State doctoral exam topics
Mathematical Engineering
- Discrete Mathematics
- Graph Theory and its applications
- Combinatorial algorithms
- Complexity Theory
- Automata Theory and its applications
- Languages and grammars, decidability
- Applied Algebra
- The notion of a group, its generalizations and applications
- Lattices, Boolean algebras and their application in logics
- Universal algebra, varieties and quasivarieties
- Matrix calculus, applications to signal processing
- The spectral decomposition of a matrix, singular value decomposition of a matrix (SVD).
- Computational algebraic geometry
- Numerical Analysis
- Basic algorithms of matrix algebra and their computational complexity
- Numerical methods of computation of the spectrum of matrix
- Principle of iterative methods, examples of applications in linear algebra and calculus
- Least Squares Method. Minimization o functions.
- Solution to the Cauchy problem for ordinary differential equations
- Functional Analysis
- Duality and linear operators on Banach spaces
- Banach algebras
- Spectral theory of operators on Hilbert spaces
- Distributions and Fourier transform
- Measures and probabilities on infinitedimensional spaces
- Theory of Operator Algebras
- C*-algebras
- Von Neumann algebras
- Noncommutative measure and probability theory
- Jordan algebras
- Axiomatic foundations of Quantum theory
- Operator-algebraic approach
- Convex approach
- Quantum Structures
- Quantum logics and effect algebras
- Measures on quantum structures
- Probability and Statistics
- Multidimensional statistical analysis
- Linear and non-linear regression
- Estimation and approximation of probability density functions
- Statistical methods based on information theory
- Mathematical Methods in Signal and System Theory
- Multidimensional signals and systems
- Wavelet basis and wavelet transform
- Algebraic Methods of Computer Science
- Domain theory and its application to semantics of programming languages
- Algebraic and coalgebraic specification
- Algebraical Structures
- Lie groups and algebras. Representations of semisimple algebras.
- Nonassociative algebras. Alternative and composition algebras.
- Logic
- Deductive systems and matrix semantics.
- Algebraisable logics.
- Modal logics and Kripke semantics.
- First-order definability of modal logics.
- Category Theory
- Adjunctions, monads, Beck's Theorem.
- Basics of enriched category theory, weighted limits and colimits, categories of presheaves.
- Free cocompletions under a class of colimits.
- Two-dimensional monads and two-dimensional Beck's Theorem.