Grants


Grant: 1905271Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation

Groups and their actions, operator algebras, and descriptive set theory


Objectives:
The goal of the project is to to find new connections and prove some interesting conjectures on the boundaries of three, currently very attractive mathematical disciplines  geometric group theory, operator algebras, and descriptive set theory.



Grant: 1907129Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation

Linearanalysis techniques in operator algebras and vice versa


Objectives:
The theory of normed spaces and their operators is at the core of the linear analysis. The idea of employing algebras of operators acting on infinitedimensional spaces originated in quantum physics and was further successfully integrated with the theory of unitary representations of locally compact groups. An operator algebra, which is also a normed space, carries intrinsically a much richer structure and therefore operator algebras are not usually viewed from the perspective of linear analysis. Nevertheless, the transfer of ideas from Banach spaces can be very fruitful as illustrated by the notion of nuclearity that was recognised as an approximation property with respect to a certain class of finiterank operators. On the other hand, operator Ktheory
was almost unknown in Banach space theory until it was spectacularly applied in the seminal work of Gowers and Maurey. Consequently, there is tremendous potential in transferring ideas between these two areas. The very aim of the project is a closer reconciliation of these two theories by interchanging ideas between them.



Grant: 1905497S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation

Complexity of mathematical proofs and structures


Objectives:
We study weak logical systems, guided by the question: what is the weakest natural theory in which we can prove a mathematical statement? This question is often fundamentally complexity theoretic in nature, as proofs in such weak systems can be associated with feasible computations. We will study this and related topics in a range of settings, including bounded arithmetic, model theory, algebraic complexity, bounded set theory, and nonclassical logics.



Grant: 1904243S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation

Partial differential equations in mechanics and thermodynamics of fluids


Objectives:
Partial differential equations in mechanics and thermodynamics of fluids are a mathematical tool which models the time evolution of basic physical quantities. The goal of this project is to study these systems of partial differential equations from the point of view of mathematical and numerical analysis and to compare these results with the properties of their numerical solutions. We will mostly deal with solvability of the problems (existence of solutions for different formulations, possibly their uniqueness), qualitative properties of the solutions, analysis of the adequate numerical methods and numerical solutions of these problems. The proposal of the project assumes a close collaboration of specialists from different branches. Such a collaboration stimulates positively the developement of all participating mathematical disciplines.



Grant: 1909659S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation

Exact solutions of gravity theories: black holes, radiative spacetimes and electromagnetic fields


Objectives:
Exact solutions to Einstein gravity play a crucial role in the understanding of many mathematical and physical aspects of the theory. In recent years, for several theoretical reasons, various modifications of Einstein gravity and their solutions have been studied. Due to the complexity of resulting field equations, very few exact solutions of such theories are known. We plan to construct and study exact solutions to Einstein gravity and various higherorder
gravities, such as quadratic gravity and Lovelock gravity, with a strong focus on black hole solutions, radiative spacetimes, and pform fields. We also intend to study generic properties of certain classes of spacetimes, such as asymptotically flat spacetimes. When appropriate, we will benefit from employing mathematical methods, such as algebraic classification and a generalized GHP formalism developed in part by our team.



Grant: 1927871X
from 01/01/2019
to 31/12/2023
Grantor: Czech Science Foundation

Efficient approximation algorithms and circuit complexity


Objectives:
The goal of this project is to understand the role of approximation in finegrained and parameterized complexity and create solid foundations for these areas by developing lower bound techniques capable of addressing the key unproven assumptions underpinning these areas. We will focus on several central problems: Edit Distance, Integer Programming, Satisfiability and study their approximation and parameterized algorithms with the aim of
designing the best possible algorithms. Additionally we will focus on several methods of proving complexity lower bounds.



Grant: Praemium Academiae
from 01/01/2019
to 31/12/2024
Grantor: Czech Academy of Sciences

Operadic categories and their applications


Objectives:
Operads are objects formalizing compositionsof operations with several inputs. They were invented to describe homotopy invariant structures on topological spaces. Later it turned out that they can be used as well for the study of sundry structures in geometry, algebra and mathematical physics.
The research supported by Praemium Academie is aimed at formulating a unifying paradigm for very general operadic structures, and using this emerging systematic approach for proving various results in algebra, geometry and mathematical physics. Our team is international from the very beginning, as emphasized by the planned positions for postdocs and foreign specialists.



Grant: MSM100191801
from 01/01/2018
to 31/12/2018
Grantor: Czech Academy of Sciences

Structure and localizations of the derived category of a commutative ring


Objectives:
The derived category of an abelian category is a meeting point between algebra and homotopical methods originally designed for topology. This project aims to extend the current knowledge about derived categories in the following directions. 1) Extend the recent description of equivalence classes of (big) tilting modules over an arbitrary commutative ring to the setting of silting complexes, or even compactly generated tstructures. 2) Describe the cotilting modules, or even the cosilting modules, over arbitrary Pruefer domain. Study the derived equivalences these modules induce. 3) Search for a purely tensor triangulated category construction of a noncompactly generated smashing localization based on an algebraic construction by Bazzoni and Šťovíček used for the module category of a valuation domain. The possible application could help solve the long open Telescope Conjecture for the homotopy category of spectra. The three topics are are strongly connected to each other and are expected to be studied simultaneously.



Grant: GJ1801472Y
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Graph limits and inhomogeneous random graphs


Objectives:
Theories of dense and sparse graph limits are one of the most important recent tools of discrete mathematics. Their emergence and development have led to many breakthroughs on old problems in extremal graph theory and random graph theory, and especially have linked discrete mathematics to areas such as probability theory, functional analysis or group theory in a profound way. Recognitions related to the development of the field include the 2012 Fulkerson Prize, the 2013 CoxeterJames Prize, and the 2013 David P. Robbins Prize.
The project will study the theories of dense a sparse graph limits as well as the related theory of inhomogeneous random graphs. Specific problems in the area of inhomogeneous random graphs include questions on key graph parameters such as the chromatic number or the independence number. In the theory of sparse graph limits our main goal is to extend our understanding of localglobal convergence. A further goal is to create a comprehensive theory of limits of subgraphs of hypercubes.



Grant: GC1801953J
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Geometric methods in statistical learning theory and applications


Objectives:
Statistical learning theory is mathematical foundation of machine learning  the currently fastest growing branch of computer sciences and artificial intelligence. Central objects of statistical learning theory are statistical models. The project is based on our results obtained jointly with N. Ay and J. Jost and covers the following topics: geometry of efficient estimations, geometry of natural gradient flows and properties of KullbackLeibler divergence on statistical models, in particular graphical models, hidden Markov models, Boltzmann machine, multilayer perceptrons and infinite dimensional exponential models.



Grant: GA1807776S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Higher structures in algebra, geometry and mathematical physics


Objectives:
It has been gradually realized that various deep problems in algebra, geometry and mathematical physics, particularly in string theory, involve previously unknown higher structures. Over the years, this originally esoteric concept has become widely recognized, with parallel breakthroughs in the foundations of derived algebraic geometry and topology, category theory, representation theory and other seemingly unrelated fields. Our project aims to increase
the understanding of the topics mentioned above, by combining the expertise of the team members in different but tightly interlaced areas of mathematics and mathematical physics. More specifically, the project aims at topics such as the terminality conjecture for spaces relevant for string field theory, higher Lie algebras and gauge theory, Mbranes, PenroseWard transform, AdamsNovikov spectral sequence, Riemann surfaces, and related issues. The common background of these themes are operads, higher category theory and homological algebra, combined with the standard methods of differential and algebraic geometry.



Grant: GA1805974S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Oscillations and concentrations versus stability in the equations of mathematical fluid dynamics


Objectives:
The project focuses on questions of possible singularities in the equations of mathematical fluid dynamics and their adequate description by means of weak and measure valued solutions. The main topics include:(i) dissipative solutions, (ii) admissibility criteria, (iii) equations with stochastic terms, (iv) applications in the numerical analysis.
The goal is to develop a consistent mathematical theory of fluids in motion in the framework of weak and measure valued solutions, developing the concept of dissipative solution, obtaining new admissibility criteria, solving problems with stochastic terms, analyzing the underlying numerical schemes.



Grant: GA1809628S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Advanced flowfield analysis


Objectives:
The research deals with an advanced flowfield analysis, particularly for transitional and turbulent flows. Local vortex identification and more general classification of flow regions based on the velocity gradient are investigated. The velocity gradient is usually decomposed in strainrate tensor and vorticity tensor, consequently the identification and classification criteria are determined by the inner velocitygradient configuration. The impact of configuration is usually significant though hidden, and will be, including representative data sets, analyzed and described. Largescale 3D numerical experiments based on the solution of the NavierStokes equations (NSE) will be performed with the help of new effective methods (parallel domain decomposition (DD) with adaptive mesh refinement) while these methods will be further developed. Some suitable qualitative properties of the solutions to the NSE will be studied and described in detail, focusing on the regularity criteria with only one or two velocity components and one or several velocitygradient entries.



Grant: GA1800496S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Singular spaces from special holonomy and foliations


Objectives:
Spaces with singularities naturally appear in differential geometry and mathematical physics. The project is based on the development of our results covering the following topics: the holonomy groups of cones over pseudoRiemannian manifolds, their relations to Lorentzian manifold admitting imaginary Killing spinors, Sasakian and other special geometries, constructions of new examples of complete G2 and Spin(7)holonomy metrics and study of their deformations, constructions of invariant KaehlerEinstein and EinsteinSasakian metrics on cohomogeneity one manifolds. We also plan to investigate geometry of the leaf space of foliations; in particular, to develop Losik's approach to these spaces and to study their characteristic classes.



Grant: GA1800580S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation

Function Spaces and Approximation


Objectives:
We shall study important properties of various function spaces and operators acting on them. We shall focus on optimality of the obtained results. We shall develop new sampling algorithms that will have important applications in theory of approximation. We shall concentrate on applications of results obtained in other fields of mathematics.



Grant: CZ.02.2.69/0.0/0.0/16_018
from 01/09/2017
to 30/09/2022
Grantor: Ministerstvo školství, mládeže a tělovýchovy České republiky

Doktorská škola pro vzdělávání v oblasti matematických metod a nástrojů v HPC


Hlavním cílem projektu je ustavení Doktorské školy pro vzdělávání v oblasti matematických metod a nástrojů v HPC integrující doktorská studia MFF UK v Praze, MÚ AV ČR a NSC IT4Innovations VŠBTUO a navazující na jejich širší spolupráci v oblasti výzkumné. Součástí projektu je modernizace a internacionalizace jednoho doktorských programů školy (Výpočetní vědy, VŠBTUO) a vytvoření programu double degree. Projekt navazuje na související projekt ERDF Vzdělávací tréninkové centrum IT4Innovations.



Grant: Neuron Impuls 24/2016
from 01/01/2017
to 31/12/2019
Grantor: Neuron Fund for Support of Science

Guaranteed bounds of eigenvalues and eigenfunctions of differential operators


We will propose new a posteriori error estimates for eigenvalue problems of symmetric elliptic partial differential operators. We will prove their reliability and local efficiency. We will use them in the adaptive finite element method for reliable error estimates of the size of the error in eigenvalues and eigenfunctions.



Grant: Neuron Impuls Junior
from 01/01/2017
to 30/06/2018
Grantor: Neuron Fund for Support of Science

Mathematical analysis of hyperbolic conservation laws


Objectives:
The goal of the project is to deepen present knowledge concerning existence of weak solutions for hyperbolic systems of partial differential equations arising from conservation laws in multiple space dimensions, study of uniqueness and nonuniqueness of entropic weak solutions, analysis of admissibility criteria (maximization of entropy production or inviscid limit) or even designing other suitable admissibility criteria.
A convenient test problem for admissibility criteria is the so called Riemann problem, i.e. problem with initial data consisting of a jump discontinuity separating two constant states. As such initial data are onedimensional, one can use standard theory to find a onedimensional selfsimilar solution which is unique in the class of BV functions. In a broader class of functions (in particular in the class of essentialy bounded functions), other weak solutions may exist as is in the case of compressible Euler equations. In the framework of this project I will study also Riemann problems for various hyperbolic systems concerning uniqueness or nonuniqueness of weak solutions in the class of essentialy bounded functions and in the case of nonuniqueness, admissibility criteria will be tested.



Grant: GA1700941S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Topological and geometrical properties of Banach spaces and operator algebras II


Objectives:
We wish to investigate the structure of Banach spaces, C*algebras and Jordan algebras and their relationship. Main topics include quantitative approach to Banach spaces, various methods of separable reduction, decompositions of Banach spaces to smaller subspaces, integral representation of affine Baire functions, descriptive properties of weak topologies, small sets in Banach spaces and Polish groups, universal spaces in various categories of Banach spaces, operators and their numerical ranges, structure of abelian subalgebras of a C*algebra, of associative subalgebras of a Jordan algebra and related structures, different types of order in operator algebras, representation of morphisms on various substructures of operator algebras, Bell's inequalities and quantum correlations. We wish to focus especially on those problems where the mentioned areas intersect each other and by solving them to contribute to clarification of connections among various areas of functional analysis.



Grant: GA1701747S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Theory and numerical analysis of coupled problems in fluid dynamics


Objectives:
The project is focused on several important fields of today's rapidly developing mathematical fluid mechanics. The aim is to derive a series of results, from new regularity criteria, stability and robustness analysis of solutions, up to the low Mach and high Reylolds limits in a compressible fluid interacting with a solid structure. Beside the qualitative analysis of flow problems, a part of the project is the development and analysis of new, accurate and robust numerical methods for the solution of important and topical models of fluid dynamics. The attention will be paid to the development and analysis of high order methods for the solution of nonstationary nonlinear partial differential equations and compressible flow, based on the discontinuous Galerkin method. Particularly we have hpversions in mind. These methods will be applied to the numerical solution of fluidstructure interaction and multiphase flow. Another subject is the study of flow model with slip boundary conditions.



Grant: GJ1701694Y
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Mathematical analysis of partial differential equations describing inviscid flows


Objectives:
The investigator and his team will develop the theory of Camillo De Lellis and Lászlo Székelyhidi which allows to prove surprising results for incompressible Euler equations in multiple space dimensions. They will focus mainly on development and applications of the theory in the field of compressible flow, both in the isentropic case and in the case of full system of partial differential equations. They will study and propose criteria to choose "physical" solutions among the infinitely many weak solutions of appropriate systems of equations. The investigator and his team will maintain already established scientific cooperations (De Lellis, Chiodaroli, Wiedemann) and establish new ones. The results of the project will be presented on international conferences and will be published as articles in impacted journals.



Grant: GA1727844S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Generic objects


Objectives:
An object can be called generic if it occurs typically, in the sense that its copies can be found in every residual set in an appropriate space of objects. The aim of the project is to study generic objects appearing in several areas of mathematics, finding new tools for constructing and detecting such objects, and exploring their combinatorial structure. Wellknown examples of generic objects in model theory are Fraisse limits. Generic structures occur also naturally in topology (e.g. the Cantor set) and various areas of mathematical analysis (e.g. generic Banach spaces). Cohen's settheoretic forcing offers a strong tool of constructing generic objects, usually needed for specific purposes and not related to Fraisse limits. One of our objectives is to explore links between the settheoretic method of forcing and modeltheoretic methods for constructing universal homogeneous structures.
