Topology, Geometry and Physics

Harvard University, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0405143 Principal Investigator: Clifford H. Taubes The research of C. H. Taubes will mostly concentrate in differential topology, but there are three mathematical physics projects of interest as well. The primary project involves the use of symplectic geometry tools to investigate the differential topology of four dimensional spaces. The long range goal is to shed light on the classification for smooth, four dimensional spaces, this one of the outstanding open problems in low dimensional topology. As there is considerable circumstantial evidence for a symplectic geometry role in the story, the immediate plan is to probe the symplectic geometry connections to the subject. The first of the mathematical physics projects aims to construct an equivariant Fredholm setting for a Dirac operator on the loop space of a manifold. Such a setting would provide a rigorous framework for various aspects of string-theoretic physics. The second mathematical physics project will study the minimal energy states for a Hartree-Fock approximation to the strong coupling Hamiltonian for quantum chromodynamics. The third mathematical physics project is undertaken jointly with a chemist; it studies a version of free energy for a DNA molecule in a salt solution. The aim here is to predict the behavior of such molecules in various biological contexts. The major project probes the possibilities for the large scale structure of four dimensional universes. For example, our universe has four dimensions, these the usual three dimensions of space plus time as the fourth; and astronomers have confirmed Einsteins prediction that our universe is curved at large scales. The research seeks to provide a complete list of the possible large scale structures for a four dimensional universe. As of now, there is no credible conjectured list for the possibilities, but there are clues to its format. There are three other research topics that are motivated by questions in mathematical physics. The first of these seeks to confirm various conjectures from unified field theories in physics for gravity and the subatomic forces. The second topic explores techniques for predicting properties of the strong force that binds subnuclear quarks as mesons, neutrons and protons. The third topic studies equations from physical chemistry that predict the behavior of DNA molecules in various biological contexts. The goal is to develop tools to simplify the calculations.