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. Analysis of dynamical systems on graphs

   Mathematical models describing population (eventually other) dynamics usually assume a spatially isolated domain where the particular populations (or phenomena) are observed. Thus we do not care about a migration (a spatial distribution) of the population in the domain. If we wish to take it into account, a typical approach consists of adding a diffusive term which, however, yields a system of partial differential equations, hence a much more complex problem. As an alternative approach, it is possible to utilize a graph model that is convenient in situations when observing dynamics on a finite number of outlying territories between which the observed populations somehow interact. The advantage of such an approach is the same mathematical description as of the original non-graph model, only the dimension is higher. The work will be focused on analysis of selected graph models (typical tasks are stability, asymptotics, bifurcations, conditions for the onset of nonlinear phenomena, etc.).

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

2. Applications of geometric algebras in engineering

   Geometric algebras (GA) have been successfully applied in many fields of theoretical and applied sciences. In the latter case, particularly image processing, computer vision, machine learning, robotics etc. are of our interest, including neural networks construction. Still new algebras are developing to match some specific area and thus apart from standard Conformal GA, we study GA for Conics or Projective GA for lines and planes manipulation. The structure of a GA must describe the problem effectively but also must provide a reduction of computational complexity and load. The applicant will be a part of an international research team and will describe a particular application of a specific algebra together with implementation and verification of the chosen approach.

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

3. Dual numbers, Weil algebras and applications

   The topic of the doctoral study is focused on research in the field of applications of quotient algebras of multivariable polynomials, where the prototypical case is the algebra of dual numbers widely used in kinematics. Weil algebras represent a more general model, which play an important role in differential geometry. Here, in particular, the case of non-homogeneous ideals has not yet been systematically investigated, and research in this area thus represents a new and demanding scientific research. Last but not least, one can focus on special subrings of the mentioned algebras, which can be suitable, for example, in a use in lattice cryptosystems.

   Tutor: Kureš Miroslav, doc. RNDr., Ph.D.

4. 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.

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

5. Modelling of Complex Dynamic Systems

   Complex systems and their dynamic behavior are at the center of interest across many scientific disciplines, ranging from physics to biology and economics. This dissertation focuses on the development and application of advanced mathematical methods for modeling and analyzing these systems. Special attention is given to fractional calculus, which enables the modeling of memory effects and internal connections in systems, and to delayed differential equations, which reflect time delays in interactions between system components. These methods offer new possibilities for the qualitative analysis and understanding of phenomena such as phase transition or chaos in complex dynamic systems.

   Tutor: Kisela Tomáš, Ing., Ph.D.

6. Numerical algroithms for fractional differential equations

   The topic of the study is focused on numerical analysis of initial value problems for fractional differential equations. Due to numerous engineering applications, the fractional differential equations theory is of great scientific interest. A number of methods that solve fractional differential equations are already described. Due to the nature of numerical schemes, we often face a great time-consuming calculation. In addition to research and analytical activities, the scope of work will also be the design and implementation of effective numerical algorithms (with the possibility of parallelization of calculations) in a suitable computing environment (Python, Matlab).

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

7. Periodic solutions to non-linear second-order ordinary differential equations

   We will study the existence and stability of periodic solutions to 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 example of such equation is the so-called Duffing differential equation, which is derived, for instance, when aproximating a non-linearity in the equation of motion of certain forced oscillators.

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

8. Some kinds of Lie groups and their physical applications

   Studies will be devoted to the general properties of some kinds of Lie groups, particularly to jet groups. In more general context, the investigations will be done on nilpotent and solvable groups. A considerable attention will be focused on applications in physics, particularly in the continuum mechanics.

   Tutor: Tomáš Jiří, doc. RNDr., Dr.

9. Topological and combinatorial mehods for the study of connectedness of digital images

   The topic is oriented on finding and studying convenient structures on the digital plane by using tools of the graph theory and general topology. We will be interested in structures providing definitions of connectedness and possessing analogues of the Jordan curve theorem. The research is motivated by applications of the obtained results for solving problems of digital image processing.

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