Search result: Catalogue data in Spring Semester 2020
Mathematics Bachelor | ||||||
Electives | ||||||
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3201-00L | Algebraic Groups | W | 8 credits | 4G | P. D. Nelson | |
Abstract | Introduction to the theory of linear algebraic groups. Lie algebras, the Jordan Chevalley decomposition, semisimple and reductive groups, root systems, Borel subgroups, classification of reductive groups and their representations. | |||||
Objective | ||||||
Literature | A. L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups | |||||
Prerequisites / Notice | Abstract algebra: groups, rings, fields, tensor product, etc. Some familiarity with the basics of Lie groups and their Lie algebras would be helpful, but is not absolutely necessary. We will develop what we need from algebraic geometry, without assuming prior knowledge. | |||||
401-3109-65L | Probabilistic Number Theory Does not take place this semester. | W | 8 credits | 4G | E. Kowalski | |
Abstract | The course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums. | |||||
Objective | The goal of the course is to present some results of probabilistic number theory in a unified manner. | |||||
Content | The main concepts will be presented in parallel with the proof of a few main theorems: (1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions; (2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line; (3) the Chebychev bias for primes in arithmetic progressions; (4) functional limit theorems for the paths of partial sums of families of exponential sums. | |||||
Lecture notes | The lecture notes for the class are available at Link | |||||
Prerequisites / Notice | Prerequisites: Complex analysis, measure and integral; some probability theory is useful but the main concepts needed will be recalled. Some knowledge of number theory is useful but the main results will be summarized. | |||||
401-3202-09L | The Representation Theory of the Finite Symmetric Groups NOTICE: No physical class for the next few weeks until further notice. Instead a video recording will be offered. | W | 4 credits | 2V | L. Wu | |
Abstract | This course is an Introduction to the Representation Theory of the Groups. | |||||
Objective | Our goal is to give an introduction of the Representation Theory using the examples of the Finite Symmetry Groups. | |||||
Literature | * Jean-Pierre Serre: Linear Representations of Finite Groups, Graduate Texts in Mathematics, Springer. * William Fulton and Joe Harris: Representation Theory A First Course, Graduate Texts in Mathematics, Springer. * G. D. James: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Springer. * Bruce E. Sagan: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, Springer. | |||||
Prerequisites / Notice | Some basic knowledge of the Group Theory and Linear Algebra. | |||||
401-8112-20L | Geometry of Numbers (University of Zurich) No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT548 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 1U | University lecturers | |
Abstract | The Geometry of Numbers studies distribution of lattice points in the n dimensional space, for instance, existence of lattice points in various domains and existence of integral solutions of polynomial inequalities. This subject is also closely related to the Theory of Diophantine Approximation, which seeks good rational approximations for real vectors. | |||||
Objective | Learn basic techniques in the Geometry of Numbers | |||||
Literature | 1. Cassels, An introduction to Diophantine Approximation 2. Cassels, An introduction to the Geometry of Numbers 3. Schmidt, Diophantine approximation 4. Siegel, Lectures on the Geometry of Numbers | |||||
401-3058-00L | Combinatorics I Does not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |
Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||
Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||
Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||
Prerequisites / Notice | Recognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008). |
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