Abstract:
A folklore result uses the Lovasz local lemma to analyze the discrepancy of hypergraphs with bounded degree and edge size. We generalize this result to the context of real matrices with bounded row and column sums.

Abstract:
Basic path-matchings, introduced by Cunningham and Geelen (FOCS 1996), are a common generalization of matroid intersection and non-bipartite matching. The main results of this paper are a new algebraic characterization of basic path-matching problems and an algorithm for constructing basic path-matchings in O(n^w) time, where n is the number of vertices and w is the exponent for matrix multiplication. Our algorithms are randomized, and our approach assumes that the given matroids are linear and can be represented over the same field. Our main results have interesting consequences for several special cases of path-matching problems. For matroid intersection, we obtain an algorithm with running time O(nr^(w-1))=O(nr^1.38), where the matroids have n elements and rank r. This improves the long-standing bound of O(nr^1.62) due to Gabow and Xu (FOCS 1989). Also, we obtain a simple, purely algebraic algorithm for non-bipartite matching with running time O(n^w). This resolves the central open problem of Mucha and Sankowski (FOCS 2004).

Abstract:
It is shown that the Type IIA superstring compactified on $K3$ has a smooth string soliton with the same zero mode structure as the heterotic string compactified on a four torus, thus providing new evidence for a conjectured exact duality between the two six-dimensional string theories. The chiral worldsheet bosons arise as zero modes of Ramond-Ramond fields of the IIA string theory and live on a signature $(20,4)$ even, self-dual lattice. Stable, finite loops of soliton string provide the charged Ramond-Ramond states necessary for enhanced gauge symmetries at degeneration points of the $K3$ surface. It is also shown that Type IIB strings toroidally compactified to six dimensions have a multiplet of string solutions with Type II worldsheets.

Abstract:
This review is based on lectures given at the 1992 Trieste Spring School on String Theory and Quantum Gravity and at the 1992 TASI Summer School in Boulder, Colorado.

Abstract:
It is shown that the scattering of spacetime axions with fivebrane solitons of heterotic string theory at zero momentum is proportional to the Donaldson polynomial.

Abstract:
The modifications of dilaton black holes which result when the dilaton acquires a mass are investigated. We derive some general constraints on the number of horizons of the black hole and argue that if the product of the black hole charge $Q$ and the dilaton mass $m$ satisfies $Q m < O(1)$ then the black hole has only one horizon. We also argue that for $Q m > O(1)$ there may exist solutions with three horizons and we discuss the causal structure of such solutions. We also investigate the possible structures of extremal solutions and the related problem of two-dimensional dilaton gravity with a massive dilaton.

Abstract:
We investigate nonperturbative effects in M-theory compactifications arising from wrapped membranes. In particular, we show that in $d=4, \CN=1$ compactifications along manifolds of $G_2$ holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize many M-theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invariants of the 3-cycle. We discuss briefly some applications of these results. For example, using mirror symmetry we derive a counting formula for supersymmetric three-cycles in certain Calabi-Yau manifolds.

Abstract:
We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized Kac-Moody algebras. Moreover, a certain limit of the formulae suggests a reformulation of heterotic string in terms of a gauge theory based on hyperbolic algebras such as $E_{10}$. We define a generalized Kac-Moody Lie superalgebra associated to the BPS states. Finally we discuss the relation of our results with string duality.

Abstract:
We study the dyon spectrum in $N=2$ Super Yang-Mills theory with gauge group $SU(2)$ coupled to $N_f$ matter multiplets in the fundamental representation. For magnetic charge one and two we determine the spectrum explicitly and show that it is in agreement with the duality predictions of Seiberg and Witten. We briefly discuss the extension to higher charge monopoles for the self-dual $N_f=4$ case and argue that the conjectured spectrum of dyons predicts the existence of certain harmonic spinors on the moduli space of higher charge monopoles.