Principal Investigator: Daniel S. Freed
The geometry group at the University of Texas proposes to carry out a variety of research projects, most of which are related to physics. Dan Freed's current research focuses on questions in geometry and topology arising from string theory. These include the construction of determinant line bundles on manifolds with boundary and the understanding of anomalies of actions which arise in quantizing lagrangian field theories. Karen Uhlenbeck's current research involves the geometric theory of integrable systems, as well as a non-linear Schroedinger equation arising in macroscopic theories of a ferro-magnetic continuum. These two researchers are starting a project aimed at understanding the appearance of integrable systems in conformal field theories. Bob Williams is applying his expertise on attractors to the construction of tiling spaces. Constantin Teleman is pursuing several projects in the cohomology of infinite dimensional Lie Algebras. He and his coworkers are making progress on the MacDonald conjectures, and are also giving geometric interpretations in terms of the Hodge cohomology of flag varieties of loop groups. He is also interested in homotopy equivalences between holomorphic and continuous mapping spaces. Postdoctoral members of the group are Nurit Krausz, who is working on direct computations for quantum field theory in Minkowski space, and Adrian Vajiac who uses equivariant localization techniques to study topological quantum field theories.
At this point in time, geometry is a rapidly developing area of mathematics. While research in geometry, like most of pure mathematics, consists of the construction and development of abstract concepts, the origins and ultimate applications for these constructions are invariably examples and applications in more applied fields. The influence of theoretical physics on geometry is strong. For example, large numbers of geometers are currently working on questions related to quantum groups, mirror symmetry and quantum cohomology. Our group attempts not to work on problems which have already been identified by mathematicians as central, but in contrast we look at current ideas in physics of all sorts and then find, clarify, and work on the mathematically interesting questions more directly. Dan Freed's work on string theory connects the physics ideas of quantization with the mathematical subject of algebraic topology. His joint project with Karen Uhlenbeck on the appearance of integrable systems in certain quantum field theories requires an understanding of field theory, integrable systems, and the very important and basic ideas of symmetry. Constantin Teleman's work deeply involves fundamental ideas of symmetry, as well as delving into the question of how closely very messy functions can be approximated qualitatively by polynomial-like objects. Some of the geometric ideas come from other branches of physics, such as the ferro-magnetic equations studied by Uhlenbeck. The tiling spaces of William's come from beautiful examples such as those constructed by Roger Penrose. Our efforts bring new ideas and techniques into mathematics, rather than concentrating on projects which are already popular.
