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. 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.

2. Asymptotics and oscillation of dynamic equations of real orders

   We shall study qualitative properties of various integer order as well as non-integer order nonlinear differential equations, which arise from aplications. The research will be focused, for example, on obtaining asymptotic formulae for solutions or establishing new oscillation criteria. Further we want to concentrate on (new) phenomena arising in equations of non-integer orders. We shall deal not only with differential equations but also with their discrete (or time scale) analogues. This will enable us to compare and explain similaritities 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.

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

3. Differential equations in control theory

   Control thory provides a significant application area of differential equations. Standard as well as novel types of differential equations are studied in the frame of this theory. Reaction time delay of a controlled system, as well as an order of this system, provide important tools in the control process. Stabilization, destabilization, synchronization, optimization a chaotification of studied system belong among basic issues. The topic of this doctoral study will be focused on analysis of these properties for corresponding types of differential equations, and on related numerical and graphical simulations.

   Tutor: Čermák Jan, prof. RNDr., CSc.

4. 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.

5. Dynamical properties of conformal geometric algebra.

   Conformal geometric algebra makes it possible to work with Euclidean geometry extended with spheres. Objects are realized as elements in algebra up to the multiple (in the projective class). However, the norm of the given object contains some information that can be used when analyzing the dynamic behavior of the object. For example, a straight line is formed by two points, but the wedge of these two points at infinity contains information about their distance. The aim of the thesis is to investigate the use of this fact with respect to traditional and new CGA applications.

   Tutor: Hrdina Jaroslav, doc. Mgr., Ph.D.

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. More precisely, we shall study oscillatory properties of solutions to the considered equations.

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

7. Geometric optimal control on Lie groups

   Lie groups and algebras are mathematical structures that appear naturally in control theory. They represent the symmetries of the given dynamical system or the symmetries of its approximation. These symmetries can be used to simplify the relevant differential equations, for geometric discretization and more stable numerical solution.

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

8. Integrated nested Laplace approximation in statistical inference

   Statistical inference in some models with latent variables can not be based on an analytical approach. An approximation needs to be taken into account. There are several possible approaches to the approximation, for example, time-demanding MCMC or Integrated nested Laplace approximation. The aim of the study would be research in the possible application and properties of the Integrated Laplace approximation in selected models.

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

9. Mathematical methods of spectral X-ray computed tomography data processing

   Spectral computed tomography (SpCT) allows for quantitative, non-destructive measurements of the chemical composition of samples. It is a well-known technique in medicine, and an emerging tool in biology, geology, and materials science. In these fields, SpCT has the potential to enhance existing image processing workflows and open entirely new possibilities of sample analysis by providing information about the 3D spatial distribution of physical parameters like atomic number, concentration of specific compounds, or density. These can then be correlated with other measurement modalities and known parameters of the scanned samples, resulting in a uniquely powerful inspection tool. However, this requires a considerable amount of development, as the use of SpCT outside of medicine is still novel and relatively unexplored. This topic focuses on the practical application of SpCT for quantitative measurements in materials science and related interdisciplinary fields, connecting mathematical theory with physics and modern engineering.

   Tutor: Štarha Pavel, doc. Ing., Ph.D.

10. Nonlinear dynamical systems and their applications

    Nonlinear dynamical systems (continuous or discrete) exhibit, in general, a more complex behavior than linear ones. The typical feature is that the change in a system's parameter can cause a complete change of the qualitative behavior of the system, the system undergoes the so-called bifurcation. These bifurcations can even lead to a very complex behavior called deterministic chaos. About the last two decades, the interest in dynamical systems has experienced a certain renaissance in the sense that models reflecting the history of the state come to the fore, either through a delayed argument or through the so-called fractional derivative. It turned out that, in many situations, such models are able to capture reality better. The topic of PhD study is focused on selected mathematical models using systems of nonlinear equations (both differential or difference ones). 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 used, e.g., in flight dynamics or control theory.

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

11. 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).

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

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

    We shall 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.

13. Rational movements in kinematics in geometric algebra approach.

    Geometric algebras as a generalization of quaternions are commonly used for both forward and inverse kinematics problems. However, the actual trajectory of the proposed movement is not addressed. The goal of the work is to open up this topic, primarily with regard to the rationality of seated movement curves, which is essential from an implementation point of view.

    Tutor: Hrdina Jaroslav, doc. Mgr., Ph.D.

14. 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.

15. Topological and combinatorial mehods for the study of connectedness in the digital plane and space

    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.