Grant: GC22-08633J
from 01/01/2022
to 31/12/2024
Grantor: Czech Science Foundation/National Research Foundation Korea
Qualitative Theory of the MHD and Related Equations
Objectives:
The project is oriented to studies of a wide class of equations of magneto-hydro-dynamics (MHD), modeling flows of electrically conductive fluids (both incompressible and compressible, without or with Hall's effect). The presence of magnetic field evokes many new challenging questions. In addition to existing mathematical models, our aims include formulation and studies of new models, which e.g. concern (a) physically and mathematically relevant boundary conditions for the magnetic field, (b) micropolar fluids and (c) motion of a body with a cavity filled in by an electrically conductive fluid. Considered mathematical problems involve questions of regularity, analysis and consistent theory for the models from items (a)-(c), problems with artificial "outflow"' boundary conditions and problems in domains with moving boundaries. Our project assumes a close cooperation with colleagues from the Yonsei University in Seoul and other respected researchers and active participation of post-docs and doctoral students on both sides.
Grant: GF22-07833K
from 01/01/2022
to 31/12/2024
Grantor: Czech Science Foundation / National Science Centre Poland
Homogeneity and Genericity of Metric Structures - Groups, Dynamical Systems, Banach Spaces and C*-Algebras
Objectives:
The project is devoted to all aspects of homogeneity and genericity in metric groups, functional analysis and dynamical systems. Various Fraïssé constructions will be investigated, e.g. for the Jacelon-Razak and Jiang-Su C*-algebras, as well as a non-compact counterpart of the Poulsen simplex. We will also construct and analyze Polish spaces of bounded linear operators, actions of finitely generated groups on compact metric spaces and representations of countable groups. Absolute homogeneity for metric spaces and dilation groups will be examined and we will also generalize Pontryagin duality to the framework of metric groups. Moreover, we will extend the theory of Katětov functors to metric Fraïssé classes and study its relations to universality of automorphism groups of Fraïssé limits and Borel reducibility of isomorphism relations. The theory of Borel reduciblity will be also investigated in the more general framework of pseudometrics.