Measures on Banach Manifolds, Random Surfaces, and Nonperturbative String Field Theory with Cut-offs

We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this partition function and Riemann Surfaces

The cohomology rings of abelian symplectic quotients

Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is torsion-free

On the cohomology rings of Hamiltonian T-spaces

Let $M$ be a symplectic manifold equipped with a Hamiltonian action of a torus $T$. Let $F$ denote the fixed point set of the $T$-action and let $i:F\hookrightarrow M$ denote the inclusion. By a theorem of F. Kirwan \cite{K} the induced map $i^*:H_T^*(M) \to H_T^*(F)$ in equivariant cohomology is an injection. We give a simple proof of a formula of Goresky-Kottwitz-MacPherson \cite{GKM} for the image of the map $i^*$.Comment: correction to reference

On the growth of solutions to the minimal surface equation over domains containing a halfplane

We consider minimal graphs u(x,y)>0 over unbounded domains D (with u vanishing on the boundary of D). Assuming D contains a sector properly containing a halfplane, we obtain estimates on growth and provide examples illustrating a range of growth.Comment: 11 pages, one figur

Relations in the cohomology ring of the moduli space of flat SO(2n+1)SO(2n+1)-connections on a Riemann surface

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalising a conjecture of Newstead.Comment: 30 pages, 10 figures. Several corrections and clarifications; final versio

On semifree symplectic circle actions with isolated fixed points

Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes of $(P^1)^n$

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincar\'e duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.Comment: added reference; to appear in Journal of Differential Geometr

The Weighted Euler-Maclaurin Formula for a simple integral polytope

We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained by Karshon, Sternberg and Weitsman in the ''Euler-Maclaurin with remainder for a simple integral polytope''.Comment: 11 pages; to appear in the Asian Journal of Mathematic

Euler Maclaurin with remainder for a simple integral polytope

We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.Comment: 25 pages. Also see our previous paper in Proc. Nat. Acad. Sci. 100 no. 2 (2003), 426-433. Revision: added Euler Maclaurin formulas for symbols and for polynomial

On geometric quantization of bmb^m-symplectic manifolds

We study the formal geometric quantization of $b^m$-symplectic manifolds equipped with Hamiltonian actions of a torus $T$ with nonzero leading modular weight. The resulting virtual $T$-modules are finite dimensional when $m$ is odd, as in [GMW2]; when $m$ is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight.Comment: 7 page