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Topological Order Complexes, Conical Resolutions, and Homology of (Non-)Discriminant Sets.


Speaker: Prof. Vitya Vassiliev, Independent University of Moscow and UC Berkeley
Location: 693 Kerr
Start time: Mon, Feb 22 1999, 4:10PM

Many important topological spaces admit the structure of a partially ordered set.

Examples: disjoint unions of Grassmannians and multi-Grassmannians; configuration and multiconfiguration spaces; spaces of ideals of a topological ring; spaces of possible singular sets of algebraic subvarieties of a given degree.

The corresponding order complexes, supplied with a natural (but non-standard) topology, usually are very interesting spaces and provide numerous good problems on generalized configuration spaces. They are the main mean in studying homology groups of {em discriminant} sets of singular geometric objects, and hence, by the Alexander duality, also of the complementary sets of nonsingular objects.

In the examples listed above, these sets are: general linear groups, spaces of symmetric or Hermitian matrices with simple spectra, spaces of nondegenerate polynomials, spaces of knots or links or curves without multiple selfintersections; spaces of nonsingular algebraic hypersurfaces (or curves) of a given degree.

In particular, an algorithm of calculating homology groups of the latter class of spaces will be described.