduli problem over the space of stable curves. I'll discuss further questions of interest, including strategies for understanding the cohomology of these moduli spaces, generalisations of the construction to higher-dimensional base schemes, and possible connections with Torelli theorems for parabolic vector bundles on marked curves.
Thu, 14/05/2009
12:15 		Niels Gammelgaard (Aarhus) Junior Geometry and Topology Seminar Add to calendar SR1
Hitchin's connection, Toeplitz operators, and deformation quantization
I will briefly describe a differential geometric construction of Hitchin's projectively flat connection in the Verlinde bundle, over Teichmüller space, formed by the Hilbert spaces arising from geometric quantization of the moduli space of flat connections on a Riemann surface. We will work on a general symplectic manifold sharing certain properties with the moduli space. Toeplitz operators enter the picture when quantizing classical observables, but they are also closely connected with the notion of deformation quantization. Furthermore, through an intimate relationship between Toeplitz operators, the Hitchin connection manifests itself in the world of deformation quantization as a connection on formal functions. As we shall see, this formal Hitchin connection can be used to construct a deformation quantization, which is independent of the Kähler polarization used for quantization. In the presence of a symmetry group, this deformation quantization can (under certain cohomological conditions) be constructed invariantly. The talk presents joint work with J. E. Andersen.
Thu, 30/04/2009
12:00 		Oscar Randal-Williams (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Spaces of surfaces and Mumford's conjecture
I will present a new proof of Mumford's conjecture on the rational cohomology of moduli spaces of curves, which is substantially different from those given by Madsen–Weiss and Galatius–Madsen–Tillmann–Weiss: in particular, it makes no use of Harer–Ivanov stability for the homology of mapping class groups, which played a decisive role in the previously known proofs. This talk represents joint work with Soren Galatius.
Thu, 12/03/2009
12:00 		Ben Davison (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Counting invariants and wall crossing
This talk concerns the relationships between Donaldson-Thomas, Pandharipande-Thomas, and Szendroi invariants established via analysis of the geometry of wall crossing phenomena of suitably general moduli spaces. I aim to give a reasonably detailed account of the simplest example, the conifold, where in fact all of the major ideas can be easily seen.
Thu, 05/03/2009
12:00 		David Baraglia (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Semi-flat fibrations of special Lagrangian and coassociative submanifolds
I will go over my paper (arXiv:0902.2135v1) which explains how semi-flat Calabi-Yau / G$ _2 $ manifolds can be constructed from minimal 3-submanifolds in a signature (3,3) vector space.
Thu, 19/02/2009
12:00 		Dirk Schlueter (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The moduli space of vector bundles on a Riemann surface
I will briefly discuss the construction of the moduli spaces of (semi)stable bundles on a given curve. The main aim of the talk will be to describe various features of the geometry and topology of these moduli spaces, with emphasis on methods as much as on results. Topics may include irreducibility, cohomology, Verlinde numbers, Torelli theorems.
Thu, 12/02/2009
12:00 		Arman Taghavi-Chabert (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
An Introduction to the Penrose transform
Thu, 05/02/2009
12:00 		João Lopes Costa (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
On uniqueness of stationary black holes
We prove uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes. (This is joint work with Piotr Chrusciel. arXiv:0806.0016)
Thu, 22/01/2009
12:00 		Jeff Giansiracusa (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Ribbon graphs and moduli spaces of curves
Thu, 04/12/2008
12:00 		Roberto Rubio (ICMAT Spain) Junior Geometry and Topology Seminar Add to calendar SR1
Hermitian G-Higgs bundles exceptionally flavoured
We introduce the notion of $ G $-Higgs bundle from studying the representations of the fundamental group of a closed connected oriented surface $ X $ in a Lie group $ G $. If $ G $ turns to be the isometry group of a Hermitian symmetric space, much more can be said about the moduli space of $ G $-Higgs bundles, but this also implies dealing with exceptional cases. We will try to face all these subjects intuitively and historically, when possible!
Thu, 27/11/2008
12:00 		Martijn Kool (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to Deformation Theory
In this talk I will discuss some elementary notions of deformation theory in algebraic geometry like Schlessinger's Criterion. I will describe obstructions and deformations of sheaves in detail and will point out relations to moduli spaces of sheaves.

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

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