Topology, Geometry and Physics

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AbstractAward: DMS-0405143Principal Investigator: Clifford H. TaubesThe research of C. H. Taubes will mostly concentrate in differentialtopology, but there are three mathematical physics projects ofinterest as well. The primary project involves the use of symplecticgeometry tools to investigate the differential topology of fourdimensional spaces. The long range goal is to shed light on theclassification for smooth, four dimensional spaces, this one of theoutstanding open problems in low dimensional topology. As there isconsiderable circumstantial evidence for a symplectic geometry role inthe story, the immediate plan is to probe the symplectic geometryconnections to the subject. The first of the mathematical physicsprojects aims to construct an equivariant Fredholm setting for a Diracoperator on the loop space of a manifold. Such a setting wouldprovide a rigorous framework for various aspects of string-theoreticphysics. The second mathematical physics project will study theminimal energy states for a Hartree-Fock approximation to the strongcoupling Hamiltonian for quantum chromodynamics. The thirdmathematical physics project is undertaken jointly with a chemist; itstudies a version of free energy for a DNA molecule in a saltsolution. The aim here is to predict the behavior of such moleculesin various biological contexts.The major project probes the possibilities for the large scalestructure of four dimensional universes. For example, our universehas four dimensions, these the usual three dimensions of space plustime as the fourth; and astronomers have confirmed Einsteinsprediction that our universe is curved at large scales. The researchseeks to provide a complete list of the possible large scalestructures for a four dimensional universe. As of now, there is nocredible conjectured list for the possibilities, but there are cluesto its format. There are three other research topics that aremotivated by questions in mathematical physics. The first of theseseeks to confirm various conjectures from unified field theories inphysics for gravity and the subatomic forces. The second topicexplores techniques for predicting properties of the strong force thatbinds subnuclear quarks as mesons, neutrons and protons. The thirdtopic studies equations from physical chemistry that predict thebehavior of DNA molecules in various biological contexts. The goal isto develop tools to simplify the calculations.