- Teaching
Past and current courses
Algebra (Bachelor) Logika és halmazelmélet (I. év Matematika, Matematika-Informatika, Matematika-Fizika) Computational algebra, Coding Theory and Cryptology (Master) Homological Algebra (Master) Modular Representation Theory of Finite Groups (Doctoral School) Algebraic Groups (Doctoral School)-
Numere complexe, quaternioni, aplicatii
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ALGEBRA JEGYZET
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Logika és halmazelmét
Jegyzet:
Marcus Andrei-Szántó Csaba-Tóth László: Logika és halmazelmét. Scientia Kiadó, Kolozsvár, 2004.
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Computational Algebra, Coding Theory and Cryptology
1. Primality Testing and Factorization
2. Fast Fourier Transform
3. Polynomials over finite fields. The Berlekamp algorithm
4. Aplications to Cryptology
5. Introduction to Coding Theory
6. Gröbner bases. The Buchberger algorithm
7. Generators and relations in groups. The Todd-Coxeter algorithm
8. Lattice reduction and the LLL algorithm
Bibliography
1. R. Lidl, G. Pilz - Applied Abstract Algebra, Springer-Verlag 1998
2. N. Koblitz - A Course in Number Theory and Cryptography, Springer-Verlag 1994
3. D. Bressoud, S. Wagon - A Course in Computational Number Theory, Springer-Verlag 2000
4. A. M. Cohen, H. Cuypers, H. Sterk - Some Tapas of Computer Algebra, Springer-Verlag 1999
5. H. Cohen - A Course in Computational Algebraic Number Theory, Springer-Verlag 2000
6. D. Knuth - The Art of Computer Programming, Addison-Wesley
7. T. H. Cormen, Ch. Leiserson, R. R. Rivest, C. Stein - Introduction to Algorithms, Second Edition, MIT Press Cambridge MA 2001
8. The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4; 2005. (http://www.gap-system.org)
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Homological Algebra
Homological algebra first arose as a language for describing topological properties of geometrical objects. Since then, it has expanded into subject on its own right, and its contemporary applications are many and diverse. A quick look at the Mathematics Subject Classification 2000 reveals application to Number Theory, Algebra and Differential Geometry, Lie Groups and Algebras, Finite Groups, Partial Differential Equations, Functional Analysis and Operator Theory. Therefore, the homological algebra methods must be in the toolbox of every mathematician.
The aim of this cause is to introduce the basic concepts and techniques in the language of categories and functors, and to present examples coming from various fields.
- 1. Triangulated spaces
2. Simplicial sets
3. Simplicial topological spaces and the Eilenberg-Zilber theorem
4. Sheaves
- 1. Complexes and morphisms of complexes
2. Coefficient systems
3. The long exact sequence
4. Homotopy
- 1. The Cech complex
2. The complex of singular chains
3. Homology and cohomology of groups
4. The de Rham complex
5. Homology and cohomology of Lie algebras
6. Hochschild (co)homology of algebras
7. Cyclic homology
8. The Koszul complex
- 1. Categories
2. Functors and natural transformations
3. Equivalences of categories
4. Adjoint functors
5. Additive and abelian categories
- 1. Injective modules and projective modules
2. Resolutions
3. Derived functors
4. Tor and Ext
5. Examples: (co)homology of sheaves, of groups, of Lie algebras, and of algebras
1. S.I. Gelfand and Yu.I. Manin - Methods of Homological Algebra, Springer-Verlag 1998
2. Ch. Weibel - An Introduction to Homological Algebra, Cambridge University Press, 1994
3. H. Cartan and S. Eilenberg - Homological Algebra, Princeton University Press, 1956
4. P. Hilton and U. Stammbach - A Course in Homological Algebra, Springer-Verlag 1971
5. S. Maclane - Homology, Springer-Verlag 1963
6. J. L. Loday - Cyclic Homology, Springer-Verlag 1992
- Modular Representation Theory of Finite Groups
- Algebraic Groups