Jerry Goldstein : Short course in Semigroups of Operators
Abstract : The first two lectures will be on strongly continuous semigroups of linear operators. The third lecture will be on nonlinear semigroups. For each subject I will review many of the main theorems in the theory. The linear lectures will emphasize modern boundary conditions. By this is meant dynamic boundary conditions, including those of acoustic and Wentzell type. The linear lectures will treat heat, wave, telegraph and other equations, and emphasize wellposedness and sharp asymptotic results. The nonlinear examples include single conservation laws, the Hamilton-Jacobi equation, the filtration equation, and an application of the porous medium equation to ground state electron density theory in quantum mechanics. The emphasis will be that a small number of closely related ideas will enable us to give a unified approach, via functional analysis, to rather diverse problems in applied mathematics arising in science.
Ryszard Rudnicki : Stochastic operators and semigroups and their applications in physics and biology
Abstract : We introduce stochastic operators and semigroups and present results concerning generators of stochastic semigroups and methods of constructing stochastic semigroups by perturbation theorems. We give some examples of stochastic operators and semigroups : Fobenius-Perron operators, diffusion semigroups, flows semigroups with jumps and switching and semigroups related to hybrid systems. Then we present some results concerning their long-time behaviour : asymptotic stability, sweeping, completely mixing and convergence to self-similar solutions. The results will be applied to study ergodic properties of dynamical systems, an integral operator appearing in the theory of cell cycle, and semigroups related to diffusion processes, birth-death processes, genome evolution, gene expression and physiologically structured models. We also present some elementary nonlinear stochastic operators and semigroups from genetics, coagulation-fragmentation processes and phenotype-structured populations.