Sztochasztika

1. Groups:
Solvable groups, generalizations of Sylow’s theorems. Finite p-groups and nilpotent groups. Simple groups: classical simple groups, sporadic simple groups, groups of Lie type. Theory of group extensions, Schur-Zassenhaus theorem. Free groups. Burnside’s problem. Multiply transitive and primitive permutation groups. Abelian groups: pure subgroups, basic subgroups.
Bibliography:
J.J. Rotman: An introduction to the theory of groups; Springer, 1995.
D.J.S. Robinson: A course in the theory of groups; Springer, 1996.
L. Fuchs: Infinite Abelian groups I-II.; Academic Press, 1970, 1973.

2. Group Representations:
Characters: orthogonality relations, induction. Frobenius reciprocity, Clifford theory, charecter degree. Applications: Burnside theorem, Frobenius kernel, charecterisation by centralizers of involutions. Projective representations, Schur multiplier. The structure of the group algebra. Snapshots from the theory of modular representation.
Bibliography:
I.M. Isaacs: Character theory of finite groups; Dover, 1994.
G. Navarro: Characters and blocks of finite groups; Cambridge Univ. Press, 1998.

3. Semigroups and automata:
Green-relations. Transformation semigroups. Completely 0-simple semigroups. Regular and inverse semigroups. Automaton mappings. Simple automata. Products of automata. Complete systems of automata. Automata and languages. Strongly connected automata.
Bibliography:
J.M. Howie: An introduction to semigroup theory; Academic Press, 1976.
A.H. Clifford, G.B. Preston: The algebraic theory of semigroups; AMS, 1961, 1967.
F. Gécseg, I. Peák: Algebraic theory of automata; Akadémiai Kiadó, 1972.
J.M. Howie: Automata and languages; Clarendon Press, 1991.
F. Gécseg: Products of automata; Springer, 1986.
Révész Gy.: Bevezetés a formális nyelvek elméletébe [Introduction to the theory of formal languages – in Hungarian], Tankönyvkiadó, 1979.
J. Hopcroft, J.D. Ullman: Introduction to automata theory, languages and computation,
Addison-Wesley, 1979.

4. Noncommutative rings:
Wedderburn–Artin theorem. Primitive rings, Jacobson density theorem, Jacobson radical. Chain conditions: Artinian and Noetherian rings. Central simple algebras: Brauer group, crossed product of algebras. Modules: projective and injective modules. Azumaya-Remak-Krull-Schmidt theorem. Morita equivalence.
Bibliography:
F.W. Anderson, K.R. Fuller: Rings and categories of modules; Springer, 1974.
R.S. Pierce: Associative algebras; Springer, 1982.

5. Homological algebra:
Derived functors: Ext and Tor. Long exact sequences of homologies. Homological dimensions: projective and global dimension.
Bibliography:
C.A. Weibel: An introduction to homological algebra; Cambridge Univ. Press, 1994.
J.J. Rotman: An introduction to homological algebra; Academic Press, 1979.

6. Commutative algebra:
Prime and primary ideals. Localization. Noether's normalization lemma. Discrete evaluation rings, Dedekind rings.
Bibliography:
M. Atiyah, I.G. Macdonald: Introduction to commutative algebra; Addison-Wesley 1969.
D. Eisenbud: Commutative algebra with a view toward algebraic geometry; Springer,
1955.
H. Matsumura: Commutative ring theory; Cambridge Univ. Press, 1988.

7. Algebraic geometry:
Affine and projective algebraic varieties, curves. Coordinate rings.
Birational mappings. Elliptic curves.
Bibliography:
R. Shafarevich: Basic algebraic geometry, Vol. I. ; Springer, 1994.
R. Hartshorne: Algebraic geometry; Springer, 1977.

8. Fields:
Galois theory. Transcendental extensions. Lüroth’s theorem. Ordered fields: Artin-Schreier theory. Finite fields, error correcting codes.
Bibliography:
P.M. Cohn: Algebra, I-III. ; Wiley, 1982, 1989, 1991.
I. Stewart: Galois theory. Chapman & Hall, 2003.

9. Lie algebras:
Nilpotent Lie-algebras: Engel’s theorem. Solvable Lie-algebras. Semisimple Lie algebras over the complex numbers. Root systems. Chevalley basis. Enveloping algebra: Poincaré-Birkhoff-Witt theorem.
Bibliography:
J. E Humphreys: Introduction to Lie algebras and representation theory; Spinger, 1972, reprinted, 1997.
J.-P. Serre: Lie algebras and Lie groups; Springer, 1992.
W. Fulton, J. Harris: Representation Theory; Springer, 1991.

10. Universal algebra:
Varieties, free algebras, identity theories and Birkhoff’s calculus for equational logics. Subdirect decomposition. Congruence lattices. Malcev type theorems for congruence distributive and for congruence modular varieties. Clones, Rosenberg’s Theorem. Completeness theorems, primal algebras. Boole representation. Discriminator varieties. The basics of commutator theory, tame congruences.
Bibliography:
G. Grätzer: Universal algebra; Springer, 1979.
R. Freese, R. McKenzie: Commutator Theory for Congruence Modular Varieties; London Math. Soc., 1987.
R. McKenzie, G. McNulty, W. Taylor: Algebras, lattices, varieties.
S. Burris-H. P. Sankappanavar: A course in universal algebra; Springer, 1981.
D. Hobby, R. McKenzie: The structure of finite algebras; AMS, 1988.

11. Lattices:
Topological representations of distributive lattices, duality of distributive lattices and posets. Free lattices. Geometric spaces and lattices, projective spaces and complemented modular lattices, Desargues’ theorem. Representations of algebraic lattices.
Bibliography:
G. Birkhoff: Lattice Theory
G. Grätzer: General lattice theory; Academic Press, 1978.
Czédli Gábor: Hálóelmélet [Lattice theory – in Hungarian]
B. Ganter-R. Wille: Concept lattices
R. Freese-J. Ježek-J.B. Nation: Free lattices
P. Crawley-R.P. Dilworth. Algebraic theory of lattices; Prentice-Hall, 1973.

12. Model theory, algebraic logic:
Model classes and their characterisations (elementary, ∆-elementary, Σ-elementary classes), positive and negative results. Model constructions: products, embeddings, reductions. Preservation and characterization theorems. Definability. Completeness. Standard and non-standard models. Non-standard analysis. Algebraizations of logics. Specifying algebraization by logical and algebraic means. Connections between algebraic and logical concepts, characterisation of important logical properties (e.g. compactness, completeness, etc.) by universal algebraic concepts. The concept of representation.
Bibliography:
W. Hodges: A Shorter Model Theory, Cambridge Univ. Press, 1997
C.C. Chang, H.J. Keisler: Model Theory, North Holland, 3th ed.,1990
L. Henkin, J.D. Monk, A. Tarski: Cylindric Algebras I-II., North Holland, 1985
J. Bell, M. Machover: A Course in Mathematical Logic, North Holland, 1977
Ferenczi M.: Matematikai Logika [Mathematical Logic - in Hungarian], Műszaki Kiadó, 2002
Csirmaz L.: Matematikai Logika [Mathematical Logic - in Hungarian], ELTE, 1994
H.B. Enderton: A Mathematical Introduction to Logic, Academic Press, 2nd ed., 2001
Serény Gy.: A modellelmélet alapfogalmai [Fundamental Concepts of Model Theory - in Hungarian], BME, 1992.

13. Proof theory and its applications:
Deduction and refutation calculi. Analytic tableaux, resolution. Algorithms in proof theory. Normal forms. The limitations of proof theory, Gödel's theorems. The general model of logic programming, PE definitions, the correct answer problem. The logical foundations of PROLOG, SLD resolution. Connections with database theory.
Bibliography:
M. Ben-Ari: Mathematical Logic in Computer Science I-II., Prentice Hall, 1996
A. Nerode, R.A. Shore: Logic for Applications, Springer, 1997
E. Burke, E. Foxley: Logic and its Application, Prentice Hall, 1996
M. Ferenczi, M. Szőts: Mathematical Logic and Formal Methods, TypoTeX, 2010
M. Ferenczi: Matematikai logika [Mathematical Logic, in Hungarian], Műszaki Kiadó, 2014
H. B. Enderton: A Mathematical Introduction to Logic, Academic Press, 2nd ed., 2001

14. Non-classical logics in computer science:
Classification of logics. Modal, multimodal, temporal, dynamic, multivalued, arrow, relational, intuitionistic, probabilistic, and non-monotonic logics. Investigation of important logical properties for non-classical logics. Applications to knowledge representation, program correctness proofs and logic programming.
Bibliography:
R. Goldblatt: Logic of Time and Computation, CSLI, Stanford, 1992
From Modal Logic to Deductive Databases, Ed. Thayse, A., Wiley, 1992
R. Turner: Logics for Artificial Intelligence, Ellis, 1984
A. Nerode, R.A. Shore: Logic for Applications, Springer, 1997
M. Ferenczi, M. Szőts: Mathematical Logic and Formal Methods, TypoTeX, 2010
H. B. Enderton: A Mathematical Introduction to Logic, Academic Press, 2nd ed.,2001

15. Linear algebra:
Vector spaces: Determinants. Linear spaces (linear independence, basis, subspace, quotient space, dual space), linear maps (image, kernel, rank). Gram-Schmidt orthogonalizatiton. Euclidean spaces and finite dimensional Hermitian spaces.
Linear maps: Linear transformations, canonical forms of matrices. Trace, eigenvalues, minimal- and characteristic polynomial. The Jordan normalform. Frobenius normalform. Polar decomposition. Lánczos decomposition. Integral matrices, Smith normalform.
Special linear transformations: Symmetric and self-adjoint matrices. Skew symmetric, orthogonal, unitary and normal matrices. The spectral theorem. Nilpotent matrices, projections, involutions.
Multilinear algebra: Multilinear maps, tensor product, tensor algebra, extrerior product. Grasmann algebra. Symmetric and skew symmetric tensors. Reducible tensors. Tensor product and wedge product of linear maps.
Matrix inequalities: Inequalities of symmetric and self-adjoint matices. Inequalities of eigenvalues and norms of matrices. Matrices with nonnegative entries, Perron-Frobenius theorem. Doubly stochastic matrices.
Matrices occuring in algebra and analysis: Commuting matrices, commutators. Quaternions, Cayley and Clifford algebras. The resultant. Witt’s theorem. Moore-Penrose inverse, matrix equations. Functions of matrices and their derivatives.
Literature:
P.M. Cohn: Algebra, I-III. ; Wiley, 1982, 1989, 1991.
V.V. Prasolov: Problems and theorems in linear algebra; Amer. Math. Soc., 1994.
E. Fried: Algebra I. [in Hungarian]; Nemzeti Tankönyvkiadó, 2000.
R. Freud: Lineáris algebra [Linear algebra – in Hungarian]; ELTE, 1996.
E. Horváth: Lineáris Algebra [Linear algebra – in Hungarian]; BME, 1995.