Faculty of Applied Mathematics

(The committee will ask several questions for the selected topic.)

1. Power series, incl. convergence and methods for expanding functions into power series.
2. Extrema (local, conditional, and global) of functions of several variables.
3. Fourier series, incl. trigonometric series and its generalizations in Hilbert space.
4. Green's and Stokes' theorems and their consequences in vector field theory.
5. Laurent series, the residue of a function, and its applications.
6. Existence and uniqueness of the solution to an ordinary differential equation of order 1.
7. The one-dimensional wave equation (the problem of a vibrating string), methods for solving under various initial-boundary conditions.
8. Transformation of a matrix to a diagonal form and to a Jordan canonical form.
9. Quadratic form, its definiteness, and its canonical form. 
10. Transport of topology (incl. quotient topology), Tychonoff's theorem.
11. Banach’s and Brouwer's fixed-point theorems, and their applications.
12. Types of convergence of sequences of random variables, limit theorems in probability theory.
13. Generating functions and their applications.
14. Max-min theorems in combinatorics (e.g., Hall's theorem).
15. Cantor-Bernstein theorem. Countable and uncountable sets, examples. Generalized continuum hypothesis.