Mathematics and Statistics

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 units, 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 or https://is.muni.cz/th/atke4/?lang=en

Supervisor

prof. RNDr. Radan Kučera, DSc.

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

Supervisor

prof. RNDr. Jiří Rosický, DrSc.

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 quantum-physical 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/1197-jan-paseka/publications

Supervisor

prof. RNDr. Jan Paseka, CSc.

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.

Supervisor

doc. Mgr. Michal Kunc, Ph.D.

Enriched category theory provides one of the ways in which we can capture higher-dimensional categories. For instance, 2-categories, as studied by the Australian school, are enriched categories. Recently, the subject of (infinity,1)-categories, aka quasicategories, has been developed by Joyal and Lurie amongst others and Riehl and Verity have shown that these can also be captured using enriched category theory. There are many open questions and problems to be explored in this area, which involves a rich mixture of homotopy theory, enriched category theory and categorical universal algebra.

Supervisor

doc. John Denis Bourke, PhD

Supervisor

prof. RNDr. Jiří Rosický, DrSc.

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 star-free languages,

- Computational complexity of basic problems for finite semigroups.

Notes

My publications: http://www.math.muni.cz/~klima/Math/publications.html

Supervisor

doc. Mgr. Ondřej Klíma, Ph.D.

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_\infty-context
• Algorithmic aspects of the extension problem from the viewpoint of rational homotopy
theory

In case of interest, contact Lukas Vokrinek at koren@math.muni.cz.

Supervisor

doc. Lukáš Vokřínek, PhD.

Supervisor

doc. RNDr. Martin Kolář, Ph.D.

Přímkové plochy představují klasické téma s bohatou historií, zajímavými souvislostmi a slibnou budoucností. Cílem práce je shromáždit, přiblížit a rozvinout některá ze zmiňovaných hledisek. Od kandidáta se předpokládá trpělivost při studiu literatury (často starší a cizojazyčné) a tvůrčí přístup při výběru a zpracování nabytých poznatků.

Supervisor

doc. Mgr. Vojtěch Žádník, 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.

Supervisor

prof. Mgr. Petr Hasil, Ph.D.

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.

Supervisor

prof. RNDr. Jan Paseka, CSc.

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

• Semi-holonomic 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.

In case of interest, contact directly Jan Slovak at slovak@muni.cz.

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 non-holonomic equations for extremals in sub-Riemannian geometry.
• Tractography in diffusion tensor imaging

In case of interest, contact directly Jan Slovak at slovak@muni.cz.

Supervisor

prof. RNDr. Jan Slovák, DrSc.

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

Supervisor

doc. RNDr. Martin Kolář, Ph.D.

RESEARCH AREA OBJECTIVES: The fundamental feature of Batanin-Markl 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 genus-graded connected graphs, wheeled PROPs are algebras over directed graphs, &c. Moreover, operadic categories provide natural environments for Batanin's n-operads, tubings on a graph, decomposition spaces, decalage comonads, and other exotic structures. Operadic categories offer a concise framework for constructing infinity versions of operad-like 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), 1630-1687. 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.

Supervisor

RNDr. Martin Markl, DrSc.

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.

Supervisor

Mgr. Robert Hakl, Ph.D.

The objective is to study asymptotic and oscillation theory of differential equations and differential systems of real orders.

Before initiating the formal application process to doctoral studies, the interested candidates are required to contact the potential advisor for informal discussion.

Supervisor

prof. RNDr. Zuzana Došlá, DSc.

Many phenomena in nature have oscillatory character and their mathematical models have led to the research of limit periodic, almost periodic, and asymptotically almost periodic sequences. In particular, the attention is paid to special constructions of such sequences in general metric spaces.

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, 515-535. ISSN 1230-3429. doi:10.12775/TMNA.2019.051. 2. M. Veselý; P. Hasil. Values of limit periodic sequences and functions. Mathematica Slovaca, 2016, 66, no. 1, 43-62. ISSN 0139-9918. doi:10.1515/ms-2015-0114. 2. M. Veselý. Construction of almost periodic sequences with given properties. Electronic Journal of Differential Equations, 2008, 2008, no. 126, 1-22. ISSN 1072-6691.

Notes

Before initiating the formal application process to doctoral studies, all interested candidates are required to contact Michal Veselý

Supervisor

prof. RNDr. Michal Veselý, Ph.D.

Partial differential equations (PDE) have important applications in science and engineering. In the realm of linear theory, solutions of PDEs obey the principle of linear superposition, and in some cases, they possess explicit analytical expressions. However, the laws of the nature are not always linear, and nonlinear PDEs play an essential role in modeling these phenomena. The research objective is to bring into light and explain nonlinear phenomena stemming from nonlinear PDEs in connection with singularity theory.

Interested candidates are required to contact directly Phuoc-Tai Nguyen (via email: ptnguyen@math.muni.cz) for informal discussions before initiating the formal application process to doctoral studies.

Supervisor

doc. Phuoc Tai Nguyen, PhD

The OBJECTIVE is to obtain new criteria of oscillation and non-oscillation 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 half-linear 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.

Supervisor

prof. Mgr. Petr Hasil, Ph.D.

The objective 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. In particular, we are interested in the oscillation and eigenvalue theory for systems without controllability condition, theory of principal solutions, comparative index theory, Riccati differential and difference equations, Sturm--Liouville equations, Jacobi equations. 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.

Before initiating the formal application process to doctoral studies, the interested candidates are required to contact the potential advisor for informal discussion.

Supervisor

prof. RNDr. Roman Šimon Hilscher, DSc.

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, self-adjoint 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.

Supervisor

doc. Mgr. Petr Zemánek, 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 support 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

Supervisor

doc. Mgr. Jan Koláček, Ph.D.

RESEARCH AREA OBJECTIVES:
The research focuses on the qualitative theory of nonlinear dynamical systems, with emphasis on bifurcation analysis, routes to chaos, and multiscale dynamics. A central theme is the study of transitions between dynamical regimes - such as equilibria, oscillations, bursting, and chaotic behavior - using tools from dynamical systems theory. Particular attention is devoted to slow–fast systems, high-dimensional models, and networks of coupled units, where complex collective dynamics and emergent phenomena arise. Applications are mainly motivated by problems in computational neuroscience and  biological systems.

AIM:
The aim is to develop and apply modern analytical and numerical methods for the study of bifurcations and invariant structures (e.g., equilibria, limit cycles, and invariant tori), including their stability and interactions, with a strong emphasis on applications in real-world systems. In particular, the research is focused on understanding dynamical mechanisms underlying neuronal activity, such as oscillations, bursting, synchronization, and transitions to pathological states.

The project emphasizes continuation techniques and geometric approaches to uncover the structure of phase space and transitions to complex dynamics in models arising from computational neuroscience and related biological systems. Selected topics may incorporate stochastic effects and, where appropriate, data-driven approaches to support model identification, parameter estimation, or validation against experimental data. Computational implementation and numerical experimentation form an integral part of the research, with the goal of linking theoretical insights to observable phenomena.

PROJECT EXAMPLES:

• Bifurcation analysis of slow–fast systems and mechanisms of bursting and transitions to chaos.
• Continuation and stability analysis of invariant objects (equilibria, limit cycles, invariant tori).
• Dynamical analysis of coupled neuronal systems and synchronization phenomena.
• Pattern formation and its bifurcation mechanisms in nonlinear systems.
• Nonlinear dynamics and pattern formation in computational neuroscience
• Bifurcation analysis of dynamical systems on networks.
• Influence of stochastic perturbations on bifurcation structure and dynamical transitions.
• Data-assisted reconstruction or continuation of dynamical regimes from time series.

In case of interest, contact directly Lenka Přibylová at pribylova@math.muni.cz.

Supervisor

doc. RNDr. Lenka Přibylová, Ph.D.

Cílem tohoto zaměření je systematicky zhodnotit a dále rozvinout moderní statistické metody pro statistickou inferenci se zaměřením na neparametrické a (semi)parametrické modely, robustní inferenci, řešení porušení modelových předpokladů a konstrukci intervalových odhadů, včetně jejich implementace v reprodukovatelném softérovém prostředí.
 
Příklady potenciálních doktorských projektů:
 
- Moderní statistické metody pro analýzu přežití
- Moderní statistické metody v klinickém výzkumu
- Moderní statistické metody v bio-medicínském výzkumu
- Statistická inference pro mnohorozměrné funkcionální data
- Statistická inference pro mnohorozměrné data

Supervisor

doc. PaedDr. RNDr. Stanislav Katina, Ph.D.