Phd program
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No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: introductory PhD Brief introduction to the course: Various classical Algorithms are presented among others about prime searching, sorting networks, or linear programming. The goals of the course: The main goal of the course is to introduce students to classical algorithms. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
References: 1. T. H. Cormen, C. L. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, Cambridge, MA, 1990. 2. W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, and A. Schrijver, Combinatorial Optimization. Wiley, 1998. 23) QUANTUM COMPUTING Course Coordinator: Miklos Simonovits
Prerequisites: Theory of Algorithms Course Level: advanced PhD Brief introduction to the course: Advanced theory of Quantum Computing is presented like Quantum parallelism, and Shor's integer factoring algorithm. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Quantum Computing. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
Reference: J. Gruska, Quantum Computing, McGrawHill, 1999. 24) RANDOM COMPUTATION Course coordinator: Miklos Simonovits No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Probability 1 Course Level: advanced PhD Brief introduction to the course: The main topic covers: how randomization helps in design of algorithms, and the analysis of randomized algorithms. Also the related parts of complexity theory will be described, and a short introduction to derandomization will be given. The goals of the course: To learn designing and analyzing of randomized algorithms. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents:
Reference: Motwani-Raghavan: Randomized Algorithms, Cambridge University Press, 1995. 25) HOMOLOGICAL ALGEBRA Course Coordinator: Pham Ngoc Anh No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: Topics in algebra Course Level: advanced PhD Brief introduction to the course: An introduction to homological algebra. A description of projective and injective modules. Torsion and extension product with application to the theory of homological dimension and extensions of groups. The goals of the course: The main goal of the course is to introduce students to the most important, basic notions and techniques of homological algebra. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents: 1. Differential graded groups, modules. Examples from simplicial homology theory. Homology of complexes. Basic properties. 2. Exact sequence of homology. Short description of the singular homology theory. 3. Hom functor and tensor products. Projective and injective resolutions. 4. Ext 5. Ext (continued) 6. Tor 7. Tor (continued) 8. Homological dimensions. 9. Rings of low dimensions. 10. Cohomology of groups 11.Cohomology of algebras. 12. Application to theory of extensions. Referencess: 1. J. Rotman: Introduction to homological algebra, Springer 2009. 2. J. P. Jains: Rings and homology, Holt, Rinehart and Winston, New York 1964. 26) HIGHER LINEAR ALGEBRA Course coordinator: Matyas Domokos No. of Credits: 3 and no. of ECTS credits:6 Prerequisites: Topics in Algebra Course Level: Intermediate PhD Brief introduction to the course: Covers advanced topic in linear algebra beyond the standard undergraduate material. The goals of the course: Learn familiarity with the representation theory of quivers and its relevance for various areas. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.
Week 1. Introduction, motivation, overview of the course. Week 2. Matrix problems and their connection to modules over path algebras. Week 3. The variety of representations, some basic properties of algebraic group actions. Week 4. Dynkin and Euclidean diagrams. Week 5. Gabriel’s Theorem, reflection functors. Week 6. Auslander-Reiten translation. Week 7. Kronecker’s classification of matrix pencils. Week 8. Indecomposable representations of tame quivers. Week 9. Kac’s Theorems for wild quivers. Week 10. Schur roots, canonical decomposition of dimension vectors. Week 11. Quivers with relations. Week 12. Perspectives, relevance for some currently active research topics. Reference: I. Assem, D. Simson, A. Skowronski: Elements of the Representation Theory of Associative Algebras, Cambridge Univ. Press, 2006. 27) REPRESENTATION THEORY I. Course coordinator: Matyas Domokos No. of Credits: 3 and no. of ECTS credits: 6 Prerequisite Topics in Algebra Course Level: Intermediate PhD Brief introduction to the course: The course gives an introduction to the theory of group representations, in a manner that provides useful background for students continuing in diverse mathematical disciplines such as algebra, topology, Lie theory, differential geometry, harmonic analysis, mathematical physics, combinatorics. The goals of the course: Develop the basic concepts and facts of the complex representation theory of finite groups, compact toplogical groups, and Lie groups. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Week 1. Definition of linear representations, irreducible representations, general constructions. Week 2. Properties of completely reducible representations. Week 3. Finite dimensional complex representations of compact groups are unitary. Week 4. Products of representations, Schur Lemma and corollaries Week 5. Spaces of matrix elements. Of representations. Week 6. Decomposition of the regular representation of a finite group. Week 7. Characters, orthogonality, character tables, a physical application. Week 8. The Peter-Weyl Theorem Week 9. Representation of the special orthogonal group of rank three. Week 10. The Laplace spherical functions. Week 11. Lie groups and their Lie algebras Week 12. Repreentations of the complex special linear Lie algebra SL(2,C) Reference: E. B. Vinberg: Linear Representations of Groups, Birkhauser Verlag, 1989. 28) REPRESENTATION THEORY II. Course coordinator: Matyas Domokos No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites:Representation Theory I. Course Level: advanced PhD Brief introduction to the course: In Representation Theory I, the basic general principles of representation theory were laid. In the present course we discuss in detail the representation theory of the symmetric group, the general linear group and other classical groups, and semisimple Lie algebras. The goals of the course: Introduce the students the representations some of the most important groups. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.
Week 1. The irreducible representations of the symmetric group (Young symmetrizers). Week 2. Partitions, the ring of symmetric functions, Schur functions. Week 3. Pieri’s rule, Kostka numbers, Jacobi-Trudi formula. Week 4. Cauchy formula, skew Schur functions. Week 5. Induced representations, Frobenius reciprocity. Week 6. Frobenius character formula, hook formula, branching rules. Week 7. Schur-Weyl duality, double centralizing theorem. Week 8. Polynomial representations of the general linear group, Schur functors. Week 9. Semisimple Lie algebras, root systems, Weyl groups. Week 10. Highest weight theory. Week 11. Weyl character formula, the classical groups. Week 12. Littlewood-Richardson rule, plethysms. References: 1. I. G. Macdonald, Symmetric functions and Hall polynomials 2. W. Fulton, J. Harris: Representation Theory (A first course) 3. C. Procesi: Lie groups (An approach through invariants and representations) 29) UNIVERSAL ALGEBRA AND CATEGORY THEORY Course coordinator: László Márki No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites:Topics in algebra Course Level: intermediate PhD Brief introduction to the course: Basic notions and some of the fundamental theorems of the two areas are presented, with examples from concrete algebraic structures. The goals of the course: The main goal of the course is to provide access to the most general parts of algebra, those on the highest level of abstraction. This also helps to understand connections between different kinds of algebraic structures. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): 1. Algebra, many-sorted algebra, related structures (subalgebra lattice, congruence lattice, automorphism group, endomorphism monoid), factoralgebra, homomorphism theorem. 2. Direct product, subdirect product, subdirectly irreducible and simple algebras, Birkhoff’s theorem. 3. Ultraproduct, Łoś lemma, Grätzer-Schmidt theorem. 4. Variety, word algebra, free algebras, identities, Birkhoff’s variety theorem. 5. Pseudovariety, implicit operation, pseudoidentity, Reiterman’s theorem. 6. Equational implication, quasivariety, Kogalovskiĭ’s theorem, fully invariant congruence, Birkhoff’s completeness theorem. 7. Mal’cev type theorems. 8. Primality, Rosenberg’s theorem, functional completeness, generalizations of these notions. 9. Category, functor, natural transformation, speciaol morphisms, duality, contravariance, opposite, product of categories, comma categories. 10. Universal arrow, Yoneda lemma, coproducts and colimits, products and limits, complete categories, groups in categories. 11. Adjoints with examples, reflective subcategory, equivalence of categories, adjoint functor theorems. 12. Algebraic theories. References: 1. S. Burris – H.P. Sankappanavar: A course in universal algebra, Springer, 1981.; available online at www.math.uwaterloo.ca/~snburris 2. G. Grätzer: Universal algebra, 2nd ed., Springer, 1979. 3. J. Almeida: Finite semigroups and universal algebra, World Scientific, 1994. 4. S. Mac Lane: Categories for the working mathematician, Springer, 1971. 5. F. Borceux: Handbook of categorical algebra, 1-2, Cambridge Univ. Press, 1994. 30) TOPICS IN GROUP THEORY Course coordinator: Péter P. Pálfy No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: Topics in Algebra Course Level: intermediate PhD Brief introduction to the course: Group theory is the abstract mathematical theory of symmetries. It is the oldest branch of abstract algebra with its basic notions introduced by Evariste Galois around 1830. Nowadays group theory is a very rich subject encompassing many different areas with applications in various branches of mathematics (algebra, topology, number theory, combinatorics, geometry) and theoretical physics (quantum mechanics). Each week during the semester a different area of group theory will be discussed. The goals of the course: The main goal of the course is to introduce the students to the many facets of modern group theory. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Week 1: Permutation groups. Transitivity, primitivity. Wreath products. Week 2: The classification of primitive permutation groups: the O’Nan-Scott Theorem. Week 3: Multiply transitive groups. The Mathieu groups. Week 4: Simple groups. The simplicity of some matrix groups. Week 5: Automorphism groups. Coherent configurations, strongly regular graphs. Week 6: Free groups. The Nielsen-Schreier Theorem about subgroups of free groups. Week 7: Groups extensions. Cohomology of groups. The Schur-Zassenhaus Theorem. Week 8: Solvable groups. Hall’s Theorems for finite solvable groups. Week 9: Nilpotent groups and finite p-groups. Commutator calculus. Week 10: The transfer homomorphism. Normal p-complements. Week 11: Frobenius groups. The structure of the Frobenius kernel and the complement. Week 12: Subgroup lattices. Distributivity, modularity, Dedekind’s chain condition. References: Peter J. Cameron, Permutation Groups, London Mathematical Society Student Texts 45, Cambridge University Press, 1999 Derek J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, Springer-Verlag, 1993 31) TOPICS IN RING THEORY. I Course Coordinator: Pham Ngoc Anh No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites:Topics in Algebra Course Level: intermediatePhD Brief introduction to the course: The main theorems of Ring Theory are presented among others about Burnside problem and Morita theory. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the Ring Theory. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
References: 1. I. Kaplansky: Fields and Rings, The University of Chicago Press, 1972. 2. Lam: A First Course in Noncommutative Rings, Springer, 1991. 32) TOPICS IN RING THEORY. II Course Coordinator: Pham Ngoc Anh
Prerequisites:Topics in Ring theory I Course Level: advanced PhD Brief introduction to the course: Advanced theorems of Ring Theory are presented among others about Burnside problem and Morita theory. The goals of the course: The main goal of the course is to introduce students to advanced methods in Ring Theory. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. Yüklə 233,54 Kb. Dostları ilə paylaş:
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