# Mathematical Analysis

Coordinated by: Department of Mathematical Analysis
Study branch coordinator: doc. RNDr. Ondřej Kalenda, Ph.D., DSc.

The mathematical analysis curriculum offers advanced knowledge of fields traditionally forming mathematical analysis (real function theory, complex analysis, functional analysis, ordinary and partial differential equations). It is characterized by a depth of insight into individual topics and emphasis on their mutual relations and interconnections. Advanced knowledge of these topics is provided by a set of obligatory courses. Elective courses deepen the knowledge of selected fields, especially those related to the diploma thesis topic. Seminars provide contact with contemporary mathematical research. Mathematical analysis has close relationships with other mathematical disciplines, such as probability theory, numerical analysis and mathematical modelling. Students become familiar with these relationships in some of the elective courses. The programme prepares students for doctoral studies in mathematical analysis and related subjects. Applications of mathematical theory, theorems and methods to applied problems broaden the qualification to etemployment in a non-research environment.

The graduate will acquire advanced knowledge in principal fields of mathematical analysis (real function theory, complex analysis, functional analysis, ordinary and partial differential equations), understand their interconnections and relations to other mathematical disciplines. He/she will be able to apply advanced theoretical methods to real problems. The programme prepares students for doctoral studies but the knowledge and abilities acquired can be put into use in practical occupations as well.

#### Assumed knowledge

It is assumed that an incoming student of this branch has sufficient knowledge of the following topics and fields:

Differential calculus of one and several real variables. Integral calculus of one real variable. Measure theory, Lebesgue measure and Lebesgue integral. Basic algebra (matrix calculus, vector spaces).
Foundations of general topology (metric and topological spaces, completeness and compactness), complex analysis (Cauchy integral theorem, residue theorem, conformal maps) and functional analysis (Banach and Hilbert spaces, dual spaces, bounded operators, compact operators, basic theory of distributions).
Elements of the theory of ordinary differential equations (basic properties of solutions and maximal solutions, linear systems, stability theory) and of partial differential equations (quasilinear first order equations, Laplace theorem and heat equation – fundamental solution and maximum principle, wave equation – fundamental solution, finite speed of wave propagation).

Should an incoming student not meet these entry requirements, the coordinator of the study programme may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.

#### 4.1 Obligatory Courses

 Code Subject Credits Winter Summer NMMA401 Functional Analysis 1 8 4/2 C+Ex — NMMA402 Functional Analysis 2 6 — 3/1 C+Ex NMMA403 Theory of Real Functions 1 4 2/0 Ex — NMMA405 Partial Differential Equations 1 6 3/1 C+Ex — NMMA406 Partial Differential Equations 2 6 — 3/1 C+Ex NMMA407 Ordinary Differential Equations 2 5 2/2 C+Ex — NMMA408 Complex Analysis 2 5 — 2/2 C+Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NSZZ024 Diploma Thesis II 9 0/6 C — NSZZ025 Diploma Thesis III 15 — 0/10 C

#### Set 1

The courses in this group introduce various research areas in mathematical analysis, illustrate their applications, and cover other fields that are related to mathematical analysis. It is required to earn at least 21 credits from this group. (Up to 8 credits can be earned for courses taken during stays at foreign universities, if the courses are approved in advance by the garant of the programme)

 Code Subject Credits Winter Summer NMAG409 Algebraic Topology 1 5 2/2 C+Ex — NMAG433 Riemann Surfaces 3 2/0 Ex — NMMA404 Theory of Real Functions 2 4 — 2/0 Ex NMMA433 Descriptive Set Theory 1 4 2/0 Ex — NMMA434 Descriptive Set Theory 2 4 — 2/0 Ex NMMA435 Topological Methods in Functional Analysis 1 4 2/0 Ex — NMMA436 Topological Methods in Functional Analysis 2 4 — 2/0 Ex NMMA437 Advanced Differentiation and Integration 1 4 2/0 Ex — NMMA438 Advanced Differentiation and Integration 2 4 — 2/0 Ex NMMA440 Differential Equations in Banach Spaces 4 — 2/0 Ex NMMA501 Nonlinear Functional Analysis 1 5 2/2 C+Ex — NMMA502 Nonlinear Functional Analysis 2 5 — 2/2 C+Ex NMMA531 Partial Differential Equations 3 4 2/0 Ex — NMMA533 Introduction to Interpolation Theory 1 4 2/0 Ex — NMMA534 Introduction to Interpolation Theory 2 4 — 2/0 Ex NMMO401 Continuum Mechanics 6 2/2 C+Ex — NMMO532 Mathematical Theory of Navier-Stokes Equations 3 — 2/0 Ex NMMO536 Mathematical Methods in Mechanics of Compressible Fluids 3 — 2/0 Ex NMNV405 Finite Element Method 1 5 2/2 C+Ex —

#### Set 2

This group includes scientific seminars and workshops. It is required to earn at least 12 credits from this group. Each seminar yields 3 credits per semester and they can be taken repeatedly.

 Code Subject Credits Winter Summer NMMA431 Seminar on Differential Equations 3 0/2 C 0/2 C NMMA452 Seminar on Partial Differential Equations 3 0/2 C 0/2 C NMMA454 Seminar on Function Spaces 3 0/2 C 0/2 C NMMA455 Seminar on Real and Abstract Analysis 3 0/2 C 0/2 C NMMA456 Seminar on Real Functions Theory 3 0/2 C 0/2 C NMMA457 Seminar on Basic Properties of Function Spaces 3 0/2 C 0/2 C NMMA458 Seminar on Topology 3 0/2 C 0/2 C NMMA459 Seminar on Fundamentals of Functional Analysis 3 0/2 C 0/2 C

#### Requirements for taking the final exam

Earning at least 120 credits during the course of the study.
Completion of all obligatory courses prescribed by the study plan.
Earning at least 21 credits by completion of elective courses from set 1.
Earning at least 12 credits by completion of elective courses from set 2.
Submission of a completed master's thesis by the submission deadline.

#### Oral part of the state final exam

The oral part of the final exam consists of five subject areas: ''Real Analysis", ''Complex Analysis", ''Functional Analysis", ''Ordinary Differential Equations", and ''Partial Differential Equations". One question is asked from each subject area.

#### Requirements for the oral part of the final exam

1. Real Analysis
Measure theory and signed measures, Radon measures. Absolutely continuous functions and functions with bounded variation. Hausdorff measure and Hausdorff dimension.

2. Complex Analysis
Meromorphic functions. Conformal mappings. Harmonic functions of two real variables. Zeros of holomorphic functions. Holomorphic functions of several complex variables. Analytic continuation.

3. Functional Analysis
Topological linear spaces. Locally convex spaces and weak topologies. Spectral theory in Banach algebras. Spectral theory of bounded and unbounded operators. Integral transformations. Theory of distributions.

4. Ordinary Differential Equations
Carathéodory theory of solutions. Systems of first order linear equations. Stability and asymptotical stability. Dynamical systems. Bifurcations.

5. Partial Differential Equations
Linear and quasilinear first order equations. Linear and nonlinear eliptic equations. Linear and nonlinear parabolic equations. Linear hyperbolic equations. Sobolev and Bochner spaces.

#### 1st year

 Code Subject Credits Winter Summer NMMA401 Functional Analysis 1 8 4/2 C+Ex — NMMA403 Theory of Real Functions 1 4 2/0 Ex — NMMA405 Partial Differential Equations 1 6 3/1 C+Ex — NMMA407 Ordinary Differential Equations 2 5 2/2 C+Ex — NMMA402 Functional Analysis 2 6 — 3/1 C+Ex NMMA406 Partial Differential Equations 2 6 — 3/1 C+Ex NMMA408 Complex Analysis 2 5 — 2/2 C+Ex NSZZ023 Diploma Thesis I 6 — 0/4 C Optional and Elective Courses 14

#### 2nd year

 Code Subject Credits Winter Summer NSZZ024 Diploma Thesis II 9 0/6 C — NSZZ025 Diploma Thesis III 15 — 0/10 C Optional and Elective Courses 36