Grants


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: 1801472Y
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: 1801953J
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: 1809628S
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: 1800496S
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: 1800580S
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: 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.



Grant: GA1603230S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)

Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution


Objectives:
Mathematical and numerical analysis and numerical solution of fluid flows belong to the most often studied problems of the theory of partial differential equations and their numerical solution. During the last decades, a big progress has been achieved in these fields which enables us to study models of complex fluids including the possibility to consider their dependence on temperature. This project is focused on the study of such models of fluid thermodynamics and mechanics with the aim to extend the knowledge in the field of the theoretical analysis of the corresponding systems of partial differential equations and numerical analysis of the methods for their solution. Computational simulations using specific numerical methods will be performed to support the analytical results concerning the wellposedness of the model problems and qualitative properties of their solutions. The proposed projects assumes a tight collaboration of specialists in these fields which is an important prerequisite for further development of mathematical and computational fluid thermodynamics.



Grant: GA1607378S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)

Nonlinear analysis in Banach spaces


Objectives:
We plan to investigate problems concerning uniformly continuous and Lipschitz mappings between Banach spaces and their possible applications in other areas of mathematics, such as theoretical computer science, differential equations etc. This project is devoted to the following aspects of the subject:
a) Uniformly continuous and coarse mappings
b) Lipschitz isomorphism
c) Lipschitz free spaces
d) Linear and descriptive properties



Grant: 1807776S
from 01/01/2016
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: 1805974S
from 01/01/2016
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.
