Full-time study

4 years

prof. RNDr. Jan Čermák, CSc.

Chairman :
prof. RNDr. Jan Čermák, CSc.
Councillor internal :
prof. RNDr. Josef Šlapal, CSc.
prof. Ing. Ivan Křupka, Ph.D.
prof. RNDr. Miloslav Druckmüller, CSc.
doc. Mgr. Petr Vašík, Ph.D.
prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.
Councillor external :
doc. RNDr. Ing. Miloš Kopa, Ph.D.
prof. RNDr. Jan Paseka, CSc.
prof. RNDr. Roman Šimon Hilscher, DSc.
doc. RNDr. Tomáš Dvořák, CSc.

The doctoral study programme in Applied Mathematics will significantly deepen students' knowledge acquired during the study of the follow-up master's study programme in Mathematical Engineering at FME BUT in Brno and other master's programmes focused on mathematics and its applications. Students of this doctoral programme can gain in-depth knowledge of the relevant mathematical apparatus in all areas of applied mathematics, in connection with the solution of demanding practical tasks (especially technical). The offer of professional subjects of the doctoral study programme in Applied Mathematics is also adapted to this, including subjects with a deeper theoretical basis, subjects related to the applications of mathematics, and finally also subjects with a special engineering focus.
The topics of doctoral theses are listed mainly by the staff of the Department of Mathematics, and depending on the nature of the topic, experts from other FME institutes or other scientific institutions may also be involved, as specialist trainers. During their doctoral studies, students become members of scientific teams led (or in which they work) by their supervisors. The assigned topic of the doctoral thesis is usually part of a more complex problem that this team solves in various professional projects. Students will gradually learn all the basic principles of scientific work, especially the creation of professional texts and their publication in scientific journals, and the presentation of the results of their scientific work at seminars or conferences. Cooperation with foreign workplaces is a matter of course, where students can gain other useful experiences. After successfully passing the prescribed state doctoral exam, which examines both the knowledge of the theoretical foundations needed to master the topic, but also the state of development of the dissertation and the direction of research conducted within it, students focus primarily on completing their work. In order to submit it for defence, they must meet the requirements related primarily to publishing activities, the purpose of which is to ensure that dissertations submitted for defence in this study programme are at a comparable level to defended works at other mathematical institutions in the Czech Republic and abroad. After defending the doctoral thesis, students obtain a Ph.D degree.
The main goal of this doctoral study programme is to educate experts in the field of applied mathematics who will be able to continue in the scientific career begun within their doctoral studies. The means to fulfil this goal is to expand students' knowledge of non-trivial mathematical tools needed for modelling and solving practice problems, as well as to deepen the principles of their mathematical, logical and critical thinking.

The graduate will gain deep expertise in a number of special areas of modern applied mathematics, focusing on selected parts of image analysis, computer graphics, applied topology, 3D image reconstruction and visualization, continuous and discrete dynamical systems, and advanced statistical methods. They will also have a high degree of geometric perception of problems related to engineering applications. They will also gain quality knowledge of engineering disciplines related to the topic of work, and will be able to work with modern programming tools (Python, C ++, ...). The language equipment enabling professional cooperation with foreign workplaces and the presentation of the obtained results at an international forum is a matter of course.

Within the scope of his/her professional competence, the graduate is able to create mathematical models of engineering problems and, according to their nature, to search for and develop suitable mathematical tools and procedures for their solution. They are able to use mathematical software at a high level and has acquired programming skills. In a broader sense, the graduate is able to participate in solving challenging tasks in the field of technical practice.

In terms of more general skills, the graduate is capable of independent creative scientific work. They will learn the principles of teamwork at a high professional level. The team will learn to manage in terms of professional and administrative, it will also be familiar with project issues. He can also work as a mathematician in multidisciplinary teams. He is able not only to participate in solving research problems, but he can find and formulate current scientific problems. He is able to present the results of his work, both in the form of scientific publications and in the form of professional lectures.

The graduate will have a developed ability of analytical thinking, which in combination with knowledge of advanced methods of applied mathematics and computer technology will allow him to seamlessly participate in scientific teams in various types of academic institutions or in the field of applications.

Graduates find a wide job in the labour market for their adaptability, which is made possible by extensive knowledge of applied mathematics. These graduates are interested in companies engaged in development in the field of autonomous systems, robotics, automation and image analysis, as well as institutions engaged in science, research and innovation in the fields of informatics, technology, quality management, finance and data processing. Graduates of this doctoral study programme also find significant employment in the academic sphere. In addition to the Institute of Mathematics, FME (among whose employees the share of graduates of the doctoral study program Applied Mathematics reaches almost a quarter), these graduates currently work as academic staff at other FME institutes, other BUT faculties and other universities. In addition to adaptability in various areas of applied mathematics, the continuing interest in these graduates is mainly due to their scientific erudition (in many cases these graduates are already habilitated, and in increasingly monitored indicators publishing activities are often at the top of relevant educational institutions).

See applicable regulations, DEAN’S GUIDELINE Rules for the organization of studies at FME (supplement to BUT Study and Examination Rules)

The rules and conditions of study programmes are determined by:
BUT STUDY AND EXAMINATION RULES
BUT STUDY PROGRAMME STANDARDS,
STUDY AND EXAMINATION RULES of Brno University of Technology (USING "ECTS"),
DEAN’S GUIDELINE Rules for the organization of studies at FME (supplement to BUT Study and Examination Rules)
DEAN´S GUIDELINE Rules of Procedure of Doctoral Board of FME Study Programmes
Students in doctoral programmes do not follow the credit system. The grades “Passed” and “Failed” are used to grade examinations, doctoral state examination is graded “Passed” or “Failed”.

Brno University of Technology acknowledges the need for equal access to higher education. There is no direct or indirect discrimination during the admission procedure or the study period. Students with specific educational needs (learning disabilities, physical and sensory handicap, chronic somatic diseases, autism spectrum disorders, impaired communication abilities, mental illness) can find help and counselling at Lifelong Learning Institute of Brno University of Technology. This issue is dealt with in detail in Rector's Guideline No. 11/2017 "Applicants and Students with Specific Needs at BUT". Furthermore, in Rector's Guideline No 71/2017 "Accommodation and Social Scholarship“ students can find information on a system of social scholarships.

The doctoral study programme in Applied Mathematics follows on from the follow-up master's study programme in Mathematical Engineering, which is accredited (and taught) at FME BUT in Brno.

1. round (applications submitted from 01.04.2026 to 31.05.2026)

1. Analýza vlastních čísel a vektorů matice

   Eigenvalues, together with eigenvectors of a matrix, have wide applications in many areas (e.g. solving systems of linear differential or difference equations, convergence criteria of iterative methods for systems of linear equations, introduction of the spectral norm, etc.). The topic will include numerical solutions of eigenvalues and eigenvectors, as well as analysis of eigenvalues of special matrices.

   Supervisor: Tomášek Petr, doc. Ing., Ph.D.

2. Asymptotics of dynamic equations of real orders

   We shall study qualitative properties of various integer order as well as non-integer order nonlinear differential equations. Attention will be paid, among others, to the derivation of asymptotic formulas, which will significantly refine information about the behavior of the solution, or to the search for criteria for oscillation. We also want to focus on (new) phenomena that occur in equations of non-integer orders. We will consider not only differential equations, but also their discrete (or time scale) analogues. This will allow to compare and explain similarities or discrepancies between the continuous case and some of its discretization, to get an extension to new time scales, or to obtain new results e.g. in the classical discrete case through a suitable transformation to other time scale. It is expected that the results will be of importance also in the theory of stability.

   Supervisor: Řehák Pavel, prof. Mgr., Ph.D.

3. Boundary value problems for non-linear second-order ordinary differential equations

   We will study the existence and uniqueness of solutions to boundary value problems for non-linear second-order ordinary differential equations. We will focus on differential equations appearing in mathematical modelling, in particular, ordinary differential equations in mechanics. Typical examples of such equations are a non-autonomous Duffing differential equation, which arises, for instance, when approximating a nonlinearity in the equation of motion of certain oscillators, and a non-autonomous pendulum-like equation.

   Supervisor: Šremr Jiří, doc. Ing., Ph.D.

4. Design of Quantum Algorithms

   The aim of this PhD study is research and development in the design of quantum algorithms with a particular emphasis on variational quantum algorithms (VQAs), which represent one of the most promising classes of algorithms for current and near-term noisy intermediate-scale quantum (NISQ) devices. The research focuses on both theoretical and practical aspects of designing variational quantum circuits, optimization strategies, and hybrid quantum–classical computational frameworks.

   The PhD candidate will investigate the expressivity and trainability of variational ansätze, including challenges such as barren plateaus, robustness to noise, and errors inherent to quantum hardware. An important part of the research will involve the design of novel circuit architectures tailored to specific classes of problems. The work will also include the development and evaluation of algorithms for selected application domains, such as quantum chemistry, combinatorial optimization, and quantum machine learning.

   The research will be conducted at both a theoretical level and through numerical simulations and experiments on available quantum computing platforms. The PhD study is expected to result in publications in leading international journals and conferences and to involve active collaboration with national and international research groups in the field of quantum computing.

   Supervisor: Vašík Petr, doc. Mgr., Ph.D.

5. Digital Jordan curves and surfaces

   The goal of the topic is to find new structures on the 2D and 3D digital spaces providing  concepts of connectedness convenient for defining digital Jordan curves and surfaces. The research will be oriented on structures based on the graph theory (adjacency relations), theory of n-ary relations, and general topology (closure operators). The results attained will be compared with each other and also with the known methods of structuring the digital space (classical adjacencies, Khalimsky topology, etc.). Their contribution will consist in new definitions of Jordan curves and surfaces enlarging thus the known variety of the curves and surfaces and increasing the efficiency of some algorithms of digital image processing (related to borders of the objects imaged like segmentation, pattern recognition, memory compression etc.).

   Supervisor: Šlapal Josef, prof. RNDr., CSc.

6. Functional differential equations

   Functional differential equations are a generalization of ordinary differential equations. One of their further specification leads to equations with delayed argument. Their advantage is that in some cases they can better model the real situation than ordinary differential equations. Apart from delayed equations we will also handle advanced differential equations because this has not been considered seriously so far. We shall mainly focus on qualitative analysis of particular functional differential equations which are derived from real models.

   Supervisor: Opluštil Zdeněk, doc. Mgr., Ph.D.

7. Geometric control of quantum systems

   The quantum process can be described by an evolutionary operator and thus viewed as a curve in a unitary Lie group. Quantum geometric control optimizes quantum operations by treating them as geodesic curves, aiming to minimize time, energy, or errors. Key problems include finding optimal control pulses and implementing sub-Riemannian, time-optimal maneuvers for manipulating qubit states.

   Supervisor: Návrat Aleš, doc. Mgr. et Mgr., Ph.D.

8. Multivariate Extreme Value Models in Financial Sector

   Abrupt and unexpected fluctuations in financial markets lead to extreme movements in asset prices and indices, with historical crises confirming their profound impact on the global economy. This study focuses on identifying and testing suitable models for capturing the simultaneous extreme behavior of multiple assets. A key objective is an in-depth analysis of these models' properties, with an emphasis on their predictive capabilities during periods of market instability.

   Supervisor: Hübnerová Zuzana, doc. Mgr., Ph.D.

9. Nonlinear dynamical systems and their applications

   Nonlinear dynamical systems (continuous or discrete) exhibit, in general, a more complex behaviour than linear ones. A typical feature is that a change in a system's parameter can cause a complete change in the system's qualitative behaviour, leading to the so-called bifurcation. These bifurcations can even lead to a very complex behaviour called deterministic chaos. Over the last two decades, interest in dynamical systems has experienced a renaissance, with models reflecting the history of the state coming to the fore, either through a delayed argument or through the so-called fractional derivative. In many situations, such models can better capture reality.

   The PhD study focuses on selected mathematical models formulated as systems of nonlinear equations (both differential and difference equations). It is also possible to take into account fractional (i.e., non-integer order) and delayed equations (recent theoretical results allow a deeper analysis of such equations, which was not possible in the past). Regarding particular applications, it is possible to focus on models having applications, e.g., in flight dynamics or control theory.

   Supervisor: Nechvátal Luděk, doc. Ing., Ph.D.

10. Topological structures on categories

    The goal of the thesis is the study and comparison of different topological structures on categories like closure and interior operators, neighbourhood operators, convergence structures, topogenous orders, uniformity, proximity etc. This will lead to finding new approaches and obtaining new results describing behaviour of the structures and their relationships. It is also expected that some other classic topological structures (e.g., metrics and their generalizations) will be promoted onto the categorical level and then studied. An important feature of topological structures on categories is that they are pointless and, therefore, the results attained will contribute to the study of pointless topological structures.

    Supervisor: Šlapal Josef, prof. RNDr., CSc.