Geometry and Analysis Seminar

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.

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.

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.

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.

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.