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Institute of Mathematics

The Institute was established on 14 Ferbuary 1947 as the Institute for Mathematics of the Czech Academy of Sciences and Arts (Česká akademie věd a umění). In 1950 it was reorganized to form the Central Mathematical Institute, which was incorporated into the Czechoslovak Academy of Sciences on 1 January 1953 as the Mathematical Institute. In 1993 the Institute became part of the newly founded Academy of Sciences of the Czech Republic. Since January 1, 2007, the Institute of Mathematics (IM in the following) received the v. v. i. status (initials of the Czech legal term *veřejná výzkumná instituce* = public research institution), whose founder is the Academy of Sciences of the Czech Republic, pursuant to the Act No. 341/2005 Coll. on public research institutions. This change leads to broader rights and responsibility of the institute as those of a legal person and allows for certain relaxation of the rather strict budget rules in the past.

The principal activity of the 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, etc.). In cooperation with 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. The IM organises scientific meetings, conferences and seminars on the national and international levels.

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, differential geometry, and didactics of mathematics.

Departments

Abstract Analysis (AA)

Theoretical concept of infinite dimensional analysis and geometry developed in functional analysis and topology is suitable for description of systems with extremely large number state variables. Members of this department focus on fundamental questions of the structure of mathematical objects in spaces created by abstraction of notion orginally defined 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.

Algebra, Geometry and Mathematical Physics (AGMP)

The department was formed in 2014 from researchers interested 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 include represenatation theory and its applications in algebraic geometry and number theory, homological algebra, algebraic topology, applied category theory, classfication of tensors, 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, ane Albert Einstein Centre for gravity and astrophysics.

Branch in Brno (BB)

The core research performed in this section in collaboration with several members of the Department of Evolution Differential Equations concerns qualitative properties of ordinary and functional differential equations. Such equations describing development of finite dimensional systems find important applications in biology and physics. The theoretical study of their solutions would help discovering mathematical causalities in real systems including singularities in time and space variables and in discontinuous processes which are modelled by means of the Kurzweil-Henstock integral as well as in terms of equations on time scales. Another important topics is studying methods of of optimal control of complex processes.

Constructive Methods of Mathematical Analysis (CMMA)

The department countinues the long tradition of investigation and use of numerical methods established in the Institute by the world leading specialist Prof. Ivo Babuška. The importance of such methods continues growing with the development of computational and experimental technique. Mathematical modelling of complex physical processes involving immense amount of data requires new methods of communication with computers, namely for optimal employment of their ever growing capacity, and for increasing speed and controlling rigour of computation by means of superconvergence and aposteriori estimate of errors. The main topic concerns analysis and optimization of the finite element method for solving partial differential equations describing physical processes in solid matters 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.

Evolution Differential Equations (EDE)

The scope of this section covers qualitative aspects of theory of partial differential equations in mechanics and thermodynamics of continuum, in biology and in other sciences. The research aims at verification of correctness of mathematical models and of the possibility to provide theoretical predictions of future development of a system without the full knowledge of the initial state. The work focuses on investigation of equations describing fluid flow including heat exchange and interaction with solid bodies. Attention is paid also to processes in solid matters focusing on mathematical modelling of memory in multifunctional materials, to dynamical behaviour of bodies in a contact with an underlay, and to phase transitions.

Members of this department are involved in the Nečas Center for Mathematical Modeling and in the network for industrial mathematics EU-MATHS-IN.CZ, a part of the European network EU-MATHS-IN.EU. E. Feireisl is principal investigator of the ERC Advanced Grant MATHEF aimed at building mathematical theory describing motion of compressible viscous heat conducting fluids.

Several members cooperate with the Branch in Brno investigating the integration theory and ordinary differential equations.

Mathematical Logic and Theoretical Computer Science (MLTCS)

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 clasification 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 and matrix theory.

P. Pudlák is the principal investigator in the ERC Advanced Grant FEALORA and members of the department are involved in the association of research institutions DIMATIA and in the ecellence research centre Institute of Theoretical Informatics.