Topology and Physics

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AbstractAward: DMS-1207817 Principal Investigator: Daniel S. FreedThe research in this proposal fits into the broad interaction between geometry and high energy theoretical physics, with a focus on algebraic topology. There are both purely mathematical projects influenced by the physics and projects which apply topology to concrete current problems in quantum field theory, string theory, and condensed matter theory. On the mathematics side we seek a categorification of the Atiyah-Singer index theorem. This requires us to develop generalized differential cohomology theory further. We will also investigate several problems in topological field theory, particularly concerning invertible theories, which have implications for algebra and low-dimensional topology. On the physics side we begin to study aspects of the maximal superconformal field theory in six dimensions, which is playing an increasingly central role in applications to mathematics. We also continue our work on topological questions in Type II superstring theory. The PI will use his experience in this area to attack topological questions in condensed matter physics, a new area for his research. There are also graduate student projects in various directions in geometry and topology.The research supported by this grant develops modern ideas in geometry which are closely connected with current work in theoretical physics. Quantum field theory was developed in the last century to explain elementary particle physics, and it has had spectacular implications for mathematics as well. String theory is more speculative from the physics point of view, even more mysterious in mathematics, yet it too provides profound geometric insights. Many of the projects to be pursued here lie at the interface between geometry and these physical theories. They are part of a much broader community effort to bring new ideas from physics into mathematics. Some projects investigate mathematical structures and problems suggested by the physics. In others the PI collaborates with physicists to apply modern techniques in geometry and topology directly to problems in physics. The longterm value of such basic research for society, above and beyond its intrinsic intellectual value, can be only be measured now by understanding how fundamentally our current technology and economy rely on basic research of past decades and centuries. This grant also supports a longstanding outreach program in Austin which engages middle and high school students in mathematics.