Junior Geometry and Topology Seminar
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Thu, 20/01/2011 13:00 |
Tom Sutherland (University of Oxford) |
Junior Geometry and Topology Seminar  |
SR1 |
| This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety. For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver. | |||
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Thu, 27/01/2011 13:00 |
Martin Palmer (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point. However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof – using an entirely different method – which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups. | |||
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Thu, 03/02/2011 13:00 |
Victoria Hoskins (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| Consider the action of a complex reductive group on a complex projective variety X embedded in projective space. Geometric Invariant Theory allows us to construct a 'categorical' quotient of an open subset of X, called the semistable set. If in addition X is smooth then it is a symplectic manifold and in nice cases we can construct a moment map for the action and the Marsden-Weinstein reduction gives a symplectic quotient of the group action on an open subset of X. We will discuss both of these constructions and the relationship between the GIT quotient and the Marsden-Weinstein reduction. The quotients we have discussed provide a quotient for only an open subset of X and so we then go on to discuss how we can construct quotients of certain subvarieties contained in the complement of the semistable locus. | |||
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Thu, 10/02/2011 13:00 |
Imran Qureshi (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties. | |||
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Thu, 17/02/2011 13:00 |
Ben Davison (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| I will talk about a recent paper of Huh, who, building on a wealth of pretty geometry and topology, has given a proof of a conjecture dating back to 1968 regarding the chromatic polynomial (the polynomial that determines how many ways there are of colouring the vertices of a graph with n colours in such a way that no vertices which are joined by an edge have the same colour). I will mainly talk about the way in which a problem that is explicitly a combinatorics problem came to be encoded in algebraic geometry, and give an overview of the geometry and topology that goes into the solution. The talk should be accessible to everyone: no stacks, I promise. | |||
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Thu, 24/02/2011 13:00 |
Dirk Schlueter (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles. I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities. | |||
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Thu, 03/03/2011 13:00 |
Junior Geometry and Topology Seminar |
SR1 | |
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Thu, 10/03/2011 13:00 |
Camilo Arias Abad (University of Zurich) |
Junior Geometry and Topology Seminar |
SR1 |
I will explain how Chen's iterated integrals can be used to construct an  -version of de Rham's theorem (originally due to Gugenheim). I will then explain how to use this result to construct generalized holonomies and integrate homotopy representations in Lie theory. |
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