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Forthcoming events in this series
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Fission varieties
Abstract
I'll recall the quasi-Hamiltonian approach to moduli spaces of flat connections on Riemann surfaces, as a nice finite dimensional algebraic version of operations with loop groups such as fusion. Recently, whilst extending this approach to meromorphic connections, a new operation arose, which we will call "fission". As will be explained, this operation enables the construction of many new algebraic symplectic manifolds, going beyond those we were trying to construct.
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Khovanov-Rozansky homology, Hilbert scheme of points on singular curve and DAHAs.
Abstract
By intersecting a small three-dimensional sphere which surrounds a singular point of a planar curve, with the curve, one obtains a link in three-dimensional space. In my talk I explain a conjectural formula for the ranks Khovanov-Rozansky homology of the link which interpretsthe ranks in terms of topology of some natural stratification on the moduli space of torsion free sheaves on the curve. In particular I will present a formula for the ranks of the Khovanov-Rozansky homology of the torus knots which generalizes Jones formula for HOMFLY invariants of the torus knots. The later formula relates Khovanov-Rozansky homology to the represenation theory of Double Affine Hecke Algebras. The talk presents joint work with Gorsky, Shende and Rasmussen.
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Invariants for non-reductive group actions
Abstract
Translation actions appear all over geometry, so it is not surprising that there are many cases of moduli problems which involve non-reductive group actions, where Mumford’s geometric invariant theory does not apply. One example is that of jets of holomorphic map germs from the complex line to a projective variety, which is a central object in global singularity theory. I will explain how to construct this moduli space using the test curve model of Morin singularities and how this can be generalized to study the quotient of projective varieties by a wide class of non-reductive groups. In particular, this gives information about the invariant ring. This is joint work with Frances Kirwan.
Autoduality of Jacobians for singular curves
Abstract
Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In otherwords, J is self-dual in the sense that J is identified with the space of topologically trivial line bundles on itself.
Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is an algebraic group which is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J.
In this talk, I consider (projective) curves C with planar singularities. The main result is that J' is self-dual: J' is identified with a space of torsion-free sheaves on itself. This autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.
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Hyperkahler implosion
Abstract
Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry.
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Fourier-Mukai transforms and deformations in generalized complex geometry
Abstract
In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations (which can all be interpreted as deformations of X as a generalized complex manifold). Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.
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Symmetries of SL(n) Hitchin fibres
Abstract
In this talk we show how the computation of the group of components of Prym varieties of spectral covers leads to cohomological results on the moduli space of stable bundles originally due to Harder-Narasimhan. This is joint work with Christian Pauly.
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Hilbert schemes, Torus Knots, and Khovanov Homology
Abstract
Khovanov homology is an invariant of knots in S^3 which categorifies the Jones polynomial. Let C be a singular plane curve. I'll describe some conjectures relating the geometry of the Hilbert scheme of points on C to a variant of Khovanov homology which categorifies the HOMFLY-PT polynomial. These conjectures suggest a relation between HOMFLY-PT homology of torus knots and the representation theory of the rational Cherednik algebra. As a consequence, we get some easily testable predictions about the Khovanov homology of torus knots.
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Periods of Cubic Surfaces
Abstract
The moduli space of cubic surfaces is known to be isomorphic to a quotient of the unit ball in C^4 by an arithmetic
group. We review this construction, then explain how to construct
an explicit inverse to the period map by using suitable theta functions. This gives a new proof of the isomorphism between the two spaces.
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Moduli of irreducible holomorphic symplectic manifolds
Abstract
We shall discuss the moduli problem for irreducible holomorphic symplectic manifolds. If these manifolds are equipped with a polarization (an ample line bundle), then they are parametrized by (coarse) moduli spaces. We shall relate these moduli spaces to arithmetic quotients of type IV domains and discuss when they are rational or not. This is joint work with V.Gritsenko and G.K.Sankaran.
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Schematic Harder Narasimhan stratification
Abstract
The Harder Narasimhan type (in the sense of Gieseker semistability)
of a pure-dimensional coherent sheaf on a projective scheme is known to vary
semi-continuously in a flat family, which gives the well-known Harder Narasimhan
stratification of the parameter scheme of the family, by locally closed subsets.
We show that each stratum can be endowed with a natural structure of a locally
closed subscheme of the parameter scheme, which enjoys an appropriate universal property.
As an application, we deduce that pure-dimensional coherent sheaves of any given
Harder Narasimhan type form an Artin algebraic stack.
As another application - jointly with L. Brambila-Paz and O. Mata - we describe
moduli schemes for certain rank 2 unstable vector bundles on a smooth projective
curve, fixing some numerical data.
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Monodromy for systems of vector bundles and multiplicative preprojective algebras
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