Principal Investigator: Sylvain Cappell
This research conference will bring together workers and students in three key and currently active geometrical areas: singular varieties, submanifolds and knotting, and stratified spaces. These three areas of topology have historically enjoyed vigorous and productive scientific interactions, spurring important innovations through sharing ideas. Moreover, developments and problems in these subjects in topology are connected to parallel questions in other branches of geometry, analysis and algebraic geometry, where there are foundational questions concerning extending "classical" methods from smooth (nonsingular) settings to more general ones in which singularities can arise. Exciting recent developments to be discussed at this meeting include investigations on how singularities of embeddings and immersions in four-dimensional manifolds are shedding new light on basic questions concerning classification and structure of four dimensional manifolds; and how such questions also interact with new approaches to deep invariants of knots and links in three dimensional manifolds and their cobordisms. Also to be discussed are various theories of characteristic clases for singular varieties and foundational questions concerning their definitions, calculational methods, and applications, e.g., to current developments in transformation groups and to geometrical combinatorics of polytopes.
This research conference on important, current geometrical approaches to topology will bring together researchers and students who are investigating some mathemtically basic, ubiquitous and related topological phenomena from several different perspectives. These involve spaces, called varieties, that can have variable geometry at different points. Such variations in geometry, called singularities, can be analyzed using mathematical theories of knots and links (in many dimnensions); to elucidate this, such varieties are decomposed into strata. Because the natural spaces that arise in many research problems in both mathematics and theoretical physics display such singularities, investigations of such varieties, of their strata and their singularities, and of related questions about knots and links are needed. Recent and exciting developments to be analyzed include relations of such questions to four dimensional geometry, to the symmetries and invariants of such spaces, and to applications to problems in enumeration and combinatorics that arise in a very wide range of scientific problems.
