Analysis on Singular Spaces (21w5222)

Organizers

(University of Illinois, Urbana-Champaign)

(Universidad Nacional Autonoma de Mexico)

Jesse Gell-Redman (University of Melbourne)

(Massachusetts Institute of Technology)

Description

The Casa Matemática Oaxaca (CMO) will host the "Analysis on Singular Spaces" workshop in Oaxaca, from May 16 to May 21, 2021.


The presence of singularity in mathematics and physics is an omnipresent phenomena which is the subject of an enormous amount of current research. One reason for this is that techniques which are useful and well understood in the context of smooth spaces do not carry over automatically to singular spaces in general. Depending on the type of singularity involved, one might find that a theorem which is true in the smooth setting requires modification even to make sense in a particular singular setting, and a singular object with extra structure may lead to a generalization of a theorem in which that extra structure plays a role.

Once one incorporates the necessity of singularity into ones perspective it becomes clear that mathematical structures and singular structures in fact reveal information about one another. In fact, the differential equations that model physical phenomena and allow for a systematic exploration of geometric structures exhibit a surprising feature; where they fail to be smooth, i.e. where they are singular, far from lacking structure, they frequently possess a type of structured degeneracy which is iterative, in the sense that it involves a sequence of processes which generate degeneracy at increasing depth. This feature is exhibited in all areas of analysis; in the study of configuration spaces of geometric structures, of states of physical systems, of algebraic singularity, and of symmetry groups and symmetric spaces. Moreover a sort of converse process occurs frequently at the infinite ends of non-compact objects, in that the natural process of compactification (of "bringing infinity in'' so that objects and processes at far distance become understandable objects) have the same sort of iterative structure. The iterative structure points to a method for approaching these objects and studying them as mathematical objects, in particular analyzing the way mathematical models for physical processes are influenced by and reveal further structure in iterative cases. This conference will bring together top international researchers in geometry and analysis to make further progress in the study of singular and non-compact manifolds using the most recently developed, powerful results in microlocal analysis and other fields.


The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). The research station in Oaxaca is funded by CONACYT