# Workshop Programme

## for period 26 May 2011

### Vector Bundles and Coherent Systems

26 May 2011

Timetable

Thursday 26 May | ||||

11:00-12:00 | Balaji, V (Chennai Mathematical Institute) |
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Algebraic approach to tensor product theorems | Sem 1 | |||

The aim of the talk is to give the general geometric invariant theoretic approach to proving tensor product theorems of semi-stable objects as highlighted in the work of Bogomolov and Ramanan & Ramanathan. This approach will be used to give algebraic proofs of the tensor product theorem for semi-stable Hitchin pairs over arbitrary ground fields. Towards this, one needs to develop a purely algebraic notion of a Hitchin scheme, an object dual in a certain sense to a Hitchin pair. |
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13:45-14:45 | Ramanan, S (Chennai) |
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Representations of the fundamental group and geometric loci of bundles | Sem 1 | |||

Starting with Weil's seminal work on vector bundles, the relation between representations of the fundamental group and vector bundles have been studied from various points of view. The work of Narasimhan and Seshadri made clear the relation of unitary representations to polystable bundles. One of Nigel Hitchin's results pertains to representations into the split real forms of semi-simple groups. I will be giving an overview of these ending up with my effort, in collaboration with Oscar Garcia-Prada, to understand these as relating to fixed point varieties under some natural involutions on the moduli of Higgs pairs. |
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15:15-16:15 | Schmitt, A (Free University of Berlin) |
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On the modular interpretation of the Nagaraj-Seshadri locus | Sem 1 | |||

We will survey constructions of moduli spaces for principal bundles on nodal curves over the complex numbers. This includes a moduli space for torsion free sheaves A of rank r and degree zero on an irreducible nodal curve X which are endowed with a homomorphism d: ∧^r A → O_X which is an isomorphism away from the node. It is a degeneration of the moduli space of SL_r(C)-bundles on a smooth curve. In many cases, this moduli space puts a scheme structure on the "Nagaraj-Seshadri locus" inside the moduli space of semistable torsion free sheaves of rank r and degree zero. |
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16:30-17:30 | Newstead, P (Liverpool) |
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New results in higher rank Brill-Noether theory | Sem 1 | |||

In the last 12 months, many new examples of rank 2 bundles (and some of rank 3 bundles) have been discovered. I will describe briefly the links with Koszul cohomology and the moduli spaces of curves, but most of the talk will be devoted to the construction of the bundles and their relevance to higher rank Brill-Noether theory. |