Analysis on Singular Spaces
August 18, 2008 to December 19, 2008
Gilles Carron (University of Nantes), Eugenie Hunsicker (Loughborough University), Richard Melrose (Massachusetts Institute of Technology), Michael Taylor (Andras VasyUniversity of North Carolina, Chapel Hill), and Jared Wunsch (Northwestern University)
A central problem in modern mathematics is that of extending analytic constructions which are well understood in the setting of smooth compact manifolds to a broader class of spaces which are allowed to be singular. Such objects arise naturally in many geometric contexts: Singular varieties in algebraic geometry not only occur naturally as fundamental objects themselves, but even the moduli spaces of smooth varieties are naturally singular. Seemingly smooth, noncompact objects often become singular spaces upon compactification: Euclidean space can be radially compactified to a manifold with boundary, the simplest possible “singular space,” while the configuration space for k-particle dynamics on Rn naturally has a compactification as an n-dimensional manifold with corners. Smooth symmetric spaces often have natural compactifications, such as the Borel-Serre compactification, that are manifolds with corners. And objects with irregular boundaries occur frequently in mathematical physics: classical problems like the scattering of waves by a slit already involve singular geometries. Singular structures are moreover thought to play an important role in the scattering of seismic waves through the interior of the earth; the associated inverse problem is of manifest practical importance.
The study of elliptic equations on singular spaces has had fruitful interaction with topology; for instance the work of Goresky-MacPherson on intersection cohomology has pointed the way toward extending de Rham and Hodge theory to broader geometric settings. Turning to hyperbolic equations, wave propagation on incomplete spaces is complicated by diffractive effects and the subtleties of glancing rays, while on complete spaces with nice compactifications, such as Schwarzschild space, energy decay near various boundary faces is the subject of intensive current study. The subject of spectral and scattering theory on singular spaces has vast reach, spanning both number theory (modular forms) and physics (many body scattering, relativity).
All of these areas of analysis on singular space have in common the use, whether explicit or implicit, of asymptotic expansions of solutions to partial differential equations near singular strata. Calculi of pseudodifferential operators are essential tools in many problems, and a bewildering menagerie of these calculi are now known. As a result there is substantial duplication of effort, heightened by language barriers, between experts in these different subjects, and we hope that a program uniting researchers in these diverse fields will have payoffs in the transfer of mathematical “technology” from one field to another, as well as unifying disparate pseudodifferential approaches. A systematic and general theory of PDEs on stratified spaces, using iterative techniques to peel away successive strata, would be a long-term goal of the program.
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