Past Geometry and Analysis Seminar

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.

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.

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.

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.

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.

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.

TBA

TBA

The modern approach to integrability proceeds via a Riemann surface, the spectral curve.

In many applications this curve is specified by transcendental constraints in terms of periods. I will highlight some of the problems this leads to in the context of monopoles, problems including integer solutions to systems of quadratic forms, questions of real algebraic geometry and conjectures for elliptic functions. Several new results will be presented including the uniqueness of the tetrahedrally symmetric monopole.

Lattices in semisimple Lie groups have been studied from the point of view of number theory, algebraic groups, topology and geometry, and geometric group theory. The Fragestellung of one line of investigation is to what extent the properties of the lattice determine, and are determined by, the properties of the group. This talk reviews a number of results about lattices, and in particular looks at Mostow--Margulis rigidity.