Conference on Submanifolds, Singular Varieties and Stratified Spaces

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AbstractAward: DMS-0413651 Principal Investigator: Sylvain CappellThis research conference will bring together workers and studentsin three key and currently active geometrical areas: singularvarieties, submanifolds and knotting, and stratifiedspaces. These three areas of topology have historically enjoyedvigorous and productive scientific interactions, spurringimportant innovations through sharing ideas. Moreover,developments and problems in these subjects in topology areconnected to parallel questions in other branches of geometry,analysis and algebraic geometry, where there are foundationalquestions concerning extending "classical" methods from smooth(nonsingular) settings to more general ones in whichsingularities can arise. Exciting recent developments to bediscussed at this meeting include investigations on howsingularities of embeddings and immersions in four-dimensionalmanifolds are shedding new light on basic questions concerningclassification and structure of four dimensional manifolds; andhow such questions also interact with new approaches to deepinvariants of knots and links in three dimensional manifolds andtheir cobordisms. Also to be discussed are various theories ofcharacteristic clases for singular varieties and foundationalquestions concerning their definitions, calculational methods,and applications, e.g., to current developments in transformationgroups and to geometrical combinatorics of polytopes.This research conference on important, current geometricalapproaches to topology will bring together researchers andstudents who are investigating some mathemtically basic,ubiquitous and related topological phenomena from severaldifferent perspectives. These involve spaces, called varieties,that can have variable geometry at different points. Suchvariations in geometry, called singularities, can be analyzedusing mathematical theories of knots and links (in manydimnensions); to elucidate this, such varieties are decomposedinto strata. Because the natural spaces that arise in manyresearch problems in both mathematics and theoretical physicsdisplay such singularities, investigations of such varieties, oftheir strata and their singularities, and of related questionsabout knots and links are needed. Recent and excitingdevelopments to be analyzed include relations of such questionsto four dimensional geometry, to the symmetries and invariants ofsuch spaces, and to applications to problems in enumeration andcombinatorics that arise in a very wide range of scientificproblems.