# State Exam

## Basic information

Go through the following information and find out all you need to know:

Study and Examination Rules, Doctoral Study Code, and the Dean's Directive for State Doctoral Examinations.

## State Exam Topics

**Algebraic Methods of Computer Scienc**- 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.

**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

**Axiomatic foundations of Quantum theory**- Operator-algebraic approach
- Convex approach

*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

**Discrete Mathematics**- Graph Theory and its applications
- Combinatorial algorithms
- Complexity Theory
- Automata Theory and its applications
- Languages and grammars, decidability

**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

**Logic**- Deductive systems and matrix semantics
- Algebraisable logics
- Modal logics and Kripke semantics
- First-order definability of modal logics

*Mathematical Methods in Signal and System Theory*- Multidimensional signals and systems
- Wavelet basis and wavelet transform

**Nonlinear Functional Analysis**

1. Structure of Banach spaces

2. Differentiability of functions and null sets in Banach spaces

3. Linearization of maps between Banach spaces

4. Lipschitz-Free spaces and their applications

5. Uniform homeomorphisms between Banach spaces**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

**Probability and Statistics**- Multidimensional statistical analysis
- Linear and non-linear regression
- Estimation and approximation of probability density functions
- Statistical methods based on information theory

**Quantum Structures**- Quantum logics and effect algebras
- Measures on quantum structures

**Theory of Operator Algebras**- C*-algebras
- Von Neumann algebras
- Noncommutative measure and probability theory
- Jordan algebras