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.