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Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on the moduli space of punctured Riemann surfaces. The presence of supermoduli has been a major obstacle for a long time in carrying out this program. Recently, this obstacle has been overcome at genus 2, which is the first loop order where it appears in all amplitudes. An important ingredient is a better understanding of the relation between geometry and supergeometry, and between holomorphicity and superholomorphicity. This talk provides a survey of these developments and a brief discussion of the directions for further investigation.
The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the
Theorem 1 (closed strings). Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain $X$ satisfying the master equation $dX + X * X = 0$ where * is the sum over all gluings. This structure is well defined up to homotopy.
The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure called a "quantum Lie bialgebra" partially resolving the involutive identity.
There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots.
Much progress has been made in the last few decades in developing the necessary mathematics for understing the full implications of the quantum-mechanical many-body problem and why the material world appears to be as stable as it is despite the serious $−1/|x|$ singularity of the Coulomb potential that attracts negative electrons to positive atomic nuclei. Many problems remain, however, especially the understanding of the interaction of matter and the quantized radiation field discovered by Planck in 1900. A short review of some of the main topics, recent progress, and open problems will be given.
Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
In this paper, I discuss the additivity conjecture in quantum information theory. The additivity conjecture was originally a set of at least four conjectures. These conjectures said that certain functions of quantum states and channels were additive under tensor products. While some of these conjectures were previously known to be stronger than others, they have recently all been proved equivalent. This conjecture is a very intriguing mathematical question which the best efforts of a large number of quantum information theorists have not been able to resolve for nearly a decade. It is a mathematically elegant question that is one of the most important open questions in the field of quantum information and computation. This paper then is intended to be both an exposition of the conjecture, its background, and some of the methods that have been used to yield partial results for it, as well as a plea for help in resolving this conjecture. Very recently (summer 2007), substantial progress has been made on this conjecture, in that counterexamples to a set of stronger conjectures have been found. These will be described briefly.