250004 SE Seminar in Algebra and Number Theory (2022S)

This is a student seminar focusing on 3 different current research topics in Algebra and Number theory, with an additional goal of helping students to develop good habits on how to listen to and understand research talks on various topics.

Throughout the semester there will be 3 sessions, each giving a glimpse into a different topic. After a short introduction, each session will consist of 4 lectures structured as follows. In the first 3 lectures, students will prepare and present topics relevant to the focus of the session. Then in the 4th lecture, we will have an invited speaker, who gives a seminar-like research talk in the focus of the session.

The topics in this semester are the following:

Topic 1. (Univ.-Prof. Mag. Dr. Leonhard Summerer)
The Subspace Theorem
This famous Theorem is due to Wolfgang Schmidt who proved it in 1972, i.e. exactly 50 years ago. It is the ultimative generalization of Roth' Theorem, which characterizes how good algebraic numbers may be approximated by rational numbers. Through the talks we will get an introduction to Diophantine Approximation, see in which way Roth' Theorem is generalized by the subspace Theorem and get an idea of the importance of the result via some applications. Prof. Clemens Fuchs will give a talk on the subject on March 31st.

Topic 2. (Univ.-Prof. Dr. Alberto Mínguez) Introduction to L-functions.

L-functions are some analytic avatars attached to algebraic structures. One can see them as generalizations of the classical Riemann zeta function. They appear naturally in the Langlands program. For example, the Langlands correspondence for GL(2) relates L-functions of elliptic curves to L-functions of modular forms.

In this workshop, we will define L-functions and see some of their analytic/algebraic properties.

Topic 3. (Ass.-Prof. Vera Vértesi, Ph.D.)

Amenable groups

The Banach-Tarski paradox states that “there is a way of decomposing a 3-dimensional ball into finitely many disjoint pieces that can be rearranged to form two balls of the same size as the original one”. The proof of this counterintuitive result rests on the notion of amenable groups. In this session, we will understand amenable groups via 3 different equivalent definitions, and then introduce the Thompson group, a group with particularly nice properties, that is conjectured to be non-amenable. The last lecture will be given by Yash Lodha, giving an example of a “nicely presented” non-amenable group that is a supergroup of the Thompson group and does not have non-abelian free groups as subgroups.