Requirements for the final state exam in Mathematical Structures (since 2025/2026)

Explanation of requirements for the state final exam for the program Mathematical Structures for students who started their study in 2025/2026 or later. Students who started in 2024/2025 or earlier can find the requirements on a page dedicated to a former study plan.

The oral part of the final exam consists of three out of the ten subject areas below according to the student’s choice. The numbering of the areas corresponds to the Study Guide, but titles are occasionally abbreviated. It is expected that the student has adapted the choice of lectures in the study to this choice of topics in order to acquire the required knowledge. The recommended lectures for the subject areas are included the explanation below.

It is not a purpose of the exam to demonstrate detailed technical knowledge, complete proofs, etc. Students have already demonstrated this in the individual subject exams. Here, students are expected to demonstrate a broader overview, understanding of main ideas, the broader context, and connections to other areas.

1. Groups and their representations

NMAG438 Group Representations 1: Representations of groups and group algebras, Maschke‘s theorem, irreducible decompositions, decomposition of the regular representation, the number of irreducible representations, characters of irreducible representations and character tables. The discrete Fourier transform on a finite group.

NMAG431 Combinatorial Group Theory: The Nielsen-Schreier theorem Presentations of groups. Tietze transformations.

NMAG464 The Theory of Groups 2: Finite simple groups (proof of the simplicity of certain groups, e.g. alternating, linear or some sporadic groups).

2. Algebraic number theory

NMAG472 Basic Algebraic Number Theory: Number fields, norm, discriminant, integral elements. Ramification and splitting of primes. Geometry of numbers, lattices, the Minkowski theorem, units and the class group. Cyclotomic fields.

NMAL431 Advanced Algebraic Number Theory: Dedekind domains, fractional ideals, prime factorization. Completion, local fields. Ramification and inertia groups, Chebotarev's theorem.

NMAG455 Quadratic Forms: Arithmetic Theory: The 3- and 4-square theorems, universal quadratic forms. Binary quadratic forms, form class group, genus theory.

3. Homological algebra and representation theory of finite-dimensional algebras

NMAG434 Categories of Modules and Homological Algebra: Tensor product, Morita equivalence, functors Ext and Tor, the relation of the functor Ext to extensions of modules.

NMAG442 Representation Theory of Finite-Dimensional Algebras: Representations of quivers and path algebras, direct sum decompositions and the Krull-Schmidt theorem. Algebras over an algebraically closed field (general finite dimensional algebras and Morita equivalence to path algebras with relations). Hereditary algebras and their representation type.

4. Algebraic geometry and commutative algebra

NMAG401 Algebraic Geometry: Zariski topology, irreducible components of algebraic sets, coordinate rings and polynomial maps, function fields and rational maps, birational equivalence, Hilbert’s Nullstellensatz and consequences, local rings of a variety in points and discrete valuation rings, projective varieties and projective elimination, Bézout’s theorem, the sheaf of regular functions on an algebraic variety.

NMAG538 Commutative Algebra: Localization, flatness, integral extensions, going up and going down theorems. Adic completion, the Artin-Rees lemma and Krull’s intersection theorem. Krull dimension and Krull’s principal ideal theorem. Regular sequences and Koszul complexes. Regular, Gorenstein and Cohen-Macaulay rings, homological characterization of regularity.

5. Algebraic topology

NMAG409 Algebraic Topology 1: Homotopy and homotopy types of topological spaces, cell complexes, fundamental group, Van Kampen's theorem, covering spaces and their classification, deck transformation group, singular homology, simplicial homology and their equivalence, exact sequences of homology groups and excision, Mayer-Vietoris sequences, cellular homology, axioms for homology.

NMAG532 Algebraic Topology 2: Cohomological groups, universal coefficient theorem, cup and cap products, cohomological ring, Künneth’s formula, orientation and homology, Poincaré duality, homotopy groups.

6. Differential geometry

NMAG411 Riemannian Geometry 1: Tensor fields on manifolds, Riemannian manifolds, Levi-Civita connection and its covariant derivative, parallel transport and geodesics, curvature and torsion of a connection and their geometrical meaning, components of the curvature tensor – their symmetries and role in geometry.

NMAG433 Riemann Surfaces: Multivalued analytical functions, Riemann surfaces, holomorphic and meromorphic functions on Riemann surfaces, divisors, global properties of holomorphic maps between Riemann surfaces, fields of meromorphic functions on Riemann surfaces, topological properties of Riemann surfaces, the Hurwitz theorem.

NMAG448 Classical Groups and Their Invariants: Classical linear algebraic groups, their Lie algebras, their structure, their regular representations, differential of a representation, complete reducibility, highest weight classification. First fundamental theorem for invariant theory of groups GL(m), Sp(n) and O(m). Schur-Weyl duality, Weyl algebra and Howe duality.

7. Universal algebra and applications

NMAG405 Universal Algebra 1 and NMAG450 Universal Algebra 2: Basic notions of universal algebra, isomorphism theorems. Direct and subdirect decompositions. Free algebras, Birkhoff’s theorem on varieties and equational theories. Algebraic and relational clones and how they are related. Maľcev conditions and congruence lattices (permutability, distributivity, Taylor term). Rewriting systems, the Knuth-Bendix algorithm. Abelian and affine algebras, the Gumm-Smith theorem.

NMAG563 Introduction to Complexity of CSP:The main idea of how to apply of methods of universal algebra to the complexity of CSP.

8. Mathematical logic and computational aspects

NMAG331 Mathematical Logic: The completeness theorem and applications. Gödel's incompleteness theorem and the Church-Gödel undecidability theorem.

NMAG407 Model Theory: The compactness theorem and applications. The Ax-Grothendieck theorem on injective polynomial maps on the field of complex numbers. Quantifier elimination, examples (the theories of dense linear orders and of algebraically closed fields) and consequences for definable subsets; o-minimality of real closed fields assuming the quantifier elimination for the theory of real closed fields.

NMMB415 Automata and Computational Complexity: Turing machines, a universal machine and undecidable problems (the halting problem, examples in algebra and logic). Complexity classes P and NP, satisfiability of boolean formulas (the Cook-Levin theorem), other examples of NP-complete problems and reductions. Automata and regular languages.

9. Extremal and structural combinatorics and graph theory

NDMI073 Combinatorics and Graph Theory 3: Szemeredi regularity lemma, (triangle) removal lemma, Hales-Jewett theorem.

NMAG403 Combinatorics: Block designs, Bruck-Ryser-Chowla theorem, finite projective planes. Graph colorings and variants, e.g. so called chooseability of graphs, chromatic index (Heawood, Vizing theorem), matching in graphs (in general and bipartite graphs, algorithms based on flows, Edmond's algorithm, perfect matchings – Hall and Tutte theorems), planar graphs and Kuratowski's theorem.

NTIN022 Probabilistic Techniques: The probabilistic method, linearity of expectation (including the Markov inequality), application of the variance (including the Chebyshev inequality), modification method, Lovasz local lemma, Chernoff bounds, threshold functions, Markov chains.

10. Algebraic, geometric and topological combinatorics

NDMI028 Linear Algebra Applications in Combinatorics: Application of linear dependence and independence, set systems with prescribed parity of intersections, spectral graph theory, interlacing of eigenvalues and consequences.

NDMI009 Introduction to Combinatorial and Computational Geometry: Basic theorems about convex sets (Helly's, Radon's, Carathéodory's, hyperplane separation theorem), incidence of points and lines, geometric duality, convex polyhedra (basic properties, combinatorial complexity), Voronoi diagrams, arrangements.

NDMI014 Topological Methods in Combinatoric: Borsuk-Ulam theorem and equivalent version, Ham-sandwich theorem, Necklace theorem, chromatic number of Kneser graphs.