Mathematical Sciences: Topology and its Applications

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9626817 Cappell Technical summary: In this project the investigator will study several problems about topological invariants of singular varieties and their relation to symplectic geometry and algebraic geometry. General formulae for invariants of singular varieties will be developed and will be applied, using toric varieties, to obtain precise comparisons of lattice summation and integration. Analytical, combinatorial and number theoretical aspects of these comparisons will be explored with a view to many possible applications. Another line of investigation will be the use of symplectic geometrical methods in the study of representation-theoretic invariants of three-manifolds. This will include a series of studies of Maslov indices and analytical formulae for Casson's SU(2) invariant as well as a study of SU(n) invariants. A third area of investigation will be the development of a satisfactory general theory for the topological classification of finite group actions on manifolds; this will encompass such foundational questions as the appropriate notion of functoriality for this classification problem in a given homotopy type. Non-technical summary: In this project, constructions and ideas drawn from topology and geometry will be used to get precise formulae that compare multidimensional integration with lattice summations. Such comparison formulae have been available for several centuries for one-dimensional integrals and sums, i.e., in the Euler-MacLaurin expansion, but have not been available in adequate generality in the higher dimensional settings needed for many applications. Through a series of intermediate steps, such problems have been translated into questions about the topology of spaces which have singularities (that is, which are not uniform) and the measurement of how such singularities contribute to the computation of invariants of such spaces. The deep methods of modern topology will be applied to resolve these geometrical ques tions and hence ultimately to solve the concrete problems of precise comparisons of integrals and sums. Envisioned applications include new approaches to rapid integration and to discrete optimization. ***