Topology, Geometry and Physics

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The foreeable future research of C. H. Taubes will concentrate on differential topology questions and on also mathematical physics. What follows lists six specific topics of concentration:· Using the Seiberg-Witten equations to study the dynamics of vector fields on manifolds of dimension three and higher. · Using symplectic geometry to define invariants of four dimensional manifolds. · Studying geometric aspects of the spectrum of the Dirac operator on 3-dimensional manifolds· Analyzing the non-compactness of moduli spaces of other Seiberg-Witten like equations.· Studying the relations between various Floer homology/cohomology theories for 3-dimensional manifolds.· Studying certain quantum field theories of maps from a manifold or the loop space of a manifold to a compact Lie group.The ultimate goal for the first four project topics on this list is to develop new tools and methods to study the differential topology of low dimensional manifolds. Questions in low dimensional differential topology are central to much of current research in both geometry and topology. Symplectic and gauge theoretic techniques provide powerful tools for studying such manifolds, but they are not known to be all powerful, More is needed, especially in dimension four where ignorance is broad and deep. Thus the focus here to develop new tools and sharpen the old ones. The goal for the fifth project topic is to settle once and for all various conjectured equivalences between differential topology invariants of three and four dimensional manifolds. The goal of the final topic is to make some rigorous quantum field theories of the sorts that are used by physicists to model elementary particle interactions, and of the sort used to define conformal and topological field theories. The construction of mathematically rigorous versions of realistic quantum field theories is one of the key concerns of mathematical physics. Our universe requires three spatial coordinates to describe the position of any given event, and then a fourth parameter, which is the time of the event. A space that requires four coordinates to describe is said to have four dimensions. Any such space looks rather boring, and much like any other at small scales, but the large scale structure can be very complicated. The surface of the earth provides a 2-dimensional example. Any given place requires 2 parameters to specify (longitude and latitude). At small scales (those visible to us pedestrians), any one place looks relatively flat (modulo a hill or two) and much like any other. However, the surface of the earth at large scales is curved as it is, after all, a sphere. A fundamental question is to determine the large scale 4-dimensional shape of our universe. Any such determination will ultimately come from Astronomy. Even so, it is a mathematical problem to determine a reasonable list of the possible 4-dimensional shapes. Most of this research project is directed towards developing the mathematical tools that will allow us to determine such a list. As it turns out, lists are known for the spaces of dimension less than 4, and tools are available that give appropriate lists for spaces of dimension greater than 4. Dimension 4 seems the toughest nut to crack. A secondary focus of this research is to investigate certain model equations that physicists use to predict the behavior of matter and energy at sub-atomic length scales. These equations are examples of quantum field theories. Realistic quantum field theories are very complicated and so calculations are often impossible to do. The models studied here may suggest improved calculational techniques.