Mathematical Structures

Coordinated by: Department of Algebra
Study branch coordinator: doc. RNDr. Jan Šťovíček, Ph.D.

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), analysis on manifolds, commutative algebra (Galois theory, integral extensions), mathematical logic (propositional and first order logic, incompleteness and undecidability), set theory and category theory.

Deeper knowledge of combinatorics, representation theory of associative algebras (finiteness conditions, projective and injective modules) and Lie theory is an advantage (but not a necessity) for individual subject areas of this branch.

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

CodeSubjectCreditsWinterSummer
NMAG401Algebraic Geometry 52/2 C+Ex
NMAG409Algebraic Topology 1 52/2 C+Ex
NMAG411Riemannian Geometry 1 52/2 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C

2.2 Elective Courses

Set 1

It is required to earn at least 48 credits from the following elective courses.

CodeSubjectCreditsWinterSummer
NDMI009Introduction to Combinatorial and Computational Geometry 52/2 C+Ex
NDMI013Combinatorial and Computational Geometry 2 52/2 C+Ex
NDMI014Topological Methods in Combinatorics 52/2 C+Ex
NDMI028Linear Algebra Applications in Combinatorics 52/2 C+Ex
NDMI045Analytic and Combinatorial Number Theory 32/0 Ex
NDMI073Combinatorics and Graph Theory 3 52/2 C+Ex
NMAG331Mathematical Logic 32/0 Ex
NMAG403Combinatorics 52/2 C+Ex
NMAG405Universal Algebra 1 52/2 C+Ex
NMAG407Model Theory 32/0 Ex
NMAG430Algebraic Number Theory 63/1 C+Ex
NMAG431Combinatorial Group Theory 63/1 C+Ex
NMAG433Riemann Surfaces 32/0 Ex
NMAG434Categories of Modules and Homological Algebra 63/1 C+Ex
NMAG435Lattice Theory 32/0 Ex
NMAG436Curves and Function Fields 63/1 C+Ex
NMAG437Seminar on Differential Geometry 30/2 C0/2 C
NMAG438Group Representations 1 52/2 C+Ex
NMAG439Introduction to Set Theory 2 32/0 Ex
NMAG442Representation Theory of Finite-Dimensional Algebras 63/1 C+Ex
NMAG444Combinatorics on Words 32/0 Ex
NMAG446Logic and Complexity 32/0 Ex
NMAG448Classical groups and their invariants 52/2 C+Ex
NMAG450Universal Algebra 2 42/1 C+Ex
NMAG454Fibre Spaces and Gauge Fields 63/1 C+Ex
NMAG455Quadratic forms and class fields I*32/0 Ex
NMAG456Quadratic forms and class fields II*32/0 Ex
NMAG458Algebraic Invariants in Knot Theory 42/1 Ex
NMAG462Modular forms and L-functions I*32/0 Ex
NMAG473Modular forms and L-functions II*32/0 Ex
NMAG475MSTR Elective Seminar 20/2 C0/2 C
NMAG481Seminar on Harmonic Analysis 30/2 C0/2 C
NMAG498MSTR Elective 1 32/0 Ex
NMAG499MSTR Elective 2 32/0 Ex
NMAG531Approximations of Modules 32/0 Ex
NMAG532Algebraic Topology 2 52/2 C+Ex
NMAG533Principles of Harmonic Analysis 63/1 C+Ex
NMAG534Non-commutative harmonic analysis 63/1 C+Ex
NMAG535Computational Logic 52/2 C+Ex
NMAG446Logic and Complexity*32/0 Ex
NMAG536Proof Complexity and the P vs. NP Problem*32/0 Ex
NMAG563Introduction to complexity of CSP 32/0 Ex
NMAG569Mathematical Methods of Quantum Field Theory 30/2 C0/2 C
NMAG538Commutative algebra 64/0 Ex
NMAG537Selected topic from Set Theory 32/0 Ex
NMAG575Forcing 32/0 Ex
NMAL430Latin Squares and Nonassociative Structures 32/0 Ex
NMMB413Algorithms on Polynomials 42/1 C+Ex
NMMB415Automata and Computational Complexity 63/1 C+Ex
NMMB430Algorithms on Eliptic curves 42/1 C+Ex
NMMB432Randomness and Calculations 42/1 Ex
NMMB433Geometry for Computer Graphics 32/0 Ex
NTIN022Probabilistic Techniques 52/2 C+Ex

* The course is taught once in two years only.

Set 2

It is required to earn at least 8 credits in 48 credits from the following short list.

CodeSubjectCreditsWinterSummer
NMAG403Combinatorics 52/2 C+Ex
NMAG405Universal Algebra 1 52/2 C+Ex
NMAG407Model Theory 32/0 Ex
NMAG438Group Representations 1 52/2 C+Ex
NMMB415Automata and Computational Complexity 63/1 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 48 credits by completion of elective courses from set 1. At least 8 credits must be from the short list of elective courses in 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 a common subject area "1. Mathematical Structures" and a choice of one of four subject areas "2. Algebra and logic", "3. Geometry", "4. Representation Theory", "5. Combinatorics". A half of the exam is focused on subject area 1 and the other half on questions from the subject area selected from among 2, 3, 4 and 5.

Requirements for the oral part of the final exam

Common requirements

1. Mathematical Structures
Algebraic geometry. Algebraic topology.

Specialization

2. Algebra a logic
Finite groups and their representations. Combinatorial group theory. Binary systems. Advanced universal algebra. Complexity and enumerability. First order logic. Undecidability in algebraic systems. Quantifier elimination.

3. Geometry
Harmonic analysis and invariants of classical groups, Riemannian surfaces. Fibre spaces and covariant derivative.

4. Representation Theory
Representations of groups, representations of finite-dimensional algebras. Combinatorial group theory. Homological algebra.

5. Combinatorics
Applications of linear algebra and 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

CodeSubjectCreditsWinterSummer
NMAG401Algebraic Geometry 52/2 C+Ex
NMAG409Algebraic Topology 1 52/2 C+Ex
NMAG411Riemannian Geometry 1 52/2 C+Ex
NSZZ023Diploma Thesis I 60/4 C
 Optional and Elective Courses 39  

2nd year

CodeSubjectCreditsWinterSummer
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C
 Optional and Elective Courses 36