Focus topic: Algebra

In algebra, one deals with the solutions of polynomial equations in a variable X, such as X^{^2} - 2 = 0 or X^{^5} + 4·X + 2 = 0. The set of all solutions of such polynomial equations with rational numbers as coefficients forms one field Q^alg. This means that you can add, subtract, multiply, and divide the numbers in it. One of the main problems of number theory is to better understand the field Q^alg. On the other hand, if polynomial equations (or systems of polynomial equations) are considered in several variables, their sets of solutions have a geometric structure (see picture). Also, e.g. the real solution set of X^{^2} + Y^{^2} - 1 = 0 exactly the circle line. In addition, these sets of solutions also have a rich algebraic structure that results from considering polynomial equations. The branch of mathematics that investigates these sets of solutions is called \bf{algebraic geometry}. A more detailed description can be found http://www.math.uni-muenster.de/u/urs.hartl/gifs/Weitere_Information_Algebra.pdf here.

"Zitrus" from Herwig Hauser is the real solution set of X^{^2} + Z^{^2} + Y^{^3}·(Y−1)^{^3} = 0.

(Own work by User: Andreasmatt. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons https://commons.wikimedia.org/wiki/File:IMAGINARY_Zitrus_Herwig_Hauser.jpg)

Other algebraic areas that are the focus in Münster are representation theory, group theory and the theory of automorphic forms. Of course, algebra is not an isolated individual science, but is closely related to other areas of mathematics (such as differential geometry, topology, logic, and operator algebras and noncommutative geometry).

Courses for the specialisation in Algebra

Winter semester 2019/2020

• Prof. Dr. Lutz Hille: Algebraic Geometry 2 (Type I, II)
• Prof. Dr. Peter Schneider: Higher Algebra (Representation theory of finite groups) (Type I, II)
• Prof. Dr. Eugen Hellmann: Moduli Spaces of Elliptic Curves (Type II)
• Prof. Dr. Eugen Hellmann: Seminar about Algebraic Geometry (Type II)
• Prof. Dr. Christopher Deninger: Seminar Adic Spaces(Type II)
• Prof. Dr. Michael Weiss: Seminar Lie-Algebras, Lie-Groups and their Representations (Type II)
• PD Dr. Jakob Scholbach: Seminar Infinity Categories (Type II)

Summer semester 2020

• Prof. Dr. Lutz Hille: Algebraic Geometry 3 (Type I, II)
• Prof. Dr. Linus Kramer: Topological groups (Type I, II)
• Prof. Dr. Katrin Tent: Stable Groups (Type I, II)

Prerequisites

A lecture "Introduction to Algebra" with the following contents:

Groups, isomorphism theorems, abelian groups, permutation groups, group operations, and Sylow theorems. Rings, ideals, polynomial rings, Euclidean rings, principal ideal rings, factorial rings, divisibility in rings. Fields, field extensions and splitting fields. Galois extensions, Galois theory and applications.

Further information

The working groups with algebraic orientation present in Münster cover large parts of modern algebra and number theory. The focus of the working groups are in detail:

• AG Arithmetische Geometrie (Prof. Deninger): Algebraic geometry, algebraic number theory
• AG Prof. Hartl: Algebraic geometry, arithmetic of function fields
• AG Prof. Hellmann: Algebraic geometry, Galois representations, p-adic automorphic forms
• AG Prof. Hille: Representation theory, toric algebraic geometry
• AG Geometrie, Topologie und Gruppentheorie (Prof. Kramer): Geometric group theory, Lie groups, theory of buildings
• AG Prof Scherotzke: Representation theory, derivatived algebraic geometry
• AG p-adische Arithmetik (Prof. Schneider): Representation theory, p-adic analysis, Langlands program
• AG Modelltheorie und Algebra (Prof. Tent): Group theory, mathematical logic, model theory

On the pages of these working groups you will find further information about lectures, seminars, master theses etc.

 Contact and module representative

Prof. Dr. Urs Hartl