Index of all courses

Courses available in English:

ALG012 Universal Algebra 1, 2
1 (winter term): Diagrams, limits, colimits, reflections, universal algebras, varieties of algebras, Birkhoff theorem, equational logic.
2 (spring term): Amalgamation and strong amalgamation, lattices of varieties, Mal'cev conditions, absolutely free algebras of terms, identities, arithmetic of terms, lattices of subalgebras and cogruences, Schreier's property.

ALG015 Commutative Algebra 1
Integral extensions, valuation domains, noetherian rings (Artin-Rees theorem), Dedekind domains, integral closures of noetherian domains (separable case, Krull-Akizuki theorem).

ALG017 Introduction to the Theory of Groups
Free groups, defining relations, action of a group on a set, free, Cartesian and direct products, semidirect product, Abelian groups, finitely generated Abelian groups, Schreier's transversal and subgroups of free groups, Zassenhaus lemma, main and composition series, solvable groups and their characterization, Sylow theorems, nilpotent groups, characterization of nilpotent groups.

ALG021 Group Representations 1, 2
Ireducible representations of groups, Shur's Lemma, representations as modules over group rings, direct sums and tensor products of representations, representations of finite groups, Maschke's Theorem, characters, orthogonality relations, Burnside's theorem, rank of irredcible representations, representations over the field of complex numbers and irreducible characters, permutation representations, representations induced by subgroups of finite index, projective representations, Shur's multiplier.

ALG028 Rings and Modules (winter term)
Ring theory (Jacobson radical, structure of semisimple modules and rings, Wedderburn-Artin theorem, artinian and noetherian rings and modules, Hopkins' theorem, Hilbert basis theorem), free and projective modules (Kaplansky theorems, structure of projective modules) injective modules (Baer criterion, injective envelopes, structure of injective modules over noetherian rings, structure of divisible abelian groups).

ALG029 Modules and Homological Algebra (spring term)
Category theory of modules (covariant and contravariant Hom functors, projective and injective modules, tensor products, flat modules, adjointness of Hom functors and tensor product, Morita equivalence of rings and its characterization, a generalization: tilting modules and tilted algebras), introduction to homological algebra (complexes, projective and injective resolutions, Extn and Torn functors, connections between Ext1 and extensions of modules.

ALG031 Algebra and Infinite Combinatorics
Extensions of modules, the Ext group, hereditary rings, non-perfect rings, the solution of the Whitehead problem for countable groups and modules, Diamond and uniformization, constructions of non-projective Whitehead modules, the solution of the Whitehead problem for regular cardinals, Shelah's compactness theorem and the general solution of the Whitehead problem, new applications in the module theory.

ALG033 Combinatorical Group Theory
Winter term: Subgroups of free groups (Nielsen's and Reidemaister's method), Tietze transformations, HNN extensions, free products with amalgamated subgroups, geometrical methods, Cayley complexes.
Spring term: Other selected topics in elementary combinatorical group theory.

ALG052 Introduction to the Theory of Finite Groups
Finite projective, symplectic and orthogonal groups, p-groups (Burnside's problem, Engel's elements, Sylow subgroups of simple groups, collecting proces), transfer.

ALG077 Approximations of Modules
General theory of module approximations, A proof of the Flat Cover Conjecture, Relations to the Finitistic Dimension Conjectures, A proof of the Conjectures for Iwanaga-Gorenstein rings.

ALG008 Student Algebra Seminar
Selected topics in modern algebra. Subjects will depend on interest of students.

ALG030 Algebra Seminar
Reports on new results in a variety of topics from contemporary algebra.