# Search result: Catalogue data in Autumn Semester 2019

Mathematics Master | ||||||

Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||

Electives: Pure Mathematics | ||||||

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

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3033-00L | Gödel's Theorems | W | 8 credits | 3V + 1U | L. Halbeisen | |

Abstract | Die Vorlesung besteht aus drei Teilen: Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe. Teil II behandelt den Gödel'schen Vollständigkeitssatz Teil III behandelt die Gödel'schen Unvollständigkeitssätze | |||||

Objective | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||

Content | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||

Literature | Ergänzende Literatur wird in der Vorlesung angegeben. | |||||

401-4037-69L | O-Minimality and Diophantine Applications | W | 4 credits | 2V | A. Forey | |

Abstract | O-minimal structures provide a framework for tame topology as envisioned by Grothendieck. Originally it was mainly a topic of interest for real algebraic geometers. However, since Pila and Wilkie proved their counting theorem for rational points of bounded height, many applications to diophantine and algebraic geometry have been found. | |||||

Objective | The overall goal of this course is to provide an introduction to o-minimality and to prove results needed for diophantine applications. | |||||

Content | The first part of the course will be devoted to the definition of o-minimal structures and to prove the cell decomposition theorem, which is crucial for describing the shape of subsets of an o-minimal structure. In the second part of the course, we will prove the Pila-Wilkie counting theorem. The last part will be devoted to diophantine applications, with the proof by Pila and Zanier of the Manin-Mumford conjecture and, if time permit, a sketch of the proof by Pila of the André-Oort conjecture for product of modular curves. | |||||

Literature | G. Jones and A. Wilkie: O-minimality and diophantine geometry, Cambridge University Press L. van den Dries: Tame topology and o-minimal structures, Cambridge University Press | |||||

Prerequisites / Notice | This course is appropriate for people with basic knowledge of commutative algebra and algebraic geometry. Knowledge of mathematical logic is welcomed but not required. | |||||

401-4117-69L | p-Adic Galois Representations | W | 4 credits | 2V | M. Mornev | |

Abstract | This course covers the structure theory of Galois groups of local fields, the rings of Witt vectors, the classification of p-adic representations via phi-modules, the tilting construction from the theory of perfectoid spaces, the ring of de Rham periods and the notion of a de Rham representation. | |||||

Objective | Understanding the construction of the ring of de Rham periods. | |||||

Content | In addition to the subjects mentioned in the abstract the course included the basic theory of local fields, l-adic local Galois representations, an oveview of perfectoid fields, the statements of the theorems of Fontaine-Winterberger and Faltings-Tsuji. | |||||

Literature | J.-M. Fontaine, Y. Ouyang. Theory of p-adic Galois representations. O. Brinon, B. Conrad. CMI summer school notes on p-adic Hodge theory. | |||||

Prerequisites / Notice | General topology, linear algebra, Galois theory. | |||||

401-3059-00L | Combinatorics II | 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. |

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