Past Geometry and Analysis Seminar

The talk will survey some results about smooth and topological 4-manifolds obtained via surgery, and discuss some contrasting information provided by gauge theory about smooth finite group actions on 4-manifolds.

The lecture will discuss some applications of topology to a number of interesting physical systems:

1. Classifications of Phases, 2. Classifications of one-dimensional textures in Nematics and Superfluid HE-3,

3. Classification of defects, 4. Phase transition in Liquid membranes.

The solution of these problems leads to interesting mathematics but the talk will also include some historical remarks.

Motived by the desire to study geometry over the 'field with one element', in the past decade several authors have constructed extensions of scheme theory to geometries locally modelled on algebraic objects more general than rings. Semi-ring schemes exist in all of these theories, and it has been suggested that schemes over the semi-ring T of tropical numbers should describe the polyhedral objects of tropical geometry. We show that this is indeed the case by lifting Payne's tropicalization functor for subvarieties of toric varieties to the category of T-schemes. There are many applications such as tropical Hilbert schemes, tropical sheaf theory, and group actions and quotients in tropical geometry. This project is joint work with N. Giansiracusa (Berkeley).

Given a flat torus, we consider certain discrete graph approximations of

it and determine the asymptotics of the number of spanning trees

("complexity") of these graphs as the mesh gets finer. The constants in the

asymptotics involve various notions of determinants such as the

determinant of the Laplacian ("height") of the torus. The analogy between

the complexity of graphs and the height of manifolds was previously

commented on by Sarnak and Kenyon. In dimension two, similar asymptotics

were established earlier by Barber and Duplantier-David in the context of

statistical physics.

Our proofs rely on heat kernel analysis involving Bessel functions, which

in the torus case leads into modular forms and Epstein zeta functions. In

view of a folklore conjecture it also suggests that tori corresponding to

densest regular sphere packings should have approximating graphs with the

largest number of spanning trees, a desirable property in network theory.

Joint work with G. Chinta and J. Jorgenson.

We describe how one can recover the Mori--Mukai

classification of smooth 3-dimensional Fano manifolds using mirror

symmetry, and indicate how the same ideas might apply to the

classification of smooth 4-dimensional Fano manifolds. This is joint

work in progress with Corti, Galkin, Golyshev, and Kasprzyk.

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.

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.

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.

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.

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 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.