Phd program
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- Bu səhifədəki naviqasiya:
- No. of Credits
- The learning outcomes of the course
- More detailed display of contents (week-by-week)
- 117) INTRODUCTION TO ASYMPTOTIC EXPANSIONS Course coordinator
- The goals of the course
- 119)HIGHER ORDER FOURIER ANALYSIS Course coordinator
- More detailed display of contents
More detailed display of contents (week-by-week): Week 1-2 Borel, analytic, projective sets, Week 3-4 Universality, reduction, separation theorems, Week 5-6 Ranks, scales, games, Week 7-8 Axiom of determinancy, Week 9-10 Large cardinals, trees. Week 11-12 Forcing References: 1. K. Kuratowski: Topology, Academic Press, 1968. 2. A.S. Kechris: Classical Descriptive Set Theory, Springer, 1995. 115) ADVANCED SET THEORY Course Coordinator: Lajos Soukup No. of Credits: 3 and no. of ECTS credits 6 Prerequisites:Modern Set Theory Course Level: advanced PhD Brief introduction to the course: The past decades have seen a spectacular development in set theory, both in applying it to other fields (like topology and analysis) and mainly as an independent discipline. The goals of the course: Our aim is to familiarize the students with the latest developments within set theory and thereby give them a chance to do independent study and research of the many open problems of set theory. More detailed display of contents: Part I: Iterated forcing and preservation theorem Weeks 1-3. Finite support iterations and Martin's Axiom. Week 4-6. Countable support iterations and PFA. Part II. Combinatorial set theory Weeks 7-9. Combinatorial set-theory and applications to topology. Large cardinals. Basic pcf theory with applications to algebra and to topology. Part III. Set theory of the reals Weeks 10-12. ZFC results and forcing constructions. Determinacy, infinite games and combinatorics. Referencess: 1. T. Bartoszynski and H. Judah, Set theory on the structure of the real line, A K Peters, 1995. 2. Thomas Jech, Set theory, Spinger-Verlag, 1997. 3. István Juhász, Cardinal functions in topology - ten years later. Amsterdam: Mathematisch Centrum, 1980. 4. Akihiro Kanamori, Higher Infinite, Springer-Verlag, 1994. 5. Kenneth Kunen: Set theory. An introduction to Independence Proofs, Elsevier, 1999. 116) SET-THEORETIC TOPOLOGY Course Coordinator: István Juhász
Prerequisites: Modern Set-Theory, Advanced Set-Theory Course Level: advanced PhD Brief introduction to the course: The main theorems of Set-theoretic Topology are presented like topological results in special forcing extensions. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Set-theoretic Topology. 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 1-3. Cardinal functions and their interrelationships Week 4-5. Cardinal functions on special classes, in particular on compact spaces Week 6-8. Independence results, consequences of CH, Diamond, MA and PFA Week 9-10. Topological results in special forcing extensions, in particular in Cohen models Week 11-12. S and L spaces, HFD and HFC type spaces References: 1. K. Kunen, J.E. Vaughan, Handbook of Set-Theoretic Topology, Noth-Holland,1995. 2. Miroslav Huvsek and Jan van Mill. Recent progress in general topology. North-Holland , 1992.
We discuss the classical methods of the asymptotic theory of integrals like the integration by parts, Watson's lemma, Laplace's method, the principle of stationary phase and the method of steepest descents. The goals of the course: The aim of the course is to introduce the students to the classical theory of asymptotic power series. The learning outcomes of the course: By the end of the course, students are experts on the topic 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: Asymptotic notations, asymptotic sequences and expansions, failure of uniqueness, asymptotic sum, uniform asymptotic expansions Week 2: Asymptotic power series, basic operations on asymptotic power series, integration and differentiation, relation to Laurent series, Love's theorem Week 3: Incomplete gamma functions, the method of integration by parts, error bounds, the first encounter with the Stokes phenomenon Week 4: Watson's lemma for real integrals, the asymptotic expansions of the Bessel functions for large argument, Digamma function, the asymptotic expansion of the logarithm of the Gamma function Week 5: Laplace's approximation, Stirling's formula, the asymptotics of the Legendre polynomials for large order, further examples Week 6: Laplace's method, the asymptotic expansion of the Gamma function, Stirling coefficients, modified Bessel functions of large order and argument Week 7: The principle of stationary phase, the asymptotic behaviour of the Airy functions, Bessel functions of large order and argument Week 8: Watson's lemma for complex integrals, the method of steepest descents Week 9: Applications of the method of steepest descents: the Gamma function revisited, asymptotic expansions for the Airy functions, Stokes' phenomenon Week 10: Debye's expansions for the Bessel functions Week 11: The saddle point method, asymptotic approximation for the Bell numbers Week 12: Brief introduction to exponential asymptotics, optimal truncation, Ursell's lemma, asymptotic approximations for the remainders
118)ALGEBRAIC LOGIC AND MODEL THEORY 3 Course coordinator: Gábor Sági No. of Credits: 3 and no. of ECTS credits: 6. Prerequisites: Algebraic logic and model theory 2 Course level: advanced PhD Brief introduction to the course: Countable Categoricity. Stable theories and their basic properties. Uncountable Categoricity. Model theoretic Spectrum Functions. Morley’s Theorem. Many models theorem. The goals of the course: The main goal is to study additional advanced methods of mathematical logic and to learn how to apply them in other fields of mathematics. 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. 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 1. Definability of types in stable theories. Week 2. Stability and order. Week 3. Topological aspects of Morley rank. Week 4. Moley rank and forking. Week 5. Basic properties of forking in stable theories. Week 6. Forking in simple theories. Week 7. Morley sequences and the independence theorem. Week 8. Bounded equivalence relations. Week 9. Pregeometries and strongly minimal sets. Weeks 10-12. The Zilber-Cherlin-Lachlan-Harrington theorem (the proof of some technical details will be omitted) and a survey on more recent progress. References: A. Pillay, An Introduction to Stability Theory, Clarendon Press, Oxford, 1983 and 2002. A. Pillay, Geometric Stability Theory, Oxford Science Publications, 1996. S. Shelah, Classification Theory,Elsevier, 2002. F. O. Wagner, Simple Theories, Kluwer, 2000.
Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemerédi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems. The goals of the course: The course gives an introduction to the emerging theory of higher order Fourier Analysis, together with its connections to number 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 1. Classical Fourier Analysis in complex analysis Week 2. Classical Fourier Analysis on groups Week 3. Equidistribution of polynomial sequences in tori Week 4. Roth’s theorem Week 5. Linear patterns Week 6. Equidistribution of polynomials over finite fields Week 7. The inverse conjecture for the Gowers norm I. The finite field case Week 8. The inverse conjecture for the Gowers norm II. The integer case Week 9. Linear equations in primes Week 10. Ultralimit analysis and quantitative algebraic geometry Week 11. Higher order Hilbert spaces Week 12. The uncertainty principle
T. Tao: Higher order Fourier Analysis. AMS. 2012. 120) SEIBERG-WITTEN INVARIANTS Course coordinator: Andras I. Stipsicz No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: Topics in Geometry and Topology Course Level: intermediate PhD Brief introduction to the course: Seiberg-Witten invariants are probably the most important differential topological invariants of smooth 4-manifolds. In the course we discuss the definitions and main properties of the geometric objects needed in the definition of the invariants and prove some of the basic properties. The goals of the course: The aim is to get a working knowledge of basic notions of the Seiberg-Witten invariants. 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. Definition and properties of spinc structures 2. Spin and spinc manifolds 3. The Dirac operator 4. Elliptic operators 5. The Seiberg-Witten equations I 5. The Seiberg-Witten equations II 6. The Seiberg-Witten moduli space 7. Compactness and smoothness 8. The dimension formula 9. The definition of the invariance 10. Independence of choices 11. The connected sum formula and blow-ups 12. The adjunction formula
Gompf-Stipsicz: 4-manifolds and Kirby calculus 121) HEEGAARD-FLOER HOMOLOGIES Course coordinator: Andras I. Stipsicz No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: Topics in Geometry and Topology Course Level: intermediate PhD Brief introduction to the course: Heegaard Floer theory provides a package of invariants for low-dimensional objects like 3- and 4-manifolds, knots, links, contact structures and Legendrian knots. In the course we go thorugh the basic construction of Heegaard Floer homologies for 3-manifolds, and show basic propoerties of the theory. The goals of the course: The aim is to get a working knowledge of basic notions of Heegaard-Floer homologies. 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. Morse functions, the Morse-Smale-Witten complex 2. Morse homology 3. Morse homology on manifolds with boundary 4. Floer homology of Legendrian submanifolds 5. Heegaard decompositions and diagrams 6. Heegaard Floer homology groups 7. Nice diagrams, their existence 8. The invariance of Heegaard Floer homologies 9. Some computations. 10. Knot Floer homology 11. The surgery formula 12. Mixed invariants of 4-manifolds References: Ozsváth, Peter S.; Stipsicz, András I.; Szabó, Zoltán Grid homology for knots and links. Mathematical Surveys and Monographs, 208. American Mathematical Society, Providence, RI, 2015. Yüklə 233,54 Kb. Dostları ilə paylaş:
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