Phd program
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No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: intermediatePhD Brief introduction to the course: The main notions and theorems about Singularities are presented like germs, or Thom's transversality theorems. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Singularity 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. V. Arnold, S. Gusein-Zade, A. Varchenko: Singularities of Differentiable Maps,Vol. I.; Birkhauser, 1985 2. J. Martinet: Singularities of Differentiable Functions and Maps; London Math.Soc. Lecture Notes Series 58, 1982 3. C.T.C. Wall: Proceedings of Liverpool Singularities, Sypmosium I.; SLNM 192,1970 56) FOUR MANIFOLDS AND KIRBY CALCULUS Course coordinator: Andras I. Stipsicz No. of Credits: 3 and no. of ECTS credits 6 Prerequisites: - Course Level:intermediate PhD Brief introduction to the course: The course introduces modern techniques of differential topology through handle calculus, and pays special attention to the description of 4-dimensional manifolds. We also show how to manipulate diagrams representing 4-manifolds. Smooth invariants of 3- and 4-manifolds (Heegaard Floer invariants and Seiberg-Witten invariants) will be also discussed. The goals of the course: The aim is to get a working knowledge of all basic (algebraic) topologic notions such as homology, cohomology theory, the theory of knots and handlebodies and some aspects of differential geometry through the rich theory of 4-manifolds. This discussion quickly leads to some important and unsolved questions in the field. 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: Knots in the 3-space Week 2: Invariants of knots, the Alexander and the Jones polynomial Week 3: Morse theory, handle decompositions Week 4: Decompositions of 3-manifolds: Heegaard diagrams Week 5: Decompositions of 4-manifolds: Kirby diagrams Week 6: Knots in 3-manifolds and Heegaard diagrams Week 7: Surfaces in 4-manifolds; Freedman’s theorem Week 8: New invariants of knots: grid homology Week 9: Heegaard Floer invariants of 3-manifolds (combinatorial approach) Week 10: Further structures on Heegaard Floer groups Week 11: Seiberg-Witten invariants Week 12: Basic properties of Seiberg-Witten invariants References:
57) SYMPLECTIC MANIFOLDS, LEFSCHETZ FIBRATION Course coordinator: Andras I. Stipsicz No. of Credits: 3 and no. of ECTS credits: 6 Prerequisites: - Course Level: intermediate PhD Brief introduction to the course: We will discuss symplectic and contact manifolds, with a special emphasis on dimensions four and three. By results of Donaldson, Gompf and Giroux, the topological counterparts of these structures are Lefschetz fibrations and open book decompositions. In the study of these structures we need to examine mapping class group of surfaces (closed and with nonempty boundary). The goals of the course: The aim is to get a working knowledge of basic notions of symplectic and contact topology, and introduce the concepts of Lefschetz fibrations and open book decompositions. If time permits, we will also discuss 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: Week 1: Linear symplectic theory Week 2: Symplectic and contact manifolds Week 3: Almost complex and almost contact structures Week 4: Constructions of symplectic manifolds Week 5: Contact surgery and Stein manifolds Week 6: Manifolds with no symplectic structure Week 7: Mapping class groups and their presentations Week 8: Lefschetz pencils and fibrations, elliptic fibrations Week 9: Open book decompositions Week 10: The Giroux correspondance Week 11: Donaldson’s almost holomorphic technique, existence of Lefschetz fibrations Week 12: Stein manifolds References:
58) COMBINATORIAL NUMBER THEORY I Course coordinator: Gergely Harcos No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: introductory PhD Brief introduction to the course: We shall discuss additive structures in natural sets such as the integers, finite fields, or the reals. The topics were motivated by wide applicability in number theory and current research activity in the field. You will learn powerful methods from probability, combinatorics, and algebra. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Combinatorial 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-by-week): Week 1: The moment method. Sidon’s problem on thin bases. Week 2: Complementary bases of the primes. Week 3: Thin bases of higher order. Week 4: Ruzsa distance and additive energy. Week 5: Covering lemmas and approximate groups. Week 6: The sum-product problem I. Week 7: The sum-product problem II. Week 8: Plünnecke’s theorem. Week 9: The Balog-Szemerédi-Gowers theorem. Week 10: Freiman homomorphisms and inverse theorems. Week 11: The combinatorial Nullstellensatz and applications. Week 12: Snevily’s conjecture and Davenport’s problem.
59) COMBINATORIAL NUMBER THEORY II Course coordinator: Gergely Harcos No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Combinatorial Number Theory Course Level: advanced PhD Brief introduction to the course: We shall discuss additive structures in the integers, with special emphasis on arithmetic progressions. As a highlight, we shall discuss Tao’s proof of Szemerédi’s theorem. You will learn powerful modern methods from combinatorics and finite Fourier analysis. The goals of the course: The main goal of the course is to introduce students to advanced topics and methods of Combinatorial 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-by-week): Week 1: Asymptotic and Schnirelmann density. The theorems of Schnirelmann and Mann. Week 2: Adding a basis. The theorems of Erdős and Plünnecke. Week 3: Van der Waerden’s theorem. Overview of Tao’s proof of Szemerédi’s theorem. Week 4: Uniformity norms, and the generalized von Neumann theorem. Week 5: Almost periodic functions. Week 6: Factors of almost periodic functions. Week 7: The energy incrementation argument. Proof of the structure theorem. Week 8: An application of van der Waerden’s theorem. Week 9: Recurrence for almost periodic functions. Week 10: Structure of sumsets I. Week 11: Structure of sumsets II. Week 12: Discussion. Minilectures by students.
60) CLASSICAL ANALYTIC NUMBER THEORY Course Coordinator: Gergely Harcos No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites:- Course Level:introductory PhD
We shall prove the classical theorems on the distribution of prime numbers in strong analytic form. You will meet the basic objects and techniques of analytic number theory such as Dirichlet L-functions and the Mellin transform. You will develop a deeper understanding of numbers, complex analysis, and the Riemann Hypothesis. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Analytic 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-by-week): Week 1: Dirichlet series. Week 2: Additive and multiplicative characters. Week 3: Primes in arithmetic progressions I. Week 4: Mellin Transform. Jensen’s Inequality. Borel-Carathéodory Lemma. Week 5: The Prime Number Theorem. Week 6: Primitive characters and Gauss sums. Week 7: Quadratic characters. The Pólya Vinogradov Inequality. Week 8: The Burgess bound. Week 9: Analytic properties of Dirichlet L-functions I. Week 10: Analytic properties of Dirichlet L-functions II. Week 11: Analytic properties of Dirichlet L-functions III. Week 12: Primes in arithmetic progressions II. Reference: Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, 2006
We will discuss probabilistic methods in number theory. You will learn about various notions of density for sets of positive integrs. You will learn about the statistical behaviour of arithmetic functions, with particular emphasis on additive and multiplicative functions. You will learn about the distribution of integers free of large prime factors which play an important role in number theoretic algorithms such as primality testing. In addition to learning beautiful results in number theory, you will deepen your intution about statistical phenomena and become more skilled in analysis. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Probabilistic 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-by-week): Week 1: Natural density, logarithmic density, analytic density. Week 2: The Hardy-Ramanujan theorem. Week 3: The Turán-Kubilius inequality. Week 4: Dual form of the Turán-Kubilius inequality. Week 5: Effective mean value estimates for multiplicative functions. Week 6: The theorems of Delange and Wirsing. Week 7: Halász’ theorem. Week 8: The Erdős-Kac theorem. Week 9: Integers free of large prime factors: Rankin’s method. Week 10: Integers free of large prime factors: the geometric method. Week 11: Integers free of large prime factors: Dickman’s function. Week 12: Integers free of large prime factors: the saddle-point method.
62) MODERN PRIME NUMBER THEORY I Course coordinator: Gergely Harcos No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Classical Analytic Number Theory Course Level: advanced PhD Brief introduction to the course: We will discuss important recent results concerning prime numbers. The highlights will include (1) Huxley’s theorem on the nonexistence of large gaps between prime numbers, (2) the theorem of Goldston-Pintz-Yildirim on the existence of small gaps between prime numbers, (3) the Agrawal-Kayal-Saxena algorithm for recognizing prime numbers in polynomial time, and (4) Linnik’s theorem on the least prime in arithmetic progressions. You will learn modern techniques of analytic number theory such as mean value theorems for Dirichlet polynomials, density results for the zeroes of the Riemann zeta function, and important ideas inspired by the Hardy-Littlewood method. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of Modern Prime 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-by-week): Week 1: Mean values of Dirichlet polynomials Week 2: Carlson’s zero density estimate Week 3: Fourth moment of the Riemann zeta function Week 4: Ingham’s zero density estimate Week 5: The Halász-Montgomery inequality Week 6: Huxley’s zero density estimate Week 7: The Goldston-Pintz-Yildirim theorem, Part 1 Week 8: The Goldston-Pintz-Yildirim theorem, Part 2 Week 9: The Goldston-Pintz-Yildirim theorem, Part 3 Week 10: The Agrawal-Kayal-Saxena algorithm Week 11: Linnik’s theorem on the least prime in arithmetic progressions, Part 1 Week 12: Linnik’s theorem on the least prime in arithmetic progressions, Part 2
63) MODERN PRIME NUMBER THEORY II Course coordinator: Gergely Harcos No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Modern Prime Number Theory Course Level: advanced PhD Brief introduction to the course: We will discuss in detail the Green-Tao theorem on the existence of long arithmetic progressions among prime numbers and Linnik’s theorem on the least prime in arithmetic progressions. You will learn modern techniques of combinatorial and analytic number theory such as the Gowers uniformity norm and density results for the zeroes of Dirichlet L-functions. The goals of the course: The main goal of the course is to introduce students to advanced topics and methods of Modern Prime 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-by-week):
Reference: Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481-547. 64) EXPONENTIAL SUMS IN COMBINATORIAL NUMBER THEORY Course Coordinator: Imre Ruzsa
Prerequisites: Harmonic Analysis 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. Contents: We learn to use Fourier-analytic techniques to solve several problems on general sets of integers. In particular: to find estimates for sets free of arithmetic progressions; methods of Roth, Szemerédi, Bourgain and Gowers. To find arithmetic progressions and Bohr sets in sumsets: methods of Bogolyubov, Bourgain, and Ruzsa's construction. Difference sets and the van der Corput property. References: There are no textbooks for these subjects, the original papers have to be used.
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