# Teaching winter term 2024/25

# Algebraic Number Theory

Tutorial Algebraic Number Theory

Time: Mo, Do 10:00 - 12:00 Uhr

Place: M 3

The lecture course will provide an introduction to Algebraic Number Theory and will cover Algebraic numbers, Dedekind rings and their extensions, Cyclotomic fields, Completions and local fields, Adeles and Ideles.Literatur:J. W. S. Cassels and A. Fröhlich, editors. Algebraic number theory. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. Reprint of the 1967 original.

Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

# Semi-simple Lie-Algebras and their representations

Semi-simple Lie-Algebras and their representations

Time: Di 14:00 - 16:00 Uhr

Place: SR 1C

Organizational meeting: Tuesday 24.9.24, 14h SR 1C

Lie algebras are ubiquitous in algebra, but also in many other areas of mathe- matic. They usually arise as linearization (i.e. as the tangent space) of Lie groups or algebraic groups, but can also be studied as objects of intrinsic interest. By def- inition they are given by a vector space g over a field k together with a Lie bracket [−, −] : g×g → g that behaves like the commutator XY −Y X on gln = Matn×n(k). In the first part of the seminar we will study Lie algebras and in particular semi- simple Lie algebras. It turns out that these objects admit a beautiful classification in terms of root systems.

In the second part of the seminar we will study representations of semi-simple Lie algebras. The category of finite dimensional representations turns out to be semi- simple and there is a complete classification of the irreducible representations. The last three talks will present an introduction to further and more advanced topics about certain infinite dimensional representations.

Literatur:

J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer, 1974.

J.E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics 94, AMS 2008.# Colloquium of Pure and Applied Mathematics

# Mittagsseminar "Arithmetic"

**Time:**Tuesday, 10 - 12 h**Place:**SRZ 216/217

**Learnweb-Link:**

A list of the talks can be found here.

# Research seminar "p-adic arithmetic"

Research Seminar "p-adic arithmetic"

**Time:**Monday, 12- 14 h**Place:**SRZ 216/217**Topic:**Vector bundles in the v-topology and Sen theoryA list of the talks can be found here.