Topology and its Applications

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DMS-0073006Sylvain CappellAmong the range of problems to be investigated in topology and geometry in low and high dimensions, some concern the study of transformation groups, that is the symmetries of manifolds and of more general spaces. New methods of classifying such group actions will be developed with a view to making good connections with methods of equivariant homotopy theory; basic "naturality" questions for the set of group actions on a manifold will be studied. Another set of problems concerns a variety of both topological and algebraic invariants of varieties, e.g. theories of characteristic classes, and new methods of explicitly computing them. A related series of questions to be investigated concerns the relations between the global topology of divisors and the local geometry of their singularities which must, in general, be regarded as "singular knots." The natural actions of mapping class groups and their Torelli subgroups on the moduli spaces of representations (which are foundational in algebraic geometry, in gauge theory, in 3-manifold topology, and in string theory) will be investigated with a view to applications in three dimensional topology. Decomposition methods for studying analytic and geometric invariants of manifolds and the relations between them will be investigated, again with a view of applying such relation to three manifolds. Computations of generalized characteristic classes of toric varieties will be combined with other topological, geometric, and analytical methods to obtain results in geometrical combinatorics and applications to problems concerning lattice sums.This research project involves several investigations in a range of problems in topology and geometry in low and high dimensions and studies of some new applications of these in other areas of mathematics. Some of the research work will involve invariants of manifolds and of more general spaces, such as singular varieties. Effective methods of computing such natural invariants will be sought. The moduli spaces and three manifold invariants to be investigated also arise in geometrical approaches to theoretical physics. A combination of geometrical, algebraic and analytical methods will be used to study possible applications of singular varieties to problems concerning comparisons of lattice sums with integrals. Such comparisons are of interest in many areas of the mathematical sciences.