Mathematical Sciences: Topological Methods in Quantum Field Theory and Methods of Quantum Field Theory in Topology

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Schwarz proposes to study various problems in quantum field theory. He will try to construct invariants of contact manifolds and of Legendre knots in three-dimensional contact manifolds by means of topological invariants from quantum field theory. There will also be a study of semi-infinite cohomology of infinite- dimensional Lie algebras and the theory of symplectic and contact supermanifolds. These problems are related to the general theory of quantization of constrained systems and to the superstring theory. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concepts of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments, collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single theory.