Prerequisite: Chapters 1 and 2 of "An Introduction to Diophantine Geometry" (Winter 2022).
Language: English.
The goal is to present a self-contained proof of the famous Mordell conjecture. We will use the Lecture Notes as reading material.
Block seminar: Irrationality and Transcendence
Prerequisite: Linear Algebra, Calculas.
Language: English.
It is a fundamental question in mathematics to determine whether a number is rational or algebraic. The goal of this block seminar is to discuss some existing tools for this purpose. We will start with the concrete examples \pi and e, and later on move to the statement of Wüstholz’s Analytic Subgroup Theorem and its applications to determine whether some numbers are transcendental. On the other direction, we will discuss tools used by Zudilin for irrationality of special values of the zeta function. In the end, we will discuss about the more recent works of Calegary, Dimitrov, and Tang on irrationality.
Weekly seminar on Number Theory and Arithmetic Geometry.
Twice per week for the main lectures, and once per week for the Exercise Class.
The goal of this course is to give an introduction to the height theory and to see some applications. We will include the proof the celebrated Roth's Theorem in the course. Then as topics we will prove the Schinzel-Zassenhaus Conjecture (Dimitrov's Theorem) and the finiteness of integral points on elliptic curves (Siegel's Thoerem).
Twice per week for the main lectures, and once per week for the Exercise Class.
The goal of this course is to give an introduction to algebraic groups. We follow the standard textbook "Linear Algebraic Groups" of A.Borel (GTM 126) and will cover Chapters I-IV.
Oberseminar Zahlentheorie und Arithmetische Geometrie
Twice per week for the main lectures, and once per week for the Exercise Class.
The goal of this course is to give an introduction to the height theory and to see some applications. The final goals are to prove Roth's Theorem and the Mordell Conjecture (Faltings's Theorem). We follow Vojta's approach for the proof of the Mordell Conjecture and take Bombieri's simplication.
The exam is oral and focuses on the first three chapters. Most proofs of the last three chapters are not required, but it is important to understand the notions and the statements of the results.
Oberseminar Zahlentheorie und Arithmetische Geometrie
