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 (ArtinRees theorem), Dedekind domains, integral closures of noetherian domains (separable case, KrullAkizuki 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,
WedderburnArtin 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, Ext_{n} and Tor_{n} functors, connections
between Ext_{1} and extensions of modules.
ALG031 Algebra and Infinite Combinatorics
Extensions of modules, the Ext group, hereditary rings, nonperfect rings, the solution of the Whitehead problem for countable groups and modules, Diamond and uniformization, constructions of nonprojective 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, pgroups (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 IwanagaGorenstein 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.
