Conference: Singularities in Analysis and Geometry

William Marsh Rice University, Houston TX

Investigators

Abstract

Abstract Award: DMS-0506270 Principal Investigator: Robert M. Hardt and Michael Wolf Singular objects and regularity theory for potentially singular objects have played and continue to play an essential role in all areas of geometry, topology and analysis. The works of F. Reese Harvey and John C. Polking, both jointly and separately,involving the analysis and structure of singularities, have ranged widely over all three areas. This conference, in recognition of their scholarly and academic service, will treat important issues involving singularities raised in their works over the last 30 years. It will focus on recent developments in singular spaces as an outgrowth of calibrated geometries, special Lagrangian submanifolds, removable singularity problems, and the relatively new area of singular connections. The conference will begin with an introductory session aimed at graduate students, recent Ph.D.'s, and non-specialist researchers. The body of the conference will involve topics from the wide expertise of the invited lecturers which encompass: singularities in calibrated geometries, exceptional holonomy, CR geometry, differential characters, geometry of singular spaces, singularities of complex submanifolds and varieties, extension problems for analytic objects, geometric aspects of singular connections, special Lagrangian theory, compactification of spaces of connections, and regularity theory of harmonic maps, special Lagrangian minimizers, and Yang-Mills connections. Finally one session of the conference will be devoted to the study of a specific implementation of technology in the classroom. Through mathematical models of a wide variety of phenomena from macro-economic systems, to cosmological models of the universe, to microscopic mechanics of solids, the notion of a "singularity" has gained prominence in the last thirty years. It is identified with the sudden disruption in space or time of a smooth media. It may occur at a point, as with two colliding particles, along a curve, as with lightning, or a surface, as with an earthquake fault, or on some higher dimensional "surface," as occurs in multi-parameter economic models. The basic fields in pure mathematics of analysis, geometry, and topology, have all attacked problems of understanding singularity formation and structure. This timely conference, on the occasion of the retirement of two academic leaders, brings together outstanding researchers and students from all these fields for a synergistic exchange of ideas. Many of the specific topics of the lectures of the conference have both their origins in and applications to mathematical physics, from the Lagrangian description of classical mechanical systems (1800's) to problems in string theory (1990's). The mathematical analysis of singularities provides important information for not only these well-established applications but also for new singularities arising from biological and computational problems.