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Abstracts:

    V. Balaji

    Stable bundles and holonomy group schemes of varieties

    This is based on my joint work with A.J. Parameswaran. For smooth projective varieties over algebraically closed fields, we construct a group scheme obtained from a Tannaka category of a large class of bundles on X. The irreducible representations of this group scheme are precisely the strongly stable bundles on X (with degree 0) as in the classical Narasimhan-Seshadri theorem. We study this
    group scheme and as an application of this group scheme, we construct strongly stable principal G-bundles on smooth projective surfaces when the characteristic of the base field is bounded below by the
    Coxeter number of G.

    U. Görtz

    Affine Springer fibers and affine Deligne-Lusztig Varieties

    Abstract.pdf

    U. Hackstein

    Principal G-bundles on p-adic curves and parallel transport

    We define functorial isomorphisms of parallel transport along étale paths for a class of principal G-bundles on a p-adic curve. Here G is a connected reductive algebraic group of finite presentation
    and the considered principal bundles are just those with potentially strongly semistable reduction of degree zero. The constructed isomorphisms yield continous functors from the étale fundamental groupoid of the given curve to the category of topological spaces with a simply transitive continous right G(ℂp)-action.
    This generalizes a construction in the case of vector bundles on a p-adic curve by Deninger and Werner. It may be viewed as a partial p-adic analogue of the classical theory by Ramanathan on principal bundles on compact Riemann surfaces, which generalizes the classical
    Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.

    N. Hoffmann

    Rationality for moduli spaces of symplectic bundles

    I'll report on joint work with I. Biswas about moduli spaces of symplectic bundles on an algebraic curve C. More precisely, we consider vector bundles E on C of rank 2n, endowed with a symplectic form with values in a fixed line bundle L, and prove that their moduli space is rational if n and deg(L) are both odd. The latter condition corresponds to the coprimeness of rank and degree for vector bundles.

    I. Kausz

    Zollstocks and gravitational correlators

    A zollstock is a two-pointed semistable curve of genus zero, whose dual graph is linear. The stack of zollstocks may be considered as a substitute for the (non-existing) moduli space of stable two-pointed curves of genus zero. In this talk we explain how this stack fits into a cyclic operad in the category of Artin stacks, and the relationship of this operad with gravitational correlators from Gromov-Witten theory.

    P.E. Newstead

    Higher rank Brill-Noether theory and coherent systems

    I will begin by reviewing the classification of vector bundles on an algebraic curve. Then I will introduce Brill-Noether loci, corresponding to vector bundles with at least a specified number of independent global sections, and coherent systems, that is pairs consisting of a vector bundle and a subspace of its space of sections. I will discuss briefly the connection between these two ideas and then describe results on Brill-Noether loci and the classifying spaces of coherent systems, particularly for bundles of rank 2.

    Ngo Dac T.

    Compactifications of stacks of G-shtukas

    The stacks of shtukas of Drinfeld play a central role in the proof of Drinfeld and Lafforgue of the Langlands correspondence over function fields for general linear groups. One major difficulty of their approach is to construct nice compactifications of these stacks. It is expected that the Langlands correspondence for a reductive group G would be realized by studying the stacks of G-shtukas. In this talk, I will explain how to construct compactifications of stacks of G-shtukas which generalize those of Drinfeld and Lafforgue.

    E. Opdam

    Degenerate double affine Hecke algebras at critical level

    The degenerate double affine Hecke algebra at critical level plays a predominant role in the integrable model of 1-dimensional quantum particles with delta function interaction and with periodic boundary conditions (and the root system generalizations of these models). We discuss the Bethe Ansatz equation for these systems, and various results and conjectures on the eigenfunctions and the representation theory of the underlying Hecke algebra.

    This is a report on joint work with J. Stokman and E. Emsiz.

    C.S. Seshadri

    Moduli spaces of bundles on nodal curves

    The purpose of this talk is to give a survey of work on moduli spaces of vector bundles or rather torsion free sheaves on nodal curves and generalisations in the context of principal bundles.

    W. Soergel

    Graded version of tensoring with finite dimensional representations

    We show how the functors of tensoring with finite dimensional representations can be lifted in a consistent way to the graded version of category O. This is done using the geometry of the affine Grassmannian.

    U. Stuhler

    Azumaya algebras and unit groups

    In the talk there will be a discussion of finiteness properties of unit
    groups
    of Azumaya algebras ,mainly in the case of surfaces over finite fields.On
    the way to doing this there will be a short treatment of the classical
    case of curves using bundles over  algebras and actions of the involved
    unit groups on certain affine buildings. Additionally there will  be a
    description of the moduli problems connected with the case of surfaces
    mentioned above.

    E. Vasserot

    Double afine Hecke algebras and affine flag manifolds

    I. Affine Hecke algebra and K-theory of the flag manifold.

    We'll review the Kazhdan-Lusztig classification of simple modules of affine Hecke algebras, using algebraic K-theory and perverse sheaves.

    II. Affine Grassmanian and affine flag manifolds.

    We'll give several approaches to affine grassmanians and affine flag manifolds, including Kashiwara's flag manifold and Beilinson-Drinfeld's one.

    III. Equivariant homology and K-theory of affine Grassmanians.

    We'll explain Bezrukavnikov-Finkelberg-Mirkovic computation of the equivariant cohomology of the affine Grassmanian, and its relation to Toda lattice.

    IV. Affine Springer fibers.

    We'll give on overview of Kazhdan-Lusztig's results on affine Springer fibers.

    V. Affine Springer fibers and Double affine Hecke algebras.

    We'll give the classification of simple modules of double affine Hecke algebras in terms of K-theory of affine flag manifold. We'll also give some application to finite dimensional modules.