Grant: GA23-05148S
from 01/01/2023
to 31/12/2025
Grantor: Czech Science Foundation
Homological and structural theory in geometric contexts
Objectives:
The project is focused on the study of objects originating in algebraic geometry and representation theory and their homological and structural properties. We are concerned with properties of both classical and recently discovered non-conventional derived categories and conditions for their equivalence as well as with homological dualities in semiinfinite algebraic geometry (in the form of formal and ind-schemes), homological and geometric approach to noncommutative algebras (coming for instance from variants of Fukaya categories) and a structural theory of the corresponding representations or (co)sheaves of representations. The project also aims at enhancing the long-term and fruitful collaboration with several European algebra centers and contributing to graduate education at MFF UK in new areas of contemporary mathematics.
Grant: GA23-06159S
from 01/01/2023
to 31/12/2025
Grantor: Czech Science Foundation
Vortical structures: advanced identification and efficient numerical simulation
Objectives:
Advanced analysis and description of vortical structures aiming at:
- development of vortex-identification methods,
- spatio-temporal analysis,
- efficient numerical methods based on parallel immersed boundary adaptive FEM,
- high-resolution flow simulations,
- experimental measurements.
Grant: GA23-04683S
from 01/01/2023
to 31/12/2025
Grantor: Czech Science Foundation
Compactness in set theory, with applications to algebra and graph theory
Objectives:
The goal of the project is to produce significant advances in the study of compactness in set theory and the application of set theoretic methods to questions from graph theory and homological algebra, and to foster collaboration, both locally and internationally, to achieve these aims.
Grant: GA23-04825S
from 01/01/2023
to 31/12/2025
Grantor: Czech Science Foundation
Logic and unsatisfiability
Objectives:
We aim to show that increasing logical depth can decrease proof size; to improve logical characterizations of combinatorial proof systems and search problems, and vice versa; to show lower bounds on algebraic proof systems and systems from SAT solving; to develop mathematics over very weak axioms.
Grant: GA23-04720S
from 01/01/2023
to 31/12/2025
Grantor: Grant Agency of the Czech Republic
Fine properties of functions, operators and function spaces
The main goal of the project will be solving open problems concerning fine properties of functions belonging to specific function spaces and their applications in the theory of approximations. We will put emphasis on finding optimal (not improvable) structures within certain categories.
Grant: L100192251
from 01/07/2022
to 30/06/2024
Grantor: Czech Academy of Sciences
On independence of combinatorial properties of ultrafilters on natural numbers
The main goal of the project is to establish under what conditions ultrafilters with particular combinatorial properties may or may not exist.
Recently, Dr. Cancino-Manríquez proved that consistently there is no I-ultrafilter for any ideal I with Borel complexity F sigma. As a result, J. Brendle has asked whether it is possible to generalize this theorem to get the continuum size bigger than omega 2, which is the second uncountable cardinal. Dr. Cancino-Manríquez has a partial advance in this question pointing to a positive answer: the model constructed by forcing with the Rational Perfect set forcing parametrized by a lower semicontinuous submeasure and then with the side by side product of the Silver’s forcing. This work will be continued within the framework of this project.
The second objective is to answer an open question about the existence of a model where there is no q-point but there is a rapid ultrafilter.
The third objective will be to solve completely the Isbell’s problem: whether it is consistent that all ultrafilters have maximal cofinal type. What remains open is what happens when there are no p-points.
The fourth objective addresses proposing the construction of a model where there is no q-point nor p-point. This has been a long standing question, and possibly the difficulty finds in the lack of suitable forcing techniques to produce appropriate models.
Problémy interakce tekutiny se strukturou: matematická analýza a aplikace
Objectives:
Hlavní náplní projektu Š. Nečasové je výzkum zaměřený na matematickou analýzu modelů mechaniky tekutin (primárně stlačitelných) a dynamiky pevných látek, včetně jejich interakce na společné hranici. Problémy tohoto typu se přirozeně vyskytují v aplikacích v průmyslu, biomedicíně (proudění krve) a ve vědách o životním prostředí (oceánografie, meteorologie).
Grant: L100192151
from 01/01/2022
to 31/12/2023
Grantor: Czech Academy of Sciences
Unilaterally constrained evolution
The project is supported by the "Programme to support prospective human resources – post Ph.D. candidates" funded by the Czech Academy of Sciences.
Objectives:
The main objective of the project is to establish a new theory of dynamics of sweeping processes with applications to natural sciences and network models. The directions (research activities) for the development of such a theory are:
1. A problem of elastoplastic evolution of mechanical networks: the case of large deformations for 2D and 3D networks and quasistatic solution as a limit of a dynamic problem.
2. Applications of non-convex sweeping process to piezoelectricity and magnetostriction.
3. Finite-time stability of the solutions of a sweeping process.
Grant: GA22-01591S
from 01/01/2022
to 31/12/2024
Grantor: Czech Science Foundation
Mathematical theory and numerical analysis for equations of viscous newtonian compressible fluids
Objectives:
Equations of compressible viscous fluids are important models in many applications. We will study the corresponding systems of partial differential equations from several points of view: existence theory and qualitative properties of solutions for different choices of boundary conditions (including open system), different types of domains (in particular varying in time), different types of solutions (weak, strong, dissipative) and different simplified models (in
particular, compressible primitive equations) as well as from the point of view of numerical mathematics (construction of benchmarks, numerical analysis of some methods, comparision of different numerical methods). The proposal of the project is based on a close collaboration of specialists from different mathematical disciplines.