Junior Geometry and Topology Seminar
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Thu, 05/05/2011 13:00 |
Moritz Rodenhausen (University of Bonn) |
Junior Geometry and Topology Seminar  |
SR1 |
| A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators. | |||
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Thu, 12/05/2011 13:00 |
David Hume (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| Mikhail Borovoi's theorem states that any simply connected compact semisimple Lie group can be understood (as a group) as an amalgam of its rank 1 and rank 2 subgroups. Here we present a recent extension of this, which allows us to understand the same objects as a colimit of their rank 1 and rank 2 subgroups under a final group topology in the category of Lie groups. Loosely speaking, we obtain not only the group structure uniquely by understanding all rank 1 and rank 2 subgroups, but also the topology. The talk will race through the elements of Lie theory, buildings and category theory needed for this proof, to leave the audience with the underlying structure of the proof. Little prior knowledge will be assumed, but many details will be left out. | |||
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Fri, 20/05/2011 12:00 |
Laura Schaposnik (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| In this talk I shall present some ongoing work on principal G-Higgs bundles, for G a simple Lie group. In particular, we will consider two non-compact real forms of GL(p+q,C) and SL(p+q,C), namely U(p,q) and SU(p,q). By means of the spectral data that principal Higgs bundles carry for these non-compact real forms, we shall give a new description of the moduli space of principal U(p,q) and SU(p,q)-Higgs bundles. As an application of our method, we will count the connected components of these moduli spaces. | |||
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Fri, 27/05/2011 12:00 |
Shane Kelly (Universite Paris 13) |
Junior Geometry and Topology Seminar |
SR1 |
| The derived category of a variety has (relatively) recently come into play as an invariant of the variety, useful as a tool for classification. As the derived category contains cohomological information about the variety, it is perhaps a natural question to ask how close the derived category is to the motive of a variety. We will begin by briefly recalling Grothendieck's category of Chow motives of smooth projective varieties, recall the definition of Fourier-Mukai transforms, and state some theorems and examples. We will then discuss some conjectures of Orlov http://arxiv.org/abs/math/0512620, the most general of which is: does an equivalence of derived categories imply an isomorphism of motives? | |||
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Fri, 03/06/2011 12:00 |
John Calabrese (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| I'll start by defining the zeta function and stating the Weil conjectures (which have actually been theorems for some time now). I'll then go on by saying things like "Weil cohomology", "standard conjectures" and "Betti numbers of the Grassmannian". Hopefully by the end we'll all have learned something, including me. | |||
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Fri, 10/06/2011 12:00 |
Michael Groechenig (University of Oxford) |
Junior Geometry and Topology Seminar |
SR1 |
| In spirit with John's talk we will discuss how topological invariants can be defined within a purely algebraic framework. After having introduced étale fundamental groups, we will discuss conjectures of Gieseker, relating those to certain "flat bundles" in finite characteristic. If time remains we will comment on the recent proof of Esnault-Sun. | |||
