Information for applicants

References to oficial pages


For whom the study branch Mathematical analysis is designed


Entry requirements


How to learn the missing topics


Information for foreign students

How to learn the missing topics

We collect below relevant literature to the advanced topics from entry requirements whose knowledge is assumed in master courses of the program Mathematical Analysis. We include also bachelor courses where these topics are taught. These courses are taught in Czech, but the respective teachers could provide consultations. The missing topics should be learnt at last during the first year of master studies using controlled self-study of the literature. It may result in the prolongation of the studies. Details should be discussed with the coordinator of the program.


For students starting their studies in 2022 or later


Elements of general topology

(topological spaces, compactness)


Taught in the bachelor course General Topology 1 (NMMA345).
Literature:
R. Engelking, General Topology, Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. (Chapter 1, Sections 2.1-2.4, 2.6, 3.1-3.2)


Elements of complex analysis

(Cauchy theorem, residue theorem)


Covered by the bachelor course Introduction to complex analysis (NMMA301).
Literature:
Rudin, W.: Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. (Chapters 10 and 12)


Elements of functional analysis

(Banach and Hilbert spaces, dual spaces, weak convergence, bounded operators, compact operators, Fourier transform)


Taught in the bachelor course Introduction to functional analysis (NMMA331).
Literature:
M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer-Verlag, New York, 2001. (Chapters 1,2,3,7)
W. Rudin, Functional Analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. (the first part of Chapter 7)


Elements of the theory of ordinary differential equations

(basic properties of solutions and maximal solutions, linear systems, stability)


Taught in the bachelor course Ordinary differential equations (NMMA336).
Literature:
G. Teschl: Ordinary differential equations and dynamical systems. AMS Graduate Studies in Mathematics, 2012. (Chapters 1-3)


Elements of the theory of partial differential equations

(quasilinear first order equations, Laplace equation and heat equation – classical solution and maximum principle, wave equation – classical solution in dimension 1,2,3, finite speed of wave propagation)


Covered by the bachelor course Introduction to partial differential equations (NMMA339.
Literature:
L.C. Evans: Partial differential equations. American Mathematical Society, Providence, RI, 2010. (Chapter 2, Sections 3.1, 3.2 and 4.6)
M. Renardy, R.C. Rogers: An introduction to partial differential equations. Springer-Verlag, New York, 2004. (Section 2.1)


For students starting their studies in 2021 or earlier


Elements of general topology

(topological spaces, compactness)


Taught in the bachelor course General Topology 1 (NMMA335, since 2021 NMMA345 ).
Literature:
R. Engelking, General Topology, Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. (Chapter 1, Sections 2.1-2.4, 2.6, 3.1-3.2)


Elements of complex analysis

(Cauchy theorem, residue theorem, conformal mappings)


Covered by the bachelor courses Introduction to complex analysis (NMMA301) and Complex analysis 1 (NMMA338).
Literature:
Rudin, W.: Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. (Chapters 10, 12, 13, 14)


Elements of functional analysis

(Banach and Hilbert spaces, dual spaces, bounded operators, compact operators, elements of the theory of distributions)


Taught in the bachelor course Introduction to functional analysis (NMMA331).
Literature:
M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer-Verlag, New York, 2001. (Chapters 1,2,7)
W. Rudin, Functional Analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. (Chapters 6 and 7)


Elements of the theory of ordinary differential equations

(basic properties of solutions and maximal solutions, linear systems, stability)


Taught in the bachelor course Ordinary differential equations (NMMA333, since 2021 NMMA336).
Literature:
G. Teschl: Ordinary differential equations and dynamical systems. AMS Graduate Studies in Mathematics, 2012. (Chapters 1-3)


Elements of the theory of partial differential equations

(quasilinear first order equations, Laplace equation and heat equation – fundamental solution and maximum principle, wave equation – fundamental solution, finite speed of wave propagation)


Covered by the bachelor course Introduction to partial differential equations (NMMA334). Since 2021 this course is replaced by a pair of courses NMMA339 and NMMA338, the required knowledge is covered by NMMA339.
Literature:
L.C. Evans: Partial differential equations. American Mathematical Society, Providence, RI, 2010. (Chapter 2, Sections 3.1, 3.2 and 4.6)
M. Renardy, R.C. Rogers: An introduction to partial differential equations. Springer-Verlag, New York, 2004. (Section 2.1)