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Host: University of Colorado
Sponsor: American Mathematical Society, National Science Foundation
Homepage: http://www.ams.org/meetings/src-cappell.html
Email: wsd@ams.org
Organizers: Sylvain E. Cappell, Courant Institute, Ronnie Lee, Yale University, Wolfgang Lück, Westfälische Wilhelms-Universität Münster
Deadline for abstracts: March 03, 1999
Description:
Recently, researchers in topology, geometry and global analysis have been encountering
some related issues in attempts to extend classical methods and results from manifolds
to more general settings of singular varieties. These are needed for applications because
many of the natural spaces that are the focus of current investigations are usually
singular. Examples include: representation spaces, such as those of fundamental groups
of Riemann surfaces arising in the study of low dimensional manifolds; moduli space
constructions in algebraic geometry; the singular spaces which arise in trying to study
and classify general finite or compact group actions on manifolds; and spaces produced
in a variety of contexts by natural compactification procedures.
The object of this conference will be to describe some recent advances in this general area and to discuss and compare questions, methods, and applications from a variety of perspectives. Among other subjects, the conference will include talks on the following topics:
1. Topological invariants and classifications of singular varieties. A toplogical motivation for studying singular varieties is the study of group actions, because their orbit spaces are stratified spaces. A goal is to make some powerful new classification methods more accessible and applicable by considering spaces arising from natural contexts. A natural question, which can be approached from several viewpoints, is: what can be said about topological classification of algebraic varietiesNULL
Important recent developments in the theory of singular varieties are related to the study of their characteristic classes. There are, in fact, several closely related theories of characteristic classes developed by a number of workers. Among questions to be addressed are: What are the relations between different theories of characteristic classes and what do they reveal about geometric and analytic structures of the varietiesNULL When can such classes, which for singular varieties generally take values in homology, be lifted back at least partially towards cohomology or to other theories; and what additional structures are necessary for such refinementsNULL
Intersection homology theory has been a powerful tool for the investigation of intersection theories and numbers in singular varieties. But its functoriality, analytical interpretation, e.g., its relation to L2 cohomolgy, as well as other issues are still not fully understood. Moreover, it doesn't cover all the now needed settings, e.g., in symplectic geometry. This meeting will be a forum for such new issues in intersection theory.
2. 3-manifold invariants and moduli spaces. Some subtle invariants of 3-manifolds are related to representation spaces. But extensions of Casson's SU(2) ideas to more general Lie groups encounter difficulties due to the more singular nature of the representation varieties and require investigations of Lagrangian subvarieties in singular symplectic varieties. These invariants shoud be compared with combinatorial invariants, e.g., those related to Vassiliev's perspective on knot theory and to "finite type invariants" of 3-manifolds.
Representation varieties have been investigated by algebraic geometers as moduli spaces of holomorphic G-bundles over a Riemann surface. Related moduli objects (e.g., of Higgs bundles, of the moduli spaces of stable k-pairs, etc.) arose through interactions with physics. Such moduli spaces exhibit similar symplectic and Kahler structures as well as gauge theory interpretations. It will be desirable to compare different treatments of singularities.
3. -Betti numbers and -torsions. -invariants such as -Betti numbers, Novikov-Shubin invariants, -torsions can be defined analytically in terms of the heat kernel of the universal covering as well as topologically. This gives fruitful links between analysis and topology with applications in differential geometry, topology, group theory and algebraic -theory.
Connections to the first topic come from the relation of intersection homology and -cohomology. The Cheeger-Goresky-MacPherson Conjecture and the Zucker Conjecture link the -cohomology of the regular part with the intersection homology of an algebraic variety. For applications in group theory and algebraic -theory it was necessary to define -Betti numbers for "very singular" spaces. This suggests extending results from actions of finite groups to (proper) actions of discrete groups.
-Betti numbers, Novikov-Shubin invariants and -torsion have been studied for -manifolds and are linked to other invariants, e.g., volume of hyperbolic manifolds and Gromov's simplicial volume. Open conjectures include the Atiyah conjecture, the Singer Conjecture and the zero-in-the-spectrum conjecture which are related to algebraic -theory, global analysis, and topological rigidity conjectures.
Mail Address:
Summer Research Conferences Coordinator American Mathematical Society P.O. Box 6887 Providence, RI 02940
Date received: February 17, 1999
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