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Moduli Spaces

Algebraic geometry is a key area of mathematical research of international significance. It has strong connections with many other areas of mathematics (differential geometry, topology, number theory, representation theory, etc.) and also with other disciplines (in the present context, particularly theoretical physics). Moduli theory is the study of the way in which objects in algebraic geometry (or in other areas of mathematics) vary in families and is fundamental to an understanding of the objects themselves. The theory goes back at least to Riemann in the mid-nineteenth century, but moduli spaces were first rigorously constructed in the 1960s by Mumford and others. The theory has continued to develop since then, perhaps most notably with the infusion of ideas from physics after 1980.

  • 52 minutes 28 seconds
    Vector bundles and coherent systems on nodal curves
    We study moduli spaces of coherent systems on an irreducible rational curve with one node. We determine the conditions for emptiness and nonemptiness of these moduli spaces. We study the properties like irreducibility, smoothness, seminormality and rationality of the moduli spaces.
    25 July 2011, 9:17 am
  • 1 hour 5 minutes
    Higgs bundles and quaternionic geometry
    The circle action on the moduli space of Higgs bundles provides a link between hyperkahler geometry and quaternionic Kahler geometry. The lecture will discuss various aspects of this.
    4 July 2011, 9:01 am
  • 1 hour 4 minutes
    Moduli spaces of locally homogeneous geometric structures
    An Ehresmann structure on a manifold is a geometric structure defined by an atlas of local coordinate charts into a fixed homogeneous space. These structures form deformation spaces which themselves are modeled on the space of representations of the fundamental group. These deformation spaces admit actions of the mapping class group, whose dynamics can be highly nontrivial. In many cases the deformation space embeds inside the space of representations of the fundamental group, and geometric structures provide a powerful tool to study representation spaces of surface groups. This talk will survey several examples of these structures and relate them to other classification problems.
    4 July 2011, 8:50 am
  • 1 hour 3 minutes
    Bridgeland stability conditions and Fourier-Mukai transforms
    Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. I will explain the relation with Gieseker stability. In particular, I will explain kown results on the birational maps of moduli spaces by using Bridgeland stability condition.
    1 July 2011, 3:43 pm
  • 29 minutes 43 seconds
    Deligne-Hodge polynomials for SL(2,C)-character varieties of genus 1 and 2.
    We give a method to compute Deligne-Hodge polinomials for various SL(2,C)-character varieties, with fixed conjugacy classes equal to Id, -Id, diagonal and Jordan type matrices. We will split them into suitable stratifications and analyze the behaviour of the polynomial for them. This is joint work with V. Mu~noz and P. Newstead.
    1 July 2011, 3:30 pm
  • 30 minutes 29 seconds
    Derived equivalences of Azumaya algebras on K3 surfaces
    We consider moduli spaces of Azumaya algebras on K3 surfaces. These correspond to twisted sheaves. We prove that when _(A) is zero and c2(A) is within 2r of its minimal bound, where r2 is the rank of A, then the moduli space if non empty is a smooth projective surface. We construct a moduli space of Azumaya algebras on the double cover of the projective plane. In some other special cases we prove a derived equivalence between K3 surfaces and moduli spaces of Azumaya algebras.
    1 July 2011, 3:23 pm
  • 26 minutes 56 seconds
    Derived categories and rationality of conic bundles
    In this talk I present a joint work with Marcello Bernardara where we show that a standard conic bundle on a rational minimal surface is rational if and only if its derived category admits a semiothogonal decomposition via derived categories of smooth projective curves and exceptional objects. In particular, even if the surface is not minimal, such a decomposition allows to reconstruct the intermediate Jacobian as the direct sum of the Jacobian of those curves.
    1 July 2011, 3:02 pm
  • 52 minutes 6 seconds
    Stability conditions for the local projective plane
    Describing the space of Bridgeland stability conditions for the local projective plane turns out to be intimately related to classical results by Drezet and Le Potier on inequalities for Chern classes of slope-stable vector bundles on P2. I will describe how this allows one to relate the geometry of this space, and the group of autoequivalences, to the congruence subgroup Gamma1(3). I will also explain a mirror symmetry statement involving the moduli space of elliptic curves with Gamma1(3)-level structure. Time permitting, I will also discuss observations on the same problem for local del Pezzo surfaces. This is based on joint work with Emanuele Macrì.
    1 July 2011, 2:53 pm
  • 1 hour 6 minutes
    Kahler-Einstein metrics and Geometric Invariant Theory
    I will discuss an approach to a version of Yau's conjecture, relating Kahler-Einstein metrics to notions of stability. The core of this approach involves estimates for the Chow invariant, obtained from asymptotic analysis. We will also describe progress on a variant of the set-up involving an anticanonical divisor, somewhat analogous to the theory of parabolic bundles. Another theme in the talk will be the importance of making progress on testing stability in explicit cases.
    1 July 2011, 2:34 pm
  • 1 hour 5 minutes
    Introduction to moduli of varieties - IV: Surfaces of general type related to abelian varieties and hyperplane arrangements
    Compact moduli of surfaces of general type derived from (1) abelian varieties, (2) line arrangements. Campedelli, Burniat, Kulikov surfaces. Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/0901.4431
    1 July 2011, 1:42 pm
  • 1 hour 6 minutes
    Monopoles on the product of a surface and the circle
    One of the important ingredients of the Witten-Kapustin approach to the geometric Langlands program is the study of singular monopoles on the product of a Riemann surface and an interval; these mediate Hecke transforms. One special case of this, the self-transforms, corresponds to monopoles on the product of a Riemann surface and a circle. We study the moduli of these, and prove a Hitchin-Kobayashi correspondence. When the surface is a torus, there is in addition an interesting Nahm transform to instantons on the product of a three-torus and the line. (with Benoit Charbonneau).
    30 June 2011, 8:56 am
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