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Topological Order Complexes, Conical Resolutions, and Homology of (Non-)Discriminant Sets.
Colloquium| Speaker: | Prof. Vitya Vassiliev, Independent University of Moscow and UC Berkeley |
| Location: | 693 Kerr |
| Start time: | Mon, Feb 22 1999, 4:10PM |
Description
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.