Felix Otto (MPI-MIS, Leipzig): Stochastic homogenization and large-scale regularity
In this mini-course, I will introduce the concept of large-scale
regularity in case of a linear elliptic equation (or system) with
heterogeneous coefficients. It is based on a smallness (on average) of the
potentials of the harmonic coordinates, and proceeds via an intrinsic
Campanato iteration. I will then apply this to the case of a random
heterogeneous coefficient field, sampled from a stationary and ergodic
ensemble. I will try to be self-contained and closely follow Theorem 1 and
Lemma 1 in Gloria, Neukamm, and Otto ``A regularity theory for random
elliptic operators'', Milan J Math 2020.
C. Schiebold(Mid Sweden University Sweden): INTEGRABLE SYSTEMS - METHODS OF MATHEMATICAL PHYSICS IN INTERACTION
Integrable systems in infinite dimension refer to an area in mathematical physics which is devoted to the study of a certain group of partial differential equations, many of them soliton equations like the classic Korteweg-de Vries equation and the nonlinear Schrodinger equation. One of the striking features is the existence of solutions with particle character, called solitons, remarkable in view of nonlinearity of the governing equations. Methodologically, integrable systems are a meeting point (melting pan) for methods from very diverse parts of mathematics. The main idea of this mini course is to highlight interactions of some of the main approaches to integrable systems, the inverse scattering method and an operator theoretic approach in the first place, and symmetry methods like Backlund transformations, recursion operators and hierarchies to a minor extent.
Throughout we will emphasise the recent topic ofnon-commutative integrable systems, like vector- and matrix soliton equations, where many fundamental questions are still open. Notably, the construction of solutions is not interesting only under the mathematical viewpoint, but also under the physical one. Indeed, very important applications of soliton equations are in nonlinear optics, for instance.