Appendix 3.4
Information on the activity of the institute
Institute
Institute of Mathematics CAS
1. Characteristics of main research directions investigated at the institute and
their achievements
Maximum length of 10 pages.
The principal mission of the Institute of Mathematics (IM) is to support fundamental
research in the fields of mathematics and its applications, and to provide necessary
infrastructure for research. The IM contributes to raising the level of knowledge and
education and to utilising the results of scientific research in practice. It acquires,
processes and disseminates scientific information, issues scientific and professional
publications (monographs, journals, proceedings, preprints, etc.). In cooperation with
Czech universities, the IM carries out doctoral study programmes and provides
training for young scientists. The IM promotes international cooperation, including the
organisation of joint research projects with foreign partners and participation in
exchange programmes and international research networks. It has been one of the
most successful Czech research institutions in the European Research Council
(ERC) Grant competition. The IM organises scientific meetings, conferences and
seminars on the national and international levels. Its experts regularly give
specialised courses at foreign universities, and are invited as plenary or key note
speakers in international congresses and conferences.
Research in the Institute focuses on mathematical analysis (differential equations,
numerical analysis, functional analysis, theory of function spaces), mathematical
physics, mathematical logic, complexity theory, combinatorics, set theory, numerical
algebra, topology (general and algebraic), optimization and control, and differential
geometry. The structure of IM consists of the following six research teams, and a
small group for Didactics of Mathematics, which is not registered as a research team
for this evaluation.
1.1 Algebra, Geometry and Mathematical Physics (AGMP)
The department was formed in 2014 upon a bottom-up initiative by gathering
researchers working in algebraic and differential geometry and on closely related
areas of mathematical physics. The research is focused on mathematical aspects of
modern theoretical models of physics of microcosmos and gravity. The research
topics range from representation theory and its applications in algebraic geometry
and number theory, over homological algebra, algebraic topology, applied category
theory, to classification of tensors, as well as investigation of Einstein's equations and
of generalized theories of gravity. Members of the department participate in two
research centres of excellence, namely Eduard Čech Institute for Algebra, Geometry
and Physics, and Albert Einstein Centre for Gravity and Astrophysics.
In mathematical relativity, the most remarkable publications of the group reflect their
systematic effort in extending the classical concepts of the relativity theory to spacetimes of higher dimensions. The main achievements include results classification of
tensors, spacetimes with vanishing curvature invariants, universal spacetimes,
asymptotic properties of Einstein spaces, uniqueness and geometric properties of
1
Appendix 3.4
solutions to Einstein's equations and their various generalisations motivated, e.g., by
quantum gravity.
In algebraic topology and geometry, the main achievement was the monograph
[379741] summarizing M. Markl's original ideas consisting in treating the subject of
deformation theory from a modern perspective, which emphasizes the role of
differential graded Lie algebras, Maurer-Cartan spaces, and operadic structures.
Other important results were obtained in string field theory, quantum groups and
vertex operator algebras, in geometric structures on manifolds, and in information
geometry and complexity.
1.2 Differential equations and theory of integral (DETI)
Most team members work in the Brno Branch of the IM. The core research performed
in this team concerns qualitative properties of ordinary and functional differential
equations. Such equations typically describe processes in finite dimensional systems,
and find important applications in biology and physics. The theoretical study of their
solutions helps to discover intrinsic structures in complex systems. Particular
attention is paid to equations with discontinuities and other types of singularities. The
classical Kurzweil theory of integration developed in the Prague part of the team
serves as a powerful tool in the theory of generalized differential equations, and also
the modern theory of equations on time scales is developed. Other important topics
studied in the group include boundary value problems, the theory of oscillation, and
asymptotic behaviour of solutions.
J. Kurzweil presented an alternative approach to generalized differential equations in
his new monograph [375634]. Emphasis is put on equations having continuous, not
necessarily absolutely continuous solutions. The abstract Kurzweil-Stieltjes integral in
Banach spaces was also one of the important topics of interest. A new method was
developed to construct upper and lower functions which help to identify periodic
regimes in equations with singularities motivated by problems in relativity theory,
quantum mechanics, and nonlinear optics. Further achievements were obtained in
the theory of non-local or singular boundary value problems for differential and
functional differential equations, in oscillation theory, and asymptotic properties of
solutions. Properties of dynamic equations on time scales were studied in detail, and
important results were obtained about the relationship between the qualitative
behaviour of their solutions and the graininess of the time scale.
1.3 Numerical Analysis (NA)
The team coincides with the administrative research department Constructive
Methods of Mathematical Analysis, which continues the long tradition of investigation
and use of numerical methods established in the Institute by the world leading
specialist Professor Ivo Babuška. The importance of such methods continues
growing with the development of computational and experimental techniques.
Mathematical modelling of complex physical processes involving immense amount of
data requires new methods of communication with computers. The main challenges
are related to an optimal exploitation of their ever growing capacity, for example by
developing fast and reliable algorithms, and by controlling rigorously the accuracy of
computation. Members of the team are experts in superconvergence methods, a
posteriori error bounds, and parallel computing. The main research topics focus on
analysis and optimization of the finite element method for solving partial differential
2
Appendix 3.4
equations describing physical processes in solids and fluids. The members of the
department are involved in the network for industrial mathematics EU-MATHS-IN.CZ
which is part of the European network EU-MATHS-IN.EU.
The main achievements of the team during the evaluation period include the
discovery of interesting and unexpected relationships between the geometry of the
simplices in finite element approximations of partial differential equations, and the
speed of convergence and accuracy of the numerical method. Further important
results are related to various improvements of the domain decomposition method for
numerical solution of problems in domains with a complicated geometry, error
bounds for finite element approximations, questions of validity of the discrete
maximum principle as a criterion for physical relevance of the computational scheme,
parallel computing, and numerical tests in biological applications.
1.4 Evolution Differential Equations (EDE)
Researchers in the team EDE focus on qualitative aspects of the theory of partial
differential equations in mechanics and thermodynamics of continuum, in biology,
and in other sciences. The main goal of the study is to check, on the one hand, the
theoretical well-posedness of mathematical models proposed by physicists and
engineers, and, on the other hand, the possibility to make reliable predictions of
future development of a system with an incomplete knowledge of the initial state. The
main topics include equations describing fluid flow with or without heat exchange,
and its interaction with rigid bodies. Attention is paid also to processes in deformable
solids, with emphasis on mathematical modelling of memory in multifunctional
materials, on dynamical behaviour of bodies in contact with an obstacle, and to
phase transitions. Members of the team are involved in the Nečas Center for
Mathematical Modeling and in the network for industrial mathematics EU-MATHSIN.CZ, a part of the European network EU-MATHS-IN.EU. E. Feireisl is principal
investigator of the ERC Advanced Grant MATHEF (Mathematical Thermodynamics of
Fluids) aimed at building a complete mathematical theory describing motion of
compressible viscous heat conducting fluids.
E. Feireisl's contribution to scientific production of the team was substantial during
the evaluation period. His novel and original approach to the interpretation of the
principles of continuum thermodynamics in modelling heat conducting fluid flow
turned out to be a rich source of results for the general theory, as for example the
concept of weak-strong uniqueness of solutions, as well as for a mathematical
justification of various particular engineering flow regimes in terms of rigorous
singular limits. Crucial results were obtained in the study of the mechanical
interactions between the fluid and the domain boundary, between fluid and moving
rigid solids, and thermodynamic interactions between the fluid and the surrounding
via radiation.
Achievements in solid mechanics were obtained in modelling contact of a body with a
rigid or deformable obstacle. Existence and uniqueness results were proved for the
difficult case of limited interpenetration. The theory of hysteresis operators was
further developed and applied in modelling material fatigue, contact of elastoplastic
bodies, and magnetostrictive hysteresis. Results in qualitative description of phase
transitions also deserve appropriate attention. In reaction-diffusion problems,
unilateral conditions were shown to substantially perturb the expected behaviour of
3
Appendix 3.4
solutions. Important progress was made in the theory of functions and function
spaces, and in particular the monograph [397224] is worth mentioning.
1.5 Logic and Theoretical Computer Science (LTCS)
The research programme of this section is connected with the questions of
information processing. The main topic is the theory of computational complexity
which is used for classification of algorithmic problems and plays important role also
in coding and electronic communication security. Further important research fields
concern general questions of logical foundations of numbers and set theory,
combinatorics, matrix theory, decidability, and control theory. P. Pudlák is the
principal investigator in the ERC Advanced Grant FEALORA (Feasibility, Logic and
Randomness in Computational Complexity), and members of the team are involved
in the association of research institutions DIMATIA and in the research centre of
excellence Institute of Theoretical Informatics.
Complexity theory is based on techniques of mathematical logic and theoretical
computer science. The method consists in identifying the combinatorial and logical
structure of the problem, and estimating the difficulty of their algorithmic solutions. An
overall goal is to provide evidence whether or not some problems have a fast
algorithmic solution. These questions have foundational importance in itself, but
potentially may also have practical applications, for example in data security.
In logical foundations of mathematics, the LTCS team is one of the world leading
centres of research in bounded arithmetic and proof complexity. The main research
strategies are related to the modern set theory. Part of the team works in this area, in
particular on forcing, complexity of equivalence relations, and independence results.
There are many themes in common with bounded arithmetic. Attention is also paid to
the development of online algorithms, databases, controlled formal models, and
supervisory control of discrete-event systems.
1.6 Topology and Functional Analysis (TFA)
Theoretical concepts of infinite dimensional analysis and geometry developed in
functional analysis and topology are suitable for description of systems with an
extremely large number of state variables. Members of the team study fundamental
questions of the structure of mathematical objects in spaces created by abstraction of
notions that were originally designed to describe natural processes. This enables
discovering hidden connections between individual elements of the system, and
helps to design methods for solution of particular problems in applied mathematics.
Theory of Banach spaces has a long tradition in the IM, and the monograph [358155]
is not only perhaps the most comprehensive survey on the state of the art in this field,
but it presents also a lot of new results and open problems as a motivation for further
research. Independently, more general objects combining the ideas of functional
analysis and the theory of categories are systematically studied in the group. Results
of the team members on structures emerging from abstract complex analysis have
drawn attention in the international community. Linear operators on Banach spaces
form another important subject of study. Questions of boundedness, convergence,
and other properties of infinite iterations of such operators often arise in applications
of mathematical methods in continuum mechanics or economics. An original abstract
approach to this kind of problems has led to a number of deep and surprising results.
4
Appendix 3.4
Specific problems of function spaces as the most common examples of Banach
spaces which have natural applications in partial differential equations were
successfully solved. Last but not least, applications of functional analysis in axiomatic
thermodynamics have been studied in detail.
1.7 Didactics of Mathematics (DM)
The continuously evolving demands on knowledge and competences of primary
school pupils and on their training for life in a society based on ever growing
information exchange lay fundamental questions on education in mathematics and
on the teachers' professional development. This very small group represents a
particular long-lasting research direction in the IM, the largest part of which has been
gradually transferred to the Charles University. The members cooperate with
specialised groups at universities in the Czech Republic and abroad in theoretical
and practical aspects of didactics of mathematics, providing a useful linkage to
primary schools. They regularly present their studies in international conferences,
and publish texts on problems of modern mathematics education in both Czech and
international journals and monographs.
2. Qualitative and quantitative description of the personnel policy of the
institute
Age structure, qualification structure, personnel structure in terms of international representation,
description of the process of hiring employees, method of evaluating researchers and teams, career
development.
Maximum length of 2 pages.
The first glance at the age structure shows that the major part (55%) of employees is
formed by those under 45. It is even slightly better if we consider only research team
members (58% under 45). A closer view reveals that there is relative gap between 45
and 60 with another peak in the category 60–65. This means that the in numbers
strong younger generation should be systematically prepared for assuming the
leadership in the Institute when the generation over 60 will get retired during the next
few years.
The qualification structure of the researchers is excellent. Naturally, all research team
members except PhD students are on PhD or DrSc level; 15% of the team members
are Professors and 26% have the DrSc degree, 17% are on the postdoc position and
6% are PhD students on the payroll of the Institute. The body of researchers and
PhD students is quite international, 24% of them are foreigners. All the percentage
shares are expressed according to FTE.
In accordance with the Act on the Czech Academy of Sciences, all posts of workers
with high-grade education in the institute are manned on the basis of open
competitions which are judged by the Qualification Audit Committee appointed by the
director. The calls for available positions are posted on the web site of the Institute
and on the job server of the European Mathematical Society. Since 2014 an
electronic system has been launched for submitting applications and referee’s letters
as well as for assessment of the candidates by the Qualification Audit Committee. In
recent years the Institute organizes 4 or 5 competitions for positions of PhD students,
postdocs or research fellows yearly.
The career development of researchers strictly follows the rules given by the Statutes
and the Career Rules of the Academy of Sciences. Every employee on a research
5
Appendix 3.4
position has a fixed term contract, in most cases ranging from 2 to 5 years. The
extension of the contract is subject to a successful evaluation of the research
performance by the Qualification Audit Committee. In this way, all researchers are
regularly evaluated.
There are a few senior researchers who are already retired but the Institute offers
them to continue in a part time job because they are experienced and still very active
mathematicians or they have got a grant. We feel that their impact on the Institute is
important and worth of their loads which altogether counts 4.25 FTE, i.e. 7% of the
total researchers FTE.
Education of PhD students is legally limited to cooperation with a university. To
recruit students and be allowed to educate them, the Institute of Mathematics is
involved in three joint doctoral study programmes (mathematics, informatics, physics)
with the Charles University, Faculty of Mathematics and Physics, one joint
programme (didactics of mathematics) with the Charles University, Faculty of
Education, and one joint programme (mathematics) with the University of West
Bohemia, Faculty of Applied Sciences. For more details of the joint study
programmes see
http://www.math.cas.cz/labo_institute/page_html.php?id_page=1&lang=0.
The Academy of Sciences launched an internal programme for development of
human resources in which the Academy institutes can compete for a two year
financial support for young talented researchers on postdoc positions. The Institute
has been very successful in this competition. Currently, there are four postdocs in the
Institute under this programme. The Institute provides them with additional support in
the form of affordable accommodation in the Institute building.
Four of seven research department heads have already approached the retirement
age. One of the departments, Didactics of Mathematics is going to be closed in a
near future. The heads of the other three will have to be replaced within a few years.
Concerning the technical staff the situation in the Institute is rather stable. There are
five departments providing technical and administrative services: the Administration
Department, the public mathematical Library, the IT Department, the Editorial Office
editing three international scientific journals, and Director’s Office. Thanks to careful
staffing policy during the previous few years there are competent and reliable people
in all these departments and with a good perspective. In the Administration
Department, a new specialist is being trained to assume the lead of the department in
one year. Concerning director’s office, we were extremely lucky to hire the very
competent and efficient project manager B. Kubiś a few years ago. She provides a
very important support to all researchers and to the management.
To stabilize the personnel situation and promote attractiveness of the Institute for
researchers we have recently transformed the salary system in order to increase the
differences between qualification degrees, and to better reward higher qualification
and competence.
2.1 Age structure of the institute
Age structure is related to 31 December 2014
6
Appendix 3.4
Age category < 25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 60-65 65-70 ≥ 70
Number of
members
0,5
6,3 12,3
9,6 16,8
7,5 6,05 5,98 10,25 4,15 2,85
18
16
14
12
10
8
6
4
2
0
16,8
12,3
10,25
9,6
7,5
6,3
6,05
5,98
4,15
2,85
0,5
< 25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
≥ 70
3. Strengths and weaknesses of the institute
Maximum length of 2 pages.
Strengths
Tradition
The research in the Institute of Mathematics is pursued in several fields representing
the best traditions of the Czech mathematics, namely theory of ordinary and partial
differential equations, theory of integral, numerical analysis, topology and functional
analysis, of the more recent, nevertheless very strong fields of mathematical logic
and theoretical computer science. The development of these fields is connected with
such personalities like E. Čech, J. Kurzweil, I. Babuška, J. Nečas, M. Fiedler, V. Pták
and P. Hájek.
Collaboration with universities
Researchers in the Institute are involved in a rich collaboration with colleagues at
universities, both in research and teaching. It is institutionalized in several cases by
formal agreements between the institutions.
International cooperation
There is a very large cooperation with foreign mathematicians which can be
documented by a big number of visits abroad and visitors to the Institute (recently
almost 200 and 150 per year, resp.).
Organizing conferences
Employees of the Institute are active in organizing several conferences and workshop
every year. Some of them are very large and have a long tradition (Equadiff,
Toposym, Winter School in Abstract Analysis, Winter School Geometry and Physics,
Nonlinear Analysis, Function Spaces and Applications, Function Spaces, Differential
Operators, Nonlinear Analysis, Programs and Algorithms of Numerical Mathematics,
Eurocomb, etc.).
Projects and centres
Currently there are two ERC Advanced Grants in the Institute, one Marie Curie
Mobility Grant, one Marie Curie Exchange Staff Grant, and several other grants. The
Institute is involved in several research centres and research consortia.
7
Appendix 3.4
Circulation of postdocs and visiting researchers
There is a well established system of regular competitions for postdocs and visiting
researchers (usually for the period from 6 to 24 months).
Young researchers
There is relatively high number of young researchers in the Institute.
Technical and material facilities
The Institute is running the largest mathematical library in the country. Besides
journals and book the library provides connection to major databases. The Institute is
maintaining the Czech Digital Mathematics Library. The buildings of the Institute are
in a good condition after a recent reconstruction. They provide enough quality space
for the employees and visitors as well as several flats and lodging capacity for
visitors. The Institute has satisfactory instrumental equipment and is not dependent
on large technical facilities.
Project manager
The Institute hired three years ago a highly competent project manager who is
relieving researchers from the major burden connected with the whole life cycle of
grants.
Weaknesses
The age structure is not perfectly balanced. There is currently a certain gap between
the younger generation (30–45) and the elder one (65+).
There are obstacles to gathering new students from universities. The legislative
obstacle is given by the Law on Higher Education which allows the non-university
institutions teaching PhD students only in a joint doctoral study programme with a
university. Another obstacle consists in unwillingness of universities to offer core
lecture courses to specialist from other institutions which decreases the chance to
attract students for a PhD study in the Institute.
4. Research plan of the institute as a whole for 2015–2019
Maximum length of 2 pages.
The core activity of the Institute of Mathematics falls in the category of fundamental
research. Due to the abstract character of mathematics, it is basically impossible to
determine, which particular discipline and which particular research direction will
deserve to be given priority at the expense of some others in several years from now.
Mathematicians in the institute do not need complex technical equipment to carry out
top quality research. A breakthrough in mathematics cannot be made to order.
Revolutionary results appear spontaneously, and it often takes years until their
significance is recognized by the community. This is why it is important to support a
maximum freedom in choosing the research subjects in mathematics, and insist only
on high quality standards resulting from continuous qualified evaluation based on
international comparison.
In the present period and in a near future, emphasis in research orientation will be
naturally put on the topics where the very high quality exists or where is the potential
to achieve it. Naturally, this concerns primarily the teams around the two ERC
Advanced Grants, MATHEF (Mathematical Thermodynamics of Fluids) of E. Feireisl,
and FEALORA (Feasibility, Logic and Randomness in Computational Complexity) of
P. Pudlák. Also memberships of our experts in Centres of Excellence and other long
8
Appendix 3.4
term national and international projects allow to predict to some extent which
directions have the most chance to be followed. All these researchers and teams will
enjoy a general support of the institute for their continuation in the excellent research.
Members of the team "Algebra, Geometry and Mathematical Physics", using the
strengthened synergy in the newly formed department, will pursue their most
successful research topics related to Einstein's equations and their generalizations,
algebraic structures, and differential geometry, in close cooperation with two research
centres of excellence, namely Eduard Čech Institute for Algebra, Geometry and
Physics, and Albert Einstein Centre for Gravity and Astrophysics.
In the group "Differential equations and theory of integration", the well established
and promising topics like differential equations on time scales or Kurzweil integration
and its applications will continue to be a rich source of challenging problems.
The team "Numerical Analysis", reinforced with the postdoc P. Kůs since April 2015
will focus on the problems of developing new efficient algorithms for an optimal
exploitation of the exponentially growing computer performance. This will include new
challenges related to the classical finite element method, in particular adaptivity,
domain decomposition, parallel computing, and possibly closer cooperation with
members of the "Evolution Differential Equations" team.
The core of the team "Evolution Differential Equations" will benefit from E. Feireisl's
ERC Advanced Grant MATHEF pursuing their successful work in thermodynamics of
fluids. Other members will continue developing cooperation with engineers on
modelling and control of multifunctional materials, phase transition modeling, as well
as systematic study of reaction-diffusion processes in biological systems.
Similarly, the programme of the team "Logic and Theoretical Computer Science" will
largely follow the plans specified in P. Pudlák's ERC Advanced Grant FEALORA
which will continue till 2018. They will also continue in fruitful cooperation with
mathematicians from the Charles University in frames of the research centre of
excellence ITI (Institute of Theoretical Informatics) and the consortium DIMATIA.
In the team "Topology and Functional Analysis", apart from running projects, new and
promising ways are proposed to study nonlinear mappings and structures in
functional analysis, and universal homogeneous structures. The team may obtain
new impulses from the two postdocs supported by the special Academy's programme
and, above all, if the recently submitted application for the prestigious Praemium
Academiae succeeds.
The Institute of Mathematics has always supported creative and independent
thinking. It is necessary to mention that the recent several success stories in
international competitions are to a large extent due to a methodical project support
system elaborated and set up by the Project Manager B. Kubiś who was hired in
2011 and who takes over most of the administrative load thus protecting the
researchers from an overall increasing bureaucracy.
For the next period, as it has been the case until now, the only criterion for future
strategy is the excellence of research. Young colleagues are encouraged to intensify
their contacts with foreign experts, to visit prestigious research institutions, and to
learn new methods and bring new ideas which will allow them to become
independent of their former supervisors and develop their own scientific career.
9
Appendix 3.4
The Institute of Mathematics will further develop all possibilities of cooperation with
universities, starting from teaching special courses at both undergraduate and
graduate levels, over supervision of Master and PhD theses, to offering the best
students a possibility of direct participation in grant teams and research projects.
Initiatives towards a better cooperation of mathematicians with the applied
community will also be encouraged. This is the case for example of our involvement
in the network for industrial mathematics EU-MATHS-IN. A workshop organized
jointly by the Institute of Mathematics, Institute for Computer Science CAS, and
Institute for Information Theory and Automation CAS, with the goal to put together
our researchers with colleagues from industry in a broad sense, is planned for the
end of 2015 with the hope to create a platform for a better communication and mutual
exchange.
The research team leaders in the Institute of Mathematics are recognized experts in
their fields, and have a good overview about the research topics studied in their
teams, as well as about the individual competences of their team members. Their
research plans manifest very good knowledge of the state of the art in their
disciplines, and well justified determination about the priorities for the near future, but
some of them will soon reach the retirement age. It will be important to encourage
younger dynamic personalities to the possibility of taking over some responsibilities
and propose new paradigms for the next decades.
10