The principal investigator proposes to further our understanding of the topology of singular spaces. Numerous invariants relevant vis-a-vis the classification theory of stratified spaces are associated to self-dual complexes of sheaves, as such objects induce a generalized Poincare-duality on the level of cohomology. On spaces with only even-codimensional strata, orthogonal decompositions of self-dual sheaves have been obtained by Beilinson-Bernstein-Deligne and Cappell-Shaneson. The investigator proposes to pursue this question on spaces that include strata of odd codimension as well. Initial evidence suggests that in many situations the odd-codimensional strata do not contribute terms in the decomposition. This new phenomenon leads to interesting consequences for associated characteristic classes and for stratified maps. In earlier work, the investigator has constructed L-classes of non-Witt spaces. He would like to explore the behavior and applications of these classes. In collaboration with S. Cappell, the investigator has obtained an Atiyah-type characteristic class formula for the twisted L-class of a singular space, assuming a ``transversality'' condition on the local system near the singularities. One is interested in extending the results beyond this assumption. This problem is already interesting on manifolds together with a codimension 2 submanifold and a local system defined on the complement. Lastly, the proposer in conjunction with R. Kulkarni, has constructed self-dual strongly perverse sheaves on the reductive Borel-Serre compactification of Hilbert modular surfaces. These are non-Witt spaces. The investigator would like to analyze the resulting invariants (particularly with a view towards their number-theoretic content), as well as generalize the results to higher dimensional modular varieties.
Topology is the study of geometrical objects focusing on the neighborhood relations between points rather than on measurement of distances. In the last century, considerable effort has been directed towards studying "manifolds" -- spaces that locally look uniform, at each point and in each direction. This effort has been immensely successful; a central insight was that crucial information about a manifold is carried by one number: its "signature," measuring intersections of geometric sub-objects within the manifold. In the last two decades, topologists have studied "singular" spaces with increasing interest, due to their numerous occurrences and applications within pure mathematics (algebraic geometry, number theory) and outside pure mathematics (mathematical physics). In contrast to a manifold, a singular space may locally look different from point to point. The proposed research tries to define, understand and compute a signature and other characteristic numbers for singular spaces.
