Mathematical analysis - topics for the oral part of the final exam

For students, who started their studies in 2022 or later

The knowledge contained in the topics are taught in the mandatory courses of the study program Mathematical analysis or  contained in the entry requirements for this program. Knowledge contained in the entry requirements are taught in mandatory or elective courses of the program General mathematics, specialization mathematical analysis. A more detailed specification is given at each area individually.

(np) = the knowledge of a proof is not required

1. Real and complex analysis

Courses covering the required knowledge

NMMA205 Measure and Integral Theory 1 (entry requirements)
NMMA343 Measure and Integral Theory 2 (entry requirements)
NMMA403 Real functions 1
NMMA410 Complex analysis

Explanation:

Measure theory: signed measures, Hahn decomposition, Luzin theorem, Jegoroff theorem, Radon-Nikodým theorem, Lebesgue decomposition of a measure, Radon measures, product measures (Fubini theorem), substitution theorem (np).

Differentiantion of measures: derivatatives of measures, Vitali covering theorem, absolutely continuous functions and functions with bounded variation, Lebesgue points, Rademacher theorem.

Hausdorff measure and dimension: outer measure, measure and dimension, relationship to the Lebesgue measure, area formula (np).

Meromorphic functions: definition of a meromorphic function, meromorphic functions on the extended complex plane, meromorphic functions on the complex plane (function Gamma, Riemann zeta function), argument principle, Rouché theorem, Mittag-Leffler theorem, Runge Theorem

Conformal mappings: preserving angles, conformal mappings, inverses of holomorphic functions, Schwarz lemma, Riemann theorem

Harmonic functions of two variables: Relationship of harmonic and holomorphic functions, Poisson integral, mean value property, Schwarz reflexion principle.

Zeros of holomorphic functions: Infinite products, Weierstrass factorization theorem

2. Functional analysis

Courses covering the required knowledge

NMMA331 Introduction to functional analysis (entry requirements)
NMMA401 Functional analysis 1
NMMA402 Functional analysis 2

Explanation:

Locally convex spaces: definition, Minkowski functionals, generating using seminorms, bounded sets, continuous linear mappings, characterizations of metrizability and normability, separatation theorems.

Weak topologies: definitions of a topology generated by a space of linear forms and duality, weak topologies, Mazur theorem, polars, bipolar theorem, Banach-Alaoglu theorem, Goldstine theorem, weak compactness and reflexivity, Eberlein-Šmulyan theorem (np).

Spectral theory in Banach algebras: Basic properties of the spectrum and of the spectral radius, Gelfand-Mazur theorem, holomorphic calculus, properties of the Gelfand transform, Gelfand-Naimark theorem.

Spectrum of bounded and unbounded operators: Compact operators, basic notions concerning unbounded operators (symmetric operator, self-adjoint operator, closed operator, closure of an operator), definition and properties of the adjoint operator (also for an unbounded operator), properties of the spectrum of an unbounded operator.

Integral transformations: Definition of the Fourier transform on L1 and its basic properties, inversion theorem, Fourier transform of a convolution and of a derivative, Plancherel theorem.

Theory of distributions: Space of test functions and convergence in it, definition of a distribution, basic examples, charakterization of a distribution, order of a distribution, operations with distributions (differentiation, multiplying by a function), Schwarz space and tempered distributions, Fourier transform on the Schwarz space and of tempered distributions, basic properties.

3. Ordinary differential equations

Courses covering the required knowledge

NMMA336 Ordinary differential equations (entry requirements)
NMMA407 Ordinary differential equations 2

Explanation:

Basic properties of solutions: existence a uniqueness, continuous dependence on initial conditions, theorem on leaving a compact, differentiable dependence on initial conditions (np), deriving the equation in variations; Carathéodory conditions, existence and uniqueness of an absolutely continuous solution

Systems of linear equations: fundamental system, Liouville formula, variation of constants, matrix exponential and its properties

Stability: theorem on linearized nonstability (np) and stability, Hartman-Grobman theorem (np), Ljapunoff functions, orbital derivative, sufficient conditions for stability and for asymptotic stability

Dynamical systems: orbit, omega-limit sets and their basic properties, topological equivalence, La Salle invariance principle, Poincaré-Bendixson theorem (np)

Bifurcations: definition, necessary condition for a bifurcation, basic types of bifurcations in R, Hopf bifurcation in R2 (np).

4. Partial differential equations

Courses covering the required knowledge

NMMA339 Introduction to partial differential equations (entry requirements)
NMMA405 Partial differential equations 1
NMMA406 Partial differential equations 2

Explanation:

Linear and quasilinear first order equations: existence a uniqueness of a solution using the method of characteristics.

Linearní and nonlinear elliptic equations: function spaces for classical and weak solutions, classical solution of the Laplace equation, existence of a weak solution (using Lax-Milgram theorem, using Fredholm alternative, using Galerkin approximation and method of monotone operators), regularity of a weak solution (using difference quotients), maximum principles and uniqueness of a solution.

Linear and nonlinear parabolic equations: function spaces for classical and weak solutions, classical solution of the  Cauchy problem for the heat equation, existence of a weak solution (using Galerkin approximation and method of monotone operators, using the theory of semigroups), maximum principles and uniqueness of a solution.

Linear hyperbolic equations: function spaces for classical and weak solutions classical solution of the  Cauchy problem for the wave equation in dimensions 1, 2, 3, existence of a weak solution for a hyperbolic equation (using Galerkin approximation, using the theory of semigroups), finite speed of wave propagation and uniqueness of a solution for the wave equation.

Auxilliary tools: Sobolev spaces and their properties (definition, basic properties, embedding theorems, theorems on traces).