Creative research conducted in a field of applied mathematics and co-operation with experts in other engineering and scientific fields are major focuses of the study that the students should master. They are encouraged in a maximum degree possible to engage in research projects of the Institute of Mathematics at the BUT Faculty of Mechanical Engineering. Computers are available to students as standard equipment.

prof. RNDr. Miloslav Druckmüller, CSc.

1. Finite commutative rings in cryptography and coding theory

   Finite commutative ring theory is a fast-developing subject and has recently been seen to have important applications in theoretical areas like combinatorics, finite geometries and in the analysis of algorithms. Moreover, in the last twenty years, there has been a growing interest in applications of commutative rings to algebraic cryptography and coding theory. PhD topic is focused on a research in polynomial automorphisms over finite commutative rings and fields, Boolean polynomials and applications in multivariate cryptography, linear and Reed-Muller codes.

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

2. Fuzzy Stochastic Models of Reliability

   Fuzzification of the probability distributions for reliability modeling of elements and systems by means of the fuzzy numerical characteristics of vague times between failures. Account of their properties, PC implementation of algorithms, and applications.

   Tutor: Karpíšek Zdeněk, doc. RNDr., CSc.

3. Mathematical Description of Electromagnetic Pulse Energy Center Velocity in the Case of Pulse Transfer of Informations in Dispersive Medium.

   Applications of tools of informatics, computer science and numerical mathematics for the description of motion of an electromagnetic pulse in dispersive medium. This approach shall be exiting from the solution of an equation describing these sorts of waving, which is identical, from the mathematical point of view, with the relativistic wave equation. It is possible to make an effort to apply the Vainshtein generalized definition of the group velocity of a pulse, eventually another definitions of this velocity, to various types of dispersive media and to different types of input pulses. The applications is expected in the pulse transfer of informations for example in waveguides, optical fibres and optical cables, especially in the case of the nanosecond pulses.

   Tutor: Klapka Jindřich, doc. RNDr., CSc.

4. Morphisms of Polynomial and Local Algebras in Geometry and Cryptography

   PhD topic is focused on a research in a field of rings of polynomials in more indereminates over fields or integral domains and local rings expressed as finite-dimensional factor rings with an emphasis to their homomorphisms, especially automorphisms. Mainly, the project follows up such properties of polynomial or local algebra automorphisms which have applications in geometry and cryptography. Hence, in particular, automorphisms of Weil algebras (with applications in the searching for natural oprators in differential geometry) and polynomial automorphisms over finite fields (with applications in multivariate cryptosystems) will be studied in more details. The topic continues both earlier research of the supervisor (especially classification results of special Weil algebras and Weil bundles) and highly up-to-date research of cryptographic community (e.g. Patarin’s attacks to TTM systems).

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

5. Nontraditional estimates of numerical characteristics of probability distribution

   Estimates of numerical characteristics of observed random variables by the help of metrics and modified random samples. Description of their properties and software implementation. Applications in the technical and economical works.

   Tutor: Karpíšek Zdeněk, doc. RNDr., CSc.