
Oscillations and concentrations versus stability in the equations of mathematical fluid dynamics (GA1805974S)
from 01/01/2018
to 31/12/2020 main investigator

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.



Singular spaces from special holonomy and foliations (GA1800496S)
from 01/01/2018
to 31/12/2020 main investigator

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.



Higher structures in algebra, geometry and mathematical physics (GA1807776S)
from 01/01/2018
to 31/12/2020 main investigator

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.



Advanced flowfield analysis (GA1809628S)
from 01/01/2018
to 31/12/2020 main investigator

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.



Geometric methods in statistical learning theory and applications (GC1801953J)
from 01/01/2018
to 31/12/2020 main investigator

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.



Graph limits and inhomogeneous random graphs (GJ1801472Y)
from 01/01/2018
to 31/12/2020 main investigator

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.



Function Spaces and Approximation (1800580S)
from 01/01/2018
to 31/12/2020 main investigator

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.



Doktorská škola pro vzdělávání v oblasti matematických metod a nástrojů v HPC (CZ.02.2.69/0.0/0.0/16_018)
from 01/09/2017
to 30/09/2022 main investigator

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.



Filters, Ultrafilters and Connections with Forcing (GF1733849L)
from 01/01/2017
to 31/12/2019 main investigator

Objectives:
The project falls within the scope of Set Theory & Foundations of Mathematics, specifically Combinatorial Set Theory and Forcing. We will investigate new combinatorial and forcing methods for constructing ultrafilters with special properties in different models of Set Theory.It will use these methods to answer independence questions about ultrafilters (no Ppoints with a large continuum or small character filters), structural questions about filters (which ultrafilters/filters contain towers) and questions about the related cardinal invariants (independence number, free sequence number). It is known that current methods cannot answer some of these questions so the project will have to come up with novel ideas. The methods used will include forcing iterations, diamondlike constructions, preservation theorems and methods from descriptive set theory.



Mathematical analysis of partial differential equations describing inviscid flows (GJ1701694Y)
from 01/01/2017
to 31/12/2019 main investigator

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.



Dynamics of mutlicomponent fluids (7AMB17FR053)
from 01/01/2017
to 31/12/2018 main investigator

The goal of the project is studying qualitative properties of a particular class of the socalled energetically weak solutions to complex system of the NavierStokesFourier type as well as Coupling of these systems with the phase transition equations of the CahnHilliard or AllenCahn type. We plan to investigate these problems also in unbounded physical domains in appropriate classes of uniformly bounded functions.
The main goal is obtaining new results in the following directions:
• applications of the relative entropy methods and the consequences concerning stability of the socalled dissipative solutions
• singular limits, in particular the sharp interface limits with rigorous mathematical justification
• longtime dynamics, with a particular emphasis on the existence of bounded absorbing sets, asymptotic compactness of greajectories and the relevant questions concerning the attractors and their structure and complexity



Theory and numerical analysis of coupled problems in fluid dynamics (GA1701747S)
from 01/01/2017
to 31/12/2019 main investigator

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.



Topological and geometrical properties of Banach spaces and operator algebras II (GA1700941S)
from 01/01/2017
to 31/12/2019 main investigator

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.



Flow of viscous fluid in time dependent domain (7AMB16PL060)
from 01/01/2017
to 31/12/2018 main investigator

Objectives:
G1. Global existence of weak solution of full system in timedependent domain.
Goals: Our goals is the study of the global existence of a weak solution in the case of a bounded or an exterior domains for the Navier or Dirichlet type of boundary conditions. We will use the method introduced in E. Feireisl: Dynamics of viscous compressible fluids, 2004 for fixed domain and we will also apply the penalization method.
G2: Relative entropy inequality.
Goals: We will focus on the derivation of the relative entropy inequality in the case of timedependent domain. We will use the results from fixed domain introduced in E. Feireisl and A. Novotný: Weakstrong uniqueness for the full NavierStokesFourier system, 2012.
G3. Singular limits.
Goals We shall study the singular limits in the regime of low Mach number. From this follows that the limit system (target system) is the system of incompressible flow in the timedependent domain. We will use the results from E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, J. Stebel: Incompressible limits of fluids excited by moving boundaries, 2014, where barotropic case was studied, and also the results for fixed domains.
G4: Weak solution of viscous flow around rotating rigid body and his asymptotic behavior.
Goals: We will focus on the problem of existence of weak solution in weighted Lorentz spaces to get the asymptotic behavior of flow around a body. Because of the lack of the regularity in Lp spaces it is necessary to go into more complicated structure of Lorentz spaces, where the integrability of convective term is satisfied. Moreover, we would like to study the relative entropy inequality in the case of compressible flow around rotating body.



Mathematical analysis of hyperbolic conservation laws (Neuron Impuls Junior)
from 01/01/2017
to 30/06/2018 main investigator

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.



Generic objects (GA1727844S)
from 01/01/2017
to 31/12/2019 main investigator

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.



Phenomenological modeling of polymeric smart foams with behavior controlled by the magnetic field (CNR1608)
from 05/04/2016
to 31/12/2018 main investigator

The aim of the collaboration is to develop and test experimentally a reliable mathematical model for magnetic foams taking into account hysteresis effects in the material, with the goal to provide, as output, parameters needed to obtain optimal performance in both "passive" (structural reinforcement and gradient properties) and "active" modes (actuation and control through magnetic field) of lightweight porous polymeric structures reinforced with aligned magnetic particles. The average distribution, size and shape of particles, their magnetic characteristics, and mechanical properties of the foam will be considered as processing parameters. Homogenization techniques will be adopted in order to derive simple constitutive relationships for low computational requirements. The developed constitutive model will be phenomenological, while the governing equations will be based on the magnetic and mechanical balance laws. The development and validation of the model will be performed by comparing predictions with experiments.



Nonlinear analysis in Banach spaces (GA1607378S)
from 01/01/2016
to 31/12/2018 main investigator

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



Performance and thermodynamic aspects of incrementally nonlinear constitutive equations of the rate type (7AMB16AT035)
from 01/01/2016
to 31/12/2017 main investigator

Objectives:
Although first models of hypoplasticity were proposed some decades ago, their properties have not yet been sufficiently understood, neither from the viewpoint of general applicability to large classes of technical problems, nor from the viewpoint of their mathematical wellposedness and numerical complexity. In special cases, however, there is evidence of their strong modeling potential for describing irreversible deformations, in particular in the case of granular or porous solids. The goal of this project is to exploit the complementary knowledge and experience of its participants, and classify the individual models within a larger thermomechanical context. The models will be investigated both analytically and numerically, and the results will be compared with known experimental data published in the literature, which will enable us to decide about their applicability for solving practical problems.



Logic and Topology in Banach spaces (GF1634860L)
from 01/01/2016
to 31/12/2018 main investigator

Objectives:
The project is devoted to the study of topological and geometric properties of Banach spaces and their duals, aiming at a better understanding of their structure. Properties of the weak topology often imply important geometric properties of the Banach space in question. On the other hand, geometric properties of the Banach space often give information about its weak topology. Similar statements are true for duals of Banach spaces with the weakstar topology. We are going to explore this interplay in detail. The main project goals are:
1. Developing new tools for constructing and studying Banach spaces, using techniques from set theory and category theory.
2. Exploring different types of networks and related concepts in weak topologies, determining connections with renorming theory.
Results of Goal 1 will lead to new examples, settling some of the problems concerning interplay between geometric and topological properties of nonseparable Banach spaces. Goal 2 will lead to a better understanding of the weak topology and its relations to the geometric structure of a Banach space.



Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution (GA1603230S)
from 01/01/2016
to 31/12/2018 main investigator

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.



Topics in set theory: Traces of large cardinals, variants of Hechler's theorem, and ultrafilters on countable sets (7AMB15AT035)
from 01/01/2015
to 31/12/2016 main investigator

Objectives:
The goal of the project is to expand the collaboration between specialists working in different areas of Set Theory. We shall, in particular, try to exploit the complementary expertise of the Czech and Austrian teams while solving problems on the borders of traditional areas of research (large cardinals, forcing, infinitary combinatorics, topology) and encourage mutual crossfertilization and inspiration sharing. The main goal is to encourage new members of both Czech and Austrian research teams to interact in join international research, to explore new possibilities for scientific collaboration, and to introduce international contacts enabling applications for funding with more ambitious join projects in future years.



Analysis of mathematical models of multifunctional materials with hysteresis (GA1512227S)
from 01/01/2015
to 31/12/2017 main investigator

Objectives:
The project topic is mathematical modeling, analysis, and numerical simulations of processes taking place in multifunctional materials with hysteresis. The results will include (1) a rigorous derivation of systems of ordinary and partial differential equations based on physical principles and experimentally verified constitutive relations, (2) proofs of existence, and possibly also uniqueness and stability of solutions to the equations, and (3) their numerical approximation including error bounds. The main applications will involve piezoelectric and magnetostrictive materials used as sensors, actuators, and energy harvestors, as well as thermoelastoplastic materials subject to material fatigue. The presence of hysteresis makes all these steps challenging, also because hysteresis nonlinearities are nondifferentiable, which creates difficulties both in the analysis and in the numerics. New algorithms will have to be developed to treat the problems in maximal complexity.



The continuum, forcing and large cardinals (GF1534700L)
from 01/01/2015
to 31/12/2017 main investigator

Objectives:
The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics:
(1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections.
(2) Second subtopic is called"Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology.
(3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on nondefinable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.



Pseudorandomness and explicit constructions in discrete mathematics (PaECiDM(628974))
from 01/09/2014
to 31/08/2016 main investigator

Programme type: PEOPLE, MARIE CURIE ACTIONS, INTRAEUROPEAN FELLOWSHIPS (IEF)
Objectives:
The project concerns research on the frontier between discrete mathematics and theoretical computer science. Discrete mathematics is an established mathematical discipline and is playing an increasing role in various fields of mathematics. Many reallife problems can be formulated using the language of discrete mathematics. Deep mathematics is hidden behind practical problems such as devising optimal schedules, or efficient routing of data packets through the internet. These problems also suggest that besides problems typical to pure mathematics (such as existence of a solution) one often seeks their efficient algorithmic counterparts (algorithm design).
The project goals are the following:
To explore constructions of specialized families of expanders, in particular of monotone expanders
To study the applications of dimension expanders (and related) in the area of explicit Ramsey graphs
To study in detail the new concept of partition expanders
To study the combinatorial construction of Mendel and Naor can yield explicit constructions of partition expanders, and to look for applications, especially in derandomization and communication complexity
To understand the role of extractors in theoretical computer science
To understand, extend, and simplify constructions of Ramsey graph of Barak et al
To find a purely combinatorial construction of Ramsey graphs



Methods of function theory and Banach algebras in operator theory V. (GA1407880S)
from 01/01/2014
to 31/12/2016 main investigator

Objectives:
The objectives of the present research project are investigations concerning:
1. orbits of operators, linear dynamics, invariant subspaces;
2. operator theory in function spaces;
3. operator positivity in matrix theory.



Albert Einstein Center for Gravitation and Astrophysics (GB1437086G)
from 01/01/2014
to 31/12/2018 main investigator

Objectives:
We propose to establish the Albert Einstein Center for Gravitation and Astrophysics, a Project of Excellence that will bring together four leading research teams from the Czech Republic to address outstanding problems in gravitation theory and its astrophysical applications. We will strive to answer questions such as: What are the properties of exact models of gravitational radiation? How will the most important physical processes near rotating black holes change in the presence of largescale magnetic fields or external sources? What are the mathematical and physical aspects of higherdimensional relativity, including its implications for other fields of physics? The applying teams have longterm expertise in the relevant areas of theoretical physics, astrophysics, and cosmology. They include internationally recognized leaders as well as young researchers working at the main universities and research institutes in the Czech Republic.



Advanced methods for flowfield analysis (GA1402067S)
from 01/01/2014
to 31/12/2016 main investigator

Objectives:
The investigation aims at developing advanced methods for flowfield analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortexidentification criterion under development at present is associated with the corotation of line segments near a point and with the socalled residual vorticity obtained after the 'removal' of local shearing motion. The new quantity  the average corotation  is a vector, thus it provides a good starting point for developing both region and linetype vortexidentification methods, especially predictorcorrector type schemes. Further, the project aims to analyze the strainrate skeleton so as to draw a more complete picture of the flow. For testing purposes, largescale numerical experiments based on the solution of the NavierStokes equations (NSE) will be performed for selected 3D flow problems using finite element method on parallel supercomputers. Some qualitative properties of the solutions to the NSE will be studied and described in detail.



Enhancing mathematics content knowledge of future primary teachers via inquiry based education (GA1401417S)
from 01/01/2014
to 31/12/2016 main investigator

Objectives:
The proposed research will focus on the opportunities to influence professional competences of future primary mathematics teachers through experienced inquiry based mathematics education (IBME). The main project activities are:
1) Clarifying the concept of IBME in the Czech context.
2) Designing and testing learning environments for future primary teachers in which they experience IBME as pupils.
3) Analyzing how students are able to evolve the gained experience into strengthening their mathematical SMK.
4) Analyzing how students are able to evolve IBME environments into own illustrations, examples, explanations and powerful analogies, that is how students are able to transform the SMK into PCK.



Singularities and impulses in boundary value problems for nonlinear ordinary differential equations (GA1406958S)
from 01/01/2014
to 31/12/2016 main investigator

Objectives:
The main objective of this project is to formulate and prove new principles for existence, uniqueness and characterization of the structure of solution sets for nonlinear problems described by ordinary differential equations and their generalizations like differential equations with impulses, Stieltjes integral equations, equations with fractional derivatives and differential inclusions. A special attention is paid to nonlinear singular problems where the nonlinearities can have singularities in all their variables.



Enriched higher category theory (DP130101172)
from 01/01/2013
to 31/12/2016 main investigator

Objectives:
Higher category theory is a very young branch of mathematics (less then 20 years old), which has already became a vital tool in many areas of mathematics and theoretical physics such as algebra, geometry, topology, mathematical logic, quantum field theory and computer science. The impact of higher category theory for the future development of mathematics and physics will be immense. In its present shape, however, this theory is technically very difficult. The challenge is to find an approach to this theory which would allow to make it transparent and accessible for the wider scientific community. In our project we are going to propose such an approach and we will study its application to important open problems in geometry and topology.



Asymptotics of Operator Semigroups (AOS(318910))
from 01/11/2012
to 31/10/2016 main investigator

Programme type: FP7 Marie Curie Action "International Research Staff Exchange Scheme"
Objectives:
The theory of asymptotic behaviour of operator semigroups is a comparatively new field serving as a common denominator for many other areas of mathematics, such as for instance the theory of partial differential equations, complex analysis, harmonic analysis and topology.
The primary interest in the study of asymptotic properties of strongly continuous operator semigroups comes from the fact that such semigroups solve abstract Cauchy problems which are often models for various phenomena arising in natural sciences, engineering and economics.
Knowledge of the asymptotics of semigroups allows one to determine the character of longtime evolution of these phenomena.
Despite an obvious importance, the asymptotic theory of oneparameter strongly continuous operator semigroups was for a very long time a collection of scattered facts rather than an organized area of research. The interest increased in the 1980s and the theory has witnessed a dramatic development over the past thirty years. Still there is a number of notorious open problems that have been left open. These missing blocks prevent the theory from being complete, slow down the development of the theory and discourage specialists from related fields to engage into the theory.
The goal of the project is to give new impetus to the theory of asymptotic behaviour of operator semigroups. To this aim we plan to extend and unify various aspects of the asymptotic theory of operator semigroups: stability, hyperbolicity, rigidity, boundedness, relations to Fredholm property, to work out new methods and to solve several longstanding open problems thus giving the theory its final shape.
We intend to create an international forum that enables and promotes a multi and cross disciplinary exchange of ideas, methods and tools under the common umbrella of asymptotic theory of operator semigroups. Thus we expect that, moreover, a wide range of modern analysis will benefit from the project.



Transport phenomena in continuum fluid dynamics (TraFlu(SCIEX 11.152))
from 01/07/2012
to 31/12/2013 main investigator

Sciex Postdoctoral Fellowship for Ondřej Kreml at University of Zurich (Host institution).
Objectives:
Ondřej Kreml will study the results of Camillo De Lellis and László Székelyhidi about illposedness of bounded weak solutions for the incompressible Euler equations and bounded admissible solutions for the compressible isentropic Euler system in multiple space dimensions. The objectives of the project are to generalize the illposedness results for compressible isentropic Euler system and to study the Riemann problem for this system. Another objective is to modify the method of De Lellis and Székelyhidi to be applicable in other systems of partial differential equations describing inviscid fluid flows.



Motivation via Natural Differentiation in Mathematics (NaDiMa)
from 01/10/2008
to 30/09/2010 main investigator

Programme type: Lifelong Learning Programme COMENIUS Multilateral project
Objectives:
The European materials on educational policy stress the importance of formation of pupils´ competences as: competence to learn, to communicate, to solve problems, to make conjectures, etc. This process should start in early age and on the primary school level it is supported mainly by the teacher.
On the general level this project aims at the development of primary school pupils:
to support the development of their learning competences;
to support the consciousness of the meaning of mathematics as a part of human culture;
to encourage pupils' motivation to learn mathematics;
to realize pupils' individual (cognitive) potentials;
to create the possibility for students to experience success in the process of problem solving.
These aims should be achieved by means of support and enhancement of teachers directed on:
to get to know examples for substantial learning environments and by that experience the nature of mathematics and recognize and use the potential for teaching and learning;
to strengthen teacher’s mathematical content knowledge in relation to and requested for these specific learning environments;
how to cope with the heterogeneity and realize natural differentiation in mathematics classrooms;
to support the change of belief on the substance and importance of mathematics for primary school level.
In concrete terms each local team in cooperation with teachers from partners and associated schools will prepare elaborated materials and examples of substantial learning environments:
its mathematical background (importance for the development of mathematical thinking, connections to crucial mathematical ideas, different ways of solutions);
different possibilities how to put it into practice in the classrooms;
our experience from field experiment (concentrating mostly on the pupils´ motivation and development of their different competences);
possibilities to develop, cultivate, and enrich mathematical ideas.



Infinite dimensional stochastic systems (201/07/0237)
from 01/01/2007
to 31/12/2009 main investigator

The project is aimed at investigating qualitative properties of stochastic infinite dimensional systems (in particular, of solutions to infinite dimensional stochastic equations) and at research in infinite dimensional stochastic control theory. The moreimmediate aims are: 1. Investigation of existence, uniqueness and pathwise properties of solutions to infinite dimensional stochastic equations, in particular of the path regularity and stability, especially for geometric wave equations, equations in Banach spaces and equations driven by fractional noises. 2. Large time behaviour and ergodicity for Markovian and nonMarkovian stochastic equations. This research will be focused on methods applicable to stochastic wave and beam equations and (in the nonMarkovian case) on equations where the driving process is a fractional Brownian motion. 3. Infinite dimensional stochastic control and the associated semilinear HamiltonJacobiBellman equations in infinite dimensions: ergodic and adaptive control.



Applied probability  stochastic geometry and metric tools (201/06/0302)
from 01/01/2006
to 31/12/2008 main investigator

In applied probability the theory of random sets (stochastic geometry) will be developed; this will lead to new estimators of intrinsic volume densities both in stationary and nonstationary case. The quality of estimators of geometrical characteristics will be described by means of probability distances. The field of probability distances will be improved theoretically, some applications in biology are expected. Stochastic simulations present a complementary tool in stochastic geometry, their development will be directed to random tessellations. Interdisciplinary use of such models will lead to the characterization of the microstructure of nanomaterials. Five PhD students will participate at the solution of the project and an international conference will be organized.



Spatial statistics of 3D confocal images of capillary network (IAA100110502)
from 01/01/2005
to 31/12/2009 main investigator

The aim of the project is quantitave evaluation of length density and spatial arrangement of brain microvessels from stacks of optical sections captured by a laser scanning confocal microscope. We will use image enhancement by deconvolution, horizontal and vertical image composition of adjacent fields of view. We will develop and test several methods for capillary detection, measure the length automatically and compare the results with state of art interactive techniques of length measurement. The detection methods will include use of tophat filters, path extraction methods by minimal path and probabilistic methods using Markov random fields of directions. Spatial arrangement will be characterized by topology, correlation functions, sphere contact distribution function and rose of directions of microvessels.
