2 Degree Plans – Mathematical Structures

Coordinated by: Department of Algebra
Study branch coordinator: prof. RNDr. Jan Krajíček, DrSc.

The curriculum is focused on extending general mathematical background (algebraic geometry and topology, Riemann geometry, universal algebra and model theory) and obtaining deeper knowledge in selected topics of algebra, geometry, logic, and combinatorics. The aim is to provide sufficient general knowledge of modern structural mathematics and to bring students up to the threshold of independent research activity. Emphasis is laid on topics taught by instructors who have achieved worldwide recognition in their field of research.

A graduate has advanced knowledge in algebra, geometry, combinatorics and logic. He/she is in close contact with the latest results of contemporary research in the selected field. The abstract approach, extensiveness and intensiveness of the programme result in the development of the ability to analyse, structure and solve complex and difficult problems. Graduates may pursue an academic career or realize themselves in jobs that involve mastering new knowledge and control of complex systems.

Assumed knowledge

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

Linear algebra, real and complex analysis, and probability theory.
Foundations of group theory (Sylow theorems, free groups, nilpotence). Lie groups, analysis on manifolds, ring and module theory (finiteness conditions, projective and injective modules), commutative algebra (Galois theory, integral extensions).
Intermediate knowledge of mathematical logic (propositional and first order logic, incompleteness and undecidability).

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.

2.1 Obligatory courses

Code Subject Credits Winter Summer
NMAG401 Algebraic Geometry   5 2/2 C+Ex
NMAG403 Combinatorics   5 2/2 C+Ex
NMAG405 Universal Algebra 1   5 2/2 C+Ex
NMAG407 Model Theory   3 2/0 Ex
NMAG409 Algebraic Topology 1   5 2/2 C+Ex
NMAG411 Riemannian Geometry 1   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

2.2 Elective Courses

It is required to earn at least 35 credits from the following elective courses.

Code Subject Credits Winter Summer
NMAG462 Modular forms and L-functions I   3 2/0 Ex
NMAG473 Modular forms and L-functions II   3 2/0 Ex
NMAG455 Quadratic forms and class fields I   3 2/0 Ex
NMAG456 Quadratic forms and class fields II   3 2/0 Ex
NMAG431 Combinatorial Group Theory 1   1 2/0 C
NMAG432 Combinatorial Group Theory 2   5 2/0 Ex
NMAG433 Riemann Surfaces   3 2/0 Ex
NMAG434 Categories of Modules and Homological Algebra   6 3/1 C+Ex
NMAG435 Lattice Theory 1   3 2/0 Ex
NMAG436 Curves and Function Fields   6 4/0 Ex
NMAG437 Seminar on Differential Geometry   3 0/2 C 0/2 C
NMAG438 Group Representations 1   5 2/2 C+Ex
NMAG440 Binary Systems   3 2/0 Ex
NMAG442 Representation Theory of Finite-Dimensional Algebras   6 3/1 C+Ex
NMAG444 Combinatorics on Words   3 2/0 Ex
NMAG446 Logic and Complexity   3 2/0 Ex
NMAG448 Invariant Theory   5 2/2 C+Ex
NMAG450 Universal Algebra 2   4 2/1 C+Ex
NMAG452 Introduction to Differential Topology   3 2/0 Ex
NMAG454 Fibre Spaces and Gauge Fields   6 3/1 C+Ex
NMAG531 Approximations of Modules   3 2/0 Ex
NMAG532 Algebraic Topology 2   5 2/2 C+Ex
NMAG533 Harmonic Analysis 1   6 3/1 C+Ex
NMAG534 Harmonic Analysis 2   6 3/1 C+Ex
NMAG536 Proof Complexity and the P vs. NP Problem   3 2/0 Ex
NMMB401 Automata and Convolutional Codes   6 3/1 C+Ex
NDMI013 Combinatorial and Computational Geometry II   6 2/2 C+Ex
NDMI028 Linear Algebra Applications in Combinatorics   6 2/2 C+Ex
NDMI045 Analytic and Combinatorial Number Theory   3 2/0 Ex
NDMI073 Combinatorics and Graph Theory III   6 2/2 C+Ex
NTIN022 Probabilistic Techniques   6 2/2 C+Ex
NTIN090 Introduction to Complexity and Computability   5 2/1 C+Ex

2.3 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 35 credits by completion of elective courses.
 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 a common subject area ''1. Mathematical Structures" and a choice of one of four subject areas ''2A. Geometry", ''2B. Representation Theory", ''2C. General and Combinatorial Algebra", or ''2D. Combinatorics". One question is asked from subject area 1 and one question is asked from the subject area selected from among 2A, 2B, 2C, or 2D.

Requirements for the oral part of the final exam

Common requirements

1. Mathematical Structures
Basics of algebraic geometry, universal algebra, Riemannian geometry, algebraic topology, model theory and combinatorics.

Specialization

2A. Geometry
Harmonic analysis and invariants of classical groups, Riemannian surfaces, algebraic topology, fibre spaces and covariant derivation.

2B. Representation Theory
Representations of groups, representations of finite-dimensional algebras. combinatorial group theory, curves and function fields, and homological algebra.

2C. General and Combinatorial Algebra
Finite groups and their representations, combinatorial group theory, binary systems (semigroups, quasigroups), advanced universal algebra (lattices, clones, Malcev conditions), complexity and enumerabilty, undecidability in algebraic systems.

2D. Combinatorics
Applications of linear algebra. combinatorics and graph theory, application of probabilistic method in combinatorics and graph theory, analytic and combinatorial number theory, combinatorial and computational geometry, structural and algorithmic graph theory.

2.4 Recommended Course of Study

1st year

Code Subject Credits Winter Summer
NMAG401 Algebraic Geometry   5 2/2 C+Ex
NMAG403 Combinatorics   5 2/2 C+Ex
NMAG405 Universal Algebra 1   5 2/2 C+Ex
NMAG409 Algebraic Topology 1   5 2/2 C+Ex
NMAG411 Riemannian Geometry 1   5 2/2 C+Ex
NMAG407 Model Theory   3 2/0 Ex
NSZZ023 Diploma Thesis I   6 0/4 C
  Optional and Elective Courses   26    

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