Requirements for the final state exam in Mathematical Structures

Explanation of requirements for the state final exam for the program Mathematical Structures for students who started their study in 2020/2021 or later.

1. Mathematical Structures (common requirements)

Explanation of the common requirements

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.

Algebraic Topology: 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.

2. Specialization (chosen by the student)

The student must choose one of the four areas 2A – 2D below and from that area select two topics for the exam. The choice must be communicated to the head of the final state exam committee.

It is assumed that the student has chosen courses in the master's study which correspond to the choice of topics for the exam in order to acquire the required knowledge.

2A. Algebra and Logic

Explanation of requirements – the student chooses two out of the following topics:

Groups: Representations of groups and group algebras, Maschke‘s theorem, characters of irreducible representations. The center of a group algebra, conjugacy classes and decomposition of the regular representation. The discrete Fourier transform on a finite group. Presentations of groups, the Nielsen-Schreier theorem.

Universal algebra: Subalgebras, congruences, homomorphisms and 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 (permutability of congruences, Jónsson terms, Taylor term). Abelian algebras and affine representations.

Combinatorial and computational aspects: Turing machines, a universal machine and undecidable problems (the halting problem, examples in algebra, the Church-Gödel theorem). 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. Rewriting systems, the Knuth-Bendix algorithm.

First-order logic: 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.

2B. Geometry

Explanation of requirements – the student chooses two out of the following topics:

Fiber bundles and covariant derivative: 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. Laplace-Beltrami operator on a Riemannian manifold. Principal and associated fiber bundles, automorphisms of fiber bundles. Bianchi identities, Maurer-Cartan equation. Chern-Weil theory, characteristic classes.

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.

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.

Harmonic analysis: Locally compact groups, Haar measure, fundamental properties of C*-algebras, Gelfand transform, universal enveloping algebras and Verma modules.

Advanced parts in algebraic topology: Cohomological groups, universal coefficient theorem, cup and cap products, cohomological ring, Künneth’s formula, orientation and homology, Poincaré duality, homotopy groups.

2C. Representation Theory

Explanation of requirements – the student chooses two out of the following topics:

Groups: Representations of groups and group algebras, Maschke‘s theorem, characters of irreducible representations. The center of a group algebra, conjugacy classes and decomposition of the regular representation. The discrete Fourier transform on a finite group. Presentations of groups, the Nielsen-Schreier theorem.

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.

Homological algebra: Tensor product, Morita equivalence, functors Ext and Tor, the relation of the functor Ext to extensions of modules.

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.

2D. Combinatorics

Explanation of requirements – the student chooses two out of the following topics:

Graph Theory: Planar graphs and graphs embeddable on surfaces of higher genus, chromatic number and chooseability of graphs, chromatic index (Vizing theorem), algorithms for maximum matching in bipartite and general graphs, perfect matchings in bipartite and general graphs (Hall and Tutte theorems). Szemeredi regularity lemma, tree-width of graphs, algorithms on graphs of bounded tree-width, spectral graph theory (eigenvalues of graphs).

Application of probabilistic techniques in combinatorics and graph theory: The probabilistic method, linearity of expectation, modification method, application of the varriance method. Lovasz local lemma. Pseudorandomness and explicit constructions. Chernoff bounds.

Discrete and computational geometry: Geometric probabilistic algorithms, combinatorial properties of hyperplane arrangements, range searching, linear programming in small dimension.

Classical combinatorial structures: Steiner triple systems, Hadamard matrices, BIBD - balanced incomplete block designs, Bruck-Ryser-Chowla theorem on szmmetric designs, finite projective planes and geometries.

Number theory: Diophantine approximation (Dirichlet's approximation theorem, Farey fractions, transcendent numbers). Diophantine equations (Pell equation, Thue equation, 4 square theorem, Hilbert's tenth problem). Prime numbers (estimates for the prime-counting function, Dirichlet's theorem on arithmetic progressions). Number fields (norm, discriminant, algebraic integers, prime factorizations). Geometry of numbers (lattices, Minkowski theorem, units and class group).