The topic Singular Stochastic Partial Differential Equations (singular SPDE) has rapidly grown to be an active research area at the interface of Stochastic Analysis and PDEs on one hand, and Mathematical Physics on the other hand. During this decade we have witnessed a series of tremendous breakthroughs in the solution theories of SPDEs, universality problems, large-scale asymptotic behaviors of solutions, and foundational relations with quantum field theories and geometry. Many long-standing problems have been resolved via newly developed methods – notably the theories of regularity structures and paracontrolled distributions – and deep connections with other fields are quickly emerging. These remarkable developments have prompted new directions of research and brought into focus the challenges of solving many remaining long-standing open problems such as global solution methods, critical dimension problems, and so on. Progress has been made independently and at more or less similar times by various research groups and individuals for a variety of different scientific reasons. Much of the progress on the theory on singular SPDE, and the invention of the theory itself, has occured in Europe. With these remarks in mind, it is a natural time to convene a large-scale semester program in order to accomplish: 1. Educate a new generation of mathematicians in these new methods and the associated research directions, bring these researchers to the central problems that lie on the cutting-edge frontier of research, and equally importantly promote further US-specific activities in the timely research area of singular SPDEs; 2. Bring together top researchers from various scientific communities, such as PDEs, probability, dynamical systems, geometry, and mathematical physics, in order to facilitate fruitful scientific interactions and cross-fertilization of problems and methods; 3. Propagate the theories, methods, results, and their applications well beyond the communities that are directly involved in the current development of singular SPDEs and closely related areas.
Keywords and Mathematics Subject Classification (MSC)
stochastic partial differential equations
quantum field theory
Statistical field theory
Theory of regularity structures