250044 SE Algebra (2024S)

This is a student seminar focusing on 3 different topics in Algebra/Number theory. Each of them will be presented in 3 or 4 lectures, to be prepared and held by the students, and building on each other. The topics in this semester are the following:

Topic 1. (H. Grobner)
Dirichlet's Prime Number Theorem

This is a topic form number theory, which describes how the prime numbers are distributed among the invertible classes modulo m. In four lectures, we are going to reprove Dirichlet's theorem that the primes are in fact "equally distributed" among such classes. This implies as a famous corollary that for each pair of coprime natural numbers (m,n), there must be infinitely many prime numbers p, which are congruent to n modulo m. Nevertheless this result is algebraic in its nature, our methods will comprise analytic tools, such as "Dirichlet L-functions", which can be seen as a generalization of the Riemann zeta-function.

Topic 2. (J. Mahnkopf)
Topics from Category Theory (for students who have a little background in category theory or are willing to acquire a little background)

a.) Freyd's theorem on the existence of adjoint functors: the theorem gives a very general criterion for a functor F to have a left adjoint functor
(note that many theorems from algebra, topology... can be reformulated as the existence of a left adjoint functor which makes this a very general theorem)

b.) Introduction to Morita equivalence: two rings are called Morita equivalent iff their respective categories of modules are equivalent (i.e. the rings "have the same representation theory"). The (first) main theorem of Morita Theory is a general criterion for rings to be Morita equivalent
and a description of the equivalence between their module categories. This can be illustrated by typical examples.

c.) The embedding Theorem for abelian categories: any abelian category embeds into a category of modules over some ring R.

The topics can be prepared by a single student or by two students.

Topic 3. (L. Summerer)
Roth's Theorem

This topic is from Diophantine Approximation and focused on Roth's celebrated result about the bound for the quality of approximation of algebraic numbers by rationals. The talks are aimed to shed light on the context of Roth's Theorem along with an outline of the proof, applications of Roth's result and the generalisation towards the subspace Theorem of W. Schmidt.