Mathematical Sciences: Topology, Geometry, and Physics

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Three investigators will study problems in the areas of geometric partial differential equations, mathematical physics, and dynamical systems. One will continue his investigation of geometric aspects of the Atiyah-Singer index theorem and the relationship between quantum field theory and low-dimensional topology. A second will continue an ongoing investigation of the harmonic map equation and its connections with Poisson-Lie structures and quantum groups. She will also investigate the Yang-Mills functional in dimensions larger than four. A third investigator will study the knot types of periodic orbits in the Lorenz attractor. Three investigators will continue ongoing projects involved in differential equations, topology, and mathematical physics. Differential equations relate the rate of change of one or more variables with respect to each other, or with respect to time. Topology is the mathematical study of shape. Both of these areas have a long history of use in mathematical physics.