# 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 |

#### 4.2 Elective Courses

*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 |

#### 4.3 State Final Exam

*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.

#### 4.4 Recommended Course of Study

*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 |