# Search result: Catalogue data in Spring Semester 2021

Mathematics Bachelor | ||||||

Electives | ||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3058-00L | Combinatorics I | 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). | |||||

401-3109-65L | Probabilistic Number Theory | 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 https://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf | |||||

Prerequisites / Notice | Prerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled). Some knowledge of number theory is useful but the main results will also be summarized. | |||||

401-3362-21L | Spectral Theory of Eisenstein Series | W | 4 credits | 2V | P. D. Nelson | |

Abstract | We plan to discuss the basic theory of Eisenstein series and the spectral decomposition of the space of automorphic forms, with focus on the groups GL(2) and GL(n). | |||||

Objective | ||||||

Prerequisites / Notice | Some familiarity with basics on Lie groups and functional analysis would be helpful, and some prior exposure to modular forms or homogeneous spaces may provide useful motivation. | |||||

401-4116-12L | Lectures on Drinfeld Modules | W | 6 credits | 3V | R. Pink | |

Abstract | Drinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc. | |||||

Objective | ||||||

Content | A central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations. A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them. | |||||

Literature | Drinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592. Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91. Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996. Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004. Further literature will be indicated during the course |

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