Forthcoming Seminars › Seminar Series

Junior Geometry and Topology Seminar

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, 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, 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 $. I will sketch a construction of projective moduli spaces which compactify the above moduli problem over the space of stable curves. I'll discuss further questions of interest, including strategies for understanding the cohomology of these moduli spaces, generalisations of the construction to higher-dimensional base schemes, and possible connections with Torelli theorems for parabolic vector bundles on marked curves.
Thu, 28/05/2009
12:15 		Dawid Kielak (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Helly's Theorem through dimension theory
Thu, 04/06/2009
12:15 		Arman Taghavi-Chabert (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Pure spinors and twistors
Thu, 11/06/2009
12:15 		Frank Gounelas (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Grothendieck's Brauer group and the Manin obstruction
In this talk I will outline the two constructions of the Brauer group Br($ 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, 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.

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

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