Conference: Singularities in Analysis and Geometry

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AbstractAward: DMS-0506270Principal Investigator: Robert M. Hardt and Michael WolfSingular objects and regularity theory for potentially singularobjects have played and continue to play an essential role in allareas of geometry, topology and analysis. The works of F. ReeseHarvey and John C. Polking, both jointly and separately,involvingthe analysis and structure of singularities, have ranged widelyover all three areas. This conference, in recognition of theirscholarly and academic service, will treat important issuesinvolving singularities raised in their works over the last 30years. It will focus on recent developments in singular spaces asan outgrowth of calibrated geometries, special Lagrangiansubmanifolds, removable singularity problems, and the relativelynew area of singular connections. The conference will begin withan introductory session aimed at graduate students, recentPh.D.'s, and non-specialist researchers. The body of theconference will involve topics from the wide expertise of theinvited lecturers which encompass: singularities in calibratedgeometries, exceptional holonomy, CR geometry, differentialcharacters, geometry of singular spaces, singularities of complexsubmanifolds and varieties, extension problems for analyticobjects, geometric aspects of singular connections, specialLagrangian theory, compactification of spaces of connections, andregularity theory of harmonic maps, special Lagrangianminimizers, and Yang-Mills connections. Finally one session ofthe conference will be devoted to the study of a specificimplementation of technology in the classroom.Through mathematical models of a wide variety of phenomena frommacro-economic systems, to cosmological models of the universe,to microscopic mechanics of solids, the notion of a"singularity" has gained prominence in the last thirtyyears. It is identified with the sudden disruption in space ortime of a smooth media. It may occur at a point, as with twocolliding particles, along a curve, as with lightning, or asurface, as with an earthquake fault, or on some higherdimensional "surface," as occurs in multi-parametereconomic models. The basic fields in pure mathematics ofanalysis, geometry, and topology, have all attacked problems ofunderstanding singularity formation and structure. This timelyconference, on the occasion of the retirement of two academicleaders, brings together outstanding researchers and studentsfrom all these fields for a synergistic exchange of ideas. Manyof the specific topics of the lectures of the conference haveboth 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 mathematicalanalysis of singularities provides important information for notonly these well-established applications but also for newsingularities arising from biological and computational problems.