Mathematics and Statistics
Přijímací řízení do doktorských programů  akad.rok 2021/2022 (zahájení: podzim 2021)
Submission deadline until midnight 30 Apr 2021
What will you learn?
The aim of the program is to prepare highquality scientific specialists in the fields covered by the Institute of Mathematics and Statistics at MU. Graduates should be prepared primarily for further scientific work in academic institutions, but also for possible application in practice. Upon graduation, graduates who wish to continue their research work are motivated to gain longterm foreign experience as postdoctoral students.
The individual research teams of the Institute of Mathematics and Statistics cover the following research themes in which PhD students are also trained:
 Categories and ordered sets, number theory, semigroup theory
 Geometric structures, geometric complex analysis, algebraic topology
 Theory of Differential and Differential Equations, Calculus of variations and optimal control
 Mathematical modeling, statistics and data analysis, applications in biology and medicine
Attention is also paid to the preparation for pedagogical work at universities. The study is based on an individual study plan and is completed by a state doctoral examination and a defense of doctoral dissertation. In addition to Czech, English is also the working language of the program.
Practical training
Practical training is not a mandatory part of this program.
Career opportunities
Graduates will apply to mathematical workplaces of basic research, universities and scientific research institutes of the Academy of Sciences of the Czech Republic.
The best graduates are fully prepared to successfully apply for postdoctoral positions at highquality universities abroad.
Graduates can also act as college teachers at universities with a technical, economic and pedagogical focus.
Graduates of applied specializations will also find use in practice, in institutions where the use of deterministic and stochastic models of real processes where specialized statistical software is being developed, and in institutions focusing on research in the field of probabilistic and mathematicalstatistical methods.
Admission requirements
The admission interview consists of two parts: expert part (max. 70 points) verifies the applicant's orientation in the selected specialization of mathematics and statistics. The language part (max. 30 points) verifies applicant's ability of independent study in English. A total of at least 80 points is required for admission.
Study options
Singlesubject studies with specialization
In the singlesubject studies, the student deepens knowledge in the concrete focus of the degree programme and chooses one specialization. The specialization is stated in the university diploma.

Algebra, Number Theory and Mathematical Logic (fulltime or combined studies in czech)
What will you learn? 
Geometry, Topology and Geometric Analysis (fulltime or combined studies in czech)
What will you learn? 
Probability, Statistics and Mathematical Modelling (fulltime or combined studies in czech)
What will you learn?
Supervisors and dissertation topics
Supervisors
 doc. Ing. Luděk Berec, Dr.
 John Denis Bourke
 doc. RNDr. Martin Čadek, CSc.
 prof. RNDr. Zuzana Došlá, DSc.
 doc. RNDr. Eduard Fuchs, CSc.
 doc. Anton Galaev, Dr. rer. nat.
 doc. Mgr. Petr Hasil, Ph.D.
 prof. Ing. Jiří Holčík, CSc.
 prof. RNDr. Ivanka Horová, CSc.
 prof. RNDr. Josef Janyška, DSc.
 doc. RNDr. Josef Kalas, CSc.
 doc. PaedDr. RNDr. Stanislav Katina, Ph.D.
 doc. Mgr. Ondřej Klíma, Ph.D.
 doc. Mgr. Jan Koláček, Ph.D.
 doc. RNDr. Martin Kolář, Ph.D.
 doc. Ilya Kossovskiy, Ph.D.
 prof. RNDr. Radan Kučera, DSc.
 doc. Mgr. Michal Kunc, Ph.D.
 doc. RNDr. Petr Lánský, DrSc.
 RNDr. Martin Markl, DrSc.
 doc. Mgr. Robert Mařík, Ph.D.
 doc. RNDr. Jan Paseka, CSc.
 doc. RNDr. Libor Polák, CSc.
 prof. RNDr. Zdeněk Pospíšil, Dr.
 prof. RNDr. Jiří Rosický, DrSc.
 prof. RNDr. Jan Slovák, DrSc.
 doc. Mgr. Josef Šilhan, Ph.D.
 prof. RNDr. Roman Šimon Hilscher, DSc.
 doc. RNDr. Jaromír Šimša, CSc.
 doc. RNDr. Michal Veselý, Ph.D.
 doc. Lukáš Vokřínek, PhD.
 prof. RNDr. Gejza Wimmer, DrSc.
 doc. Mgr. Petr Zemánek, Ph.D.
Dissertation topics
Specialization: Algebra, Number Theory and Mathematical Logic
Abelian extensions of number fields
Supervisor: prof. RNDr. Radan Kučera, DSc.
The main theme is devoted to the study of abelian extensions of the field of rational numbers, possibly of an imaginary quadratic field. The attention is focused on objects related to the ideal class groups (e.g., the group of circular unit, Stickelberger ideal, the group of elliptic units).
Notes
Examples of some older dissertations: https://is.muni.cz/th/mwiet/?lang=en or https://is.muni.cz/th/jbpxt/?lang=en
Accessible categories and their applications
Supervisor: prof. RNDr. Jiří Rosický, DrSc.
Accessible categories and their applications in algebra, model theory and homotopy theory. For example: Abstract elementary classes, Accessible model categories.
My publications: https://arxiv.org/find/grp_math/1/au:+rosicky/0/1/0/all/0/1?skip=0&query_id=8094c174213ee61e
Algebraic structures and their applications
Supervisor: doc. RNDr. Jan Paseka, CSc.
OBJECTIVES: The research deals with connections of algebra with logic, in particular quantum, tense, and fuzzy. The basic tool are residuated posets, enriched categories, and orthogonal structures but the emphasis is also on quantales in connection with C*algebras and noncommutative geometry. The practical part of the research is oriented to simulation and validation of value streams using formal words, trees, and categorical concepts. We study algebraic methods for aggregation of processes and their effects, in particular in a probabilistic environment.
AIM: a) For example, one of our research goals is a characterization of the basic quantumphysical model by means of automorphisms of its underlying orthogonality space.
b) The theoretical aspects of aggregation of multidimensional data, rankings, relations and strings will be developed in more detail, especially connected to practical situations. The mathematical model is designed primarily for industrial planning but could be used for a wider range of applications (bioinformatics etc.).
My publications:
https://www.muni.cz/en/people/1197janpaseka/publications
Combinatorial and algebraic properties of formal languages
Supervisor: doc. Mgr. Michal Kunc, Ph.D.
Research Area:The theory of automata and formal languages is an active research field on the borderline between mathematics and theoretical computer science. It combines ideas and techniques of combinatorics, algebra, logic or topology in order to tackle difficult questions about decidability and computational complexity of problems concerning sets of objects definable by diverse models of computation.
Focus:
Doctoral research projects may focus on various aspects of formal languages where techniques of combinatorics on words or semigroup theory can be applied.
Examples of potential doctoral projects:
* Decidability of properties of regular languages and semigroups.
* Computational power of formal machines and grammars.
* Solvability of language equations.
* Algorithmic characterizations of classes of formal languages.
Finite semigroup theory and algebraic theory of regular languages
Supervisor: doc. Mgr. Ondřej Klíma, Ph.D.
RESEARCH TOPIC:The modern theory of finite semigroups links universal algebra and topology with the theory of formal languages and logic in theoretical computer science. The main motivation of that research is decidability of concatenation hierarchies of regular languages. The algebraic objects in the centre of our interest are the lattice of pseudovarieties of finite ordered semigroups and the free profinite semigroups in these pseudovarieties.
FOCUS:Doctoral research project may focus on the theory of varieties of regular languages or on the theory of profinite semigroups. However, there are also other questions combining theoretical computer science and algebra, for example questions concerning computational complexity of identity checking problem for a fixed finite semigroup.
EXAMPLES of potential doctoral projects: The equational characterizations of pseudovarieties,
 Completeness of the equational logic for psedovarieties of finite algebras,
 Concatenation hierarchies of starfree languages,
 Computational complexity of basic problems for finite semigroups.
Notes
My publications: http://www.math.muni.cz/~klima/Math/publications.html
Homotopy coherent structures and computational topology
Supervisor: doc. Lukáš Vokřínek, PhD.
RESEARCH TOPIC: Homotopy coherent structures are structures, where the constrains are relaxed to hold only up to a coherent system of homotopies. They turn up when an object equipped with a strict structure is replaced by a homotopy euivalent one, e.g. by a small model of the original. For this reason, homotopy coherent structures arise quite naturally in computational topology, where small models are used for computations. They are studied via abstract homotopy theory, e.g. model categories or homological algebra but, for computational purposes, concrete formulas are preferrable; these are efficiently provided by homological perturbation theory.PROJECT EXAMPLES:
 Homotopy coherent structures through homological pertubation theory
 Effective homology in the A_\inftycontext
 Algorithmic aspects of the extension problem from the viewpoint of rational homotopy theory
Specialization: General Mathematics
History and present of topics from mathematical analysis
Supervisor: doc. Mgr. Petr Hasil, Ph.D.
The OBJECTIVE is to describe a topic from mathematical analysis. It is necessary to go through many books, search the internet and libraries to be able to fully describe a given topic from its beginning, to follow its development, and explain methods of teaching of the given subject in the past. Moreover, the present state of the studied topic and modern teaching methods should be given as well as the comparison of the modern methods with the older (ancient) ones.For EXAMPLE, the studied topic can cover sequences and their limits, infinite series, differential equations, difference equations, and many others.
BEFORE initiating the formal application process to doctoral studies, all interested candidates are required to contact the potential supervisor because of the preliminary agreement.
Ordered structures – past, present, applications
Supervisor: doc. RNDr. Jan Paseka, CSc.
Uspořádané algebraické struktury tvoří jedny z nejvíce studovaných struktur v algebře. Pozornost je věnována hlavně problematice 19. a 20. století a speciálně české matematice; nejsou však opomíjeny ani biografické a bibliografické aspekty.
Perception of mathematical concepts and results in the context of personal typology
Supervisor: prof. RNDr. Jan Slovák, DrSc.
ZÁMĚR VÝZKUMU: Je všeobecně známo, že vnímání a zpracování informací při učení i práci velmi závisí na typologii osobnosti. Výzkum by se měl zaměřit na specifický dopad typologie v kontextu matematiky.CÍLE VÝZKUMU: Na základě Jungovy osobnostní typologie a jejího rozpracování v personalistice (případně jiných přístupů k typům osobnosti, viz https://cs.wikipedia.org/wiki/Temperament) bude provedeno a vyhodnoceno šetření rozdílností ve vnímání, chápání a používání matematických nástrojů v závislosti na typologii žáků i učitelů.
Dle zájmu a možností budou zahnuty různé typy činností (středoškolská/vysokoškolská úroveň vzdělávání, přednášky/prezentace/samostatná práce apod.)
PŘEDPOKLADY: Pro výzkum bude potřebná alespoň rámcová orientace v teori osobnostních typů, např. původní teorie Junga a indikátory MyersovéBriggsové (viz https://cs.wikipedia.org/wiki/MyersBriggs_Type_Indicator) a přiměřená znalost statistických metod pro vyhodnocování šetření.
V případě zájmu kontaktujte přímo Jana Slováka na slovak@muni.cz.
Popularization of Mathematics
Supervisor: doc. Mgr. Petr Zemánek, Ph.D.
Matematika není (asi ani nikdy nebyla a nebude) mezi nejoblíbenějšími předměty většiny žáků a studentů základních či středních škol. Vzdělávacích pořadů/programů (nejev v TV) o matematice je málo, a těch kvalitních je jen hrstka. To je možná jeden z důvodů, proč se matematika postupem času dostává na okraj zájmu studentů.Prvním cílem výzkumu je zmapování současného stavu způsobu výuky matematiky (včetně nadstandardních kroužků, seminářů atd.) na vybraných školách, nabídku popularizační aktivit a odezvu ze strany škol. Druhý cíl výzkumu je pak zcela praktický a může zahrnovat např. vytvoření popularizačních interaktivních přednášek, programu matematického kroužku, matematického kalendáře pro veřejnost či popularizačního videokanálu.
V případě zájmu o toto téma kontaktujte přímo Petra Zemánka.
Specialization: Geometry, Topology and Geometric Analysis
Algebraic methods in geometric analysis
Supervisor: prof. RNDr. Jan Slovák, DrSc.
RESEARCH AREA OBJECTIVES: The geometric structures allow treating differential operators and their symmetries in a systematic way, both locally and globally. The projects will mainly enhance fundamental understanding of the features of particular geometric structures and differential operators enjoying the relevant symmetries or develop applications based on understanding the role of such (hidden) symmetries.AIM: The research could be based on Cartan geometries on filtered manifolds, extending the applications of tractor calculi and BGG machinery, including the relevant representation theory. The algebraic tools typically extend the features of analytical objects on homogeneous spaces to curved situations.
PROJECT EXAMPLES
 Semiholonomic Verma modules and tractor calculi for parabolic geometries.
 The cohomological structure of BGGs in singular characters.
 Extension of tractor calculi for particular Cartan geometries.
 Geometry of PDEs.
Geometric analysis in applications
Supervisor: prof. RNDr. Jan Slovák, DrSc.
RESEARCH AREA OBJECTIVES: In many applications, various concepts of symmetries are at the core of the available methods. The goal of the research will be to elaborate methods of differential geometry in various areas, e.g., Optimal Control Theory, and Mathematical Imaging and Vision.AIM: Based on specific geometries on filtered manifolds, we should like to develop new approaches to standard problems in Geometric Control Theory or in Imaging and Vision, including software implementation of the relevant procedures.
PROJECT EXAMPLES:
 Exploitation of nonholonomic equations for extremals in subRiemannian geometry.
 Tractography in diffusion tensor imaging
Invariants and symmetries of CR manifolds
Supervisor: doc. RNDr. Martin Kolář, Ph.D.
RESEARCH AREA:
Complex analysis in several variables leads naturally to geometric problems concerning boundaries of domains, and more generally real submanifolds of the complex space (so called CR manifolds). One of the main objectives is to understand symmetries of such manifolds and invariants with respect to holomorphic transformations.
PROJECT EXAMPLES:
 Classification problems for hypersurfaces of finite type in C^N
 Invariants and symmetries of uniformly Levi degenerate manifolds
 Dynamics of CR vector fields
Resolvents of hyperoperads
Supervisor: RNDr. Martin Markl, DrSc.
RESEARCH AREA OBJECTIVES: The fundamental feature of BataninMarkl operadic categories is that the objects under study are viewed as algebras over (generalized) operads in a specific operadic category. For instance, operads are algebras over the terminal operad in the operadic category of rooted trees, modular operads are algebras over the terminal operad in the operadic category of genusgraded connected graphs, wheeled PROPs are algebras over directed graphs, &c. Moreover, operadic categories provide natural environments for Batanin's noperads, tubings on a graph, decomposition spaces, decalage comonads, and other exotic structures. Operadic categories offer a concise framework for constructing infinity versions of operadlike objects. Operadic Grothendieck's is available a powerful tool for obtaining new operadic categories from old ones.AIM: Investigate applications of operadic categories in homological algebra, category theory and differential topology.
PROJECT EXAMPLES:
Explicit formulas for strongly homotopy operads of various particular types.
Connection between free hyperoperads and blob complexes.
LITERATURE:
[1] M. Markl, S. Schnider, J. Stasheff: Operads in Algebra, Topology and Physics, Series Mathematical Surveys and Monographs, volume 96. American Mathematical Society, Providence, Rhode Island, 2002.
[2] M. Markl, M. Batanin: Operadic categories and duoidal Deligne's conjecture, Advances in Mathematics 285(2015), 16301687. Available as preprint arXiv:1404.3886.
[3] M. Markl, M. Batanin: Koszul duality in operadic categories, arXiv:1404.3886.
[4] M. Batanin, M. Markl, J. Obradovič: Minimal models for (hyper)operads governing operads, and PROPs, work in progress.
[5] S. Morrison, K. Walker: The blob complex, preprint arXiv:1009.5025.
Specialization: Mathematical Analysis
Boundary value problems for functional differential equations
Supervisor: Mgr. Robert Hakl, Ph.D.
Differential equations with argument deviations are important for applied science and arise frequently in population dynamics, epidemiology, economy (in particular, as models of capital growth) and many other fields. Models of various real dynamical phenomena are frequently described by boundary value problems for system of functional differential equations. For such equations, the theory of boundary value problems, while very important by itself, is also of much interest in relation to the study of asymptotic properties of solutions on unbounded intervals.
The objectives include the investigation of the existence and uniqueness of a solution to boundary value problems for functional differential equations and systems in R^n and more general spaces and the study of their properties.
WWW: http://www.math.cas.cz/homepage/main_page.php?id_membre=19
Notes
The research topic and supervisor needs to be approved by the Scientific Board of the Faculty of Science.
Limit periodic and almost periodic linear difference systems
Supervisor: doc. RNDr. Michal Veselý, Ph.D.
Many phenomena in nature have oscillatory character and their mathematical models have led to the research of limit periodic and almost periodic sequences. Linear difference equations are often used in such models. Thus, the aim is to analyse the behaviour of solutions of limit periodic and almost periodic homogeneous linear difference systems whose coefficient matrices belong to transformable groups or to commutative groups whose boundedness is not required. In particular, the attention is paid to special systems with solutions which vanish at infinity or which are not asymptotically almost periodic.
Concerning examples, see:
1. M. Veselý; P. Hasil. Asymptotically almost periodic solutions of limit periodic difference systems with coefficients from commutative groups. Topological Methods in Nonlinear Analysis, 2019, 54, no. 2, 515535. ISSN 12303429. doi:10.12775/TMNA.2019.051. 2. M. Veselý; P. Hasil. Limit periodic homogeneous linear difference systems. Applied Mathematics and Computation, 2015, 265, August, 958972. ISSN 00963003. doi:10.1016/j.amc.2015.06.008. 3. M. Chvátal. Almost periodic solutions of limit periodic and almost periodic linear difference systems. Ph.D. thesis, MU, Brno, 2016.
Notes
Before initiating the formal application process to doctoral studies, all interested candidates are required to contact Michal Veselý.
Oscillation and spectral theory of Hamiltonian and symplectic systems
Supervisor: prof. RNDr. Roman Šimon Hilscher, DSc.
The objecttive is to study qualitative theory of linear Hamiltonian differential systems (also called canonical systems of differential equations) and their discrete time counterparts  symplectic difference systems. The obtained results may also contribute to other related areas of mathematics, such as to the theory of caluclus of variations, optimal control theory, or matrix analysis. Particular topics include oscillation and eigenvalue theory for systems without controllability condition, theory of principal solutions, comparative index theory, Riccati differential and difference equations, SturmLiouville equations, Jacobi equations.Before initiating the formal application process to doctoral studies, the interested candidates are required to contact the potential advisor for informal discussion.
Oscillation theory of differential and difference equations
Supervisor: doc. Mgr. Petr Hasil, Ph.D.
The OBJECTIVE is to obtain new criteria of oscillation and nonoscillation for differential and/or difference equations. Of course, there is a close connection to asymptotic theory and it is possible to focus to dynamic equations on time scales which cover differential and difference equations as their special cases.For EXAMPLE, the focus can be to halflinear equations, conditional oscillation of dynamic equations on time scales, etc.
BEFORE initiating the formal application process to doctoral studies, all interested candidates are required to contact the potential supervisor because of the preliminary agreement.
Spectral Theory of Discrete Symplectic Systems
Supervisor: doc. Mgr. Petr Zemánek, Ph.D.
The spectral theory of linear operators acting on a (finite/infinite dimensional) Hilbert space is a classical topic in functional analysis. The development of this theory for operators associated with differential equations or systems can be seen (from the mathematical point of view) as one of the cornerstones in mathematical physics. Roughly speaking, quantum mechanics is Hilbert space theory (or vice versa). Although several natural phenomena show that difference equations or systems should not be ignored in this direction, the spectral theory of linear operators associated with difference equations or systems remains currently underdeveloped. Fortunately, this topic attracts more and more attention in the last two decades and it is the main object of this research project. In particular, selfadjoint extensions and their spectrum or boundary triplets for these operators can be studied.Before initiating the formal application process to doctoral studies, all interested candidates are required to contact Petr Zemánek.
Specialization: Probability, Statistics and Mathematical Modelling
Functional Data Analysis
Supervisor: doc. Mgr. Jan Koláček, Ph.D.
Objectives: Statistical methodologies dealing with functional data are called Functional Data Analysis (FDA), where the term “functional” emphasizes the
fact that the data are functions characterizing the curves and surfaces.
Aim: The theoretical aspects of FDA will be developed in more detail,
especially connected to practical situations. Our aim is to take up these challenges by giving both theoretical and practical supports for more flexible models.
Examples of potential student doctoral projects:
 Semiparametric models in functional data analysis
 Discriminant analysis for functional data
 Nonparametric regression in functional data analysis
 Functional data analysis for irregular data
Multivariate statistical methods in (proteo)mics
Supervisor: doc. PaedDr. RNDr. Stanislav Katina, Ph.D.
Cílem výzkumného zaměření je studium a vývoj vybraných mnohorozměrných statistických metod v (proteo)mice, např. analýza hlavních komponent, parciální metoda nejmenších čtverců, problematika chybějící pozorování, a to jak z pohledu numerickématematického, tak z pohledu mnohorozměrných statistických vizualizací. Vlastnosti těchto metod budou hodnoceny pomocí různých simulačních studií. Metody budou implementovány v jazyce R a aplikovány na reálná data z oblasti medicíny. Toto zaměření vznikolo ve spolupráci s Oddělením biostatistiky FNUSAICRC v Brně.
Study information
Provided by  Faculty of Science  

Type of studies  Doctoral  
Mode  fulltime  Yes 
combined  Yes  
Study options  singlesubject studies  No 
singlesubject studies with specialization  Yes  
major/minor studies  No  
Standard length of studies  4 years  
Language of instruction  Czech  
Doctoral board and doctoral committees 