Prof. Dr. Jens Franke

Material related to Mihailescu's CIDE primality proof

In a first use of these ideas of Mihailescu, certificates for the Leyland numbers 311063+633110 and 86562929+29298656 were calculated in late 2012. The description of the format, together with a (in my opinion) complete mathematical proof that it is indeed a valid primality proof, is here. While the terminology in fmt-0.1.pdf has been chosen to be disjoint from the terminology of the Mihailescu preprints quoted there, all crucial ideas are Mihailescu's.

Lecture Algebra II (Real Algebra) WS25/26

The lectures are on Mondays 16:00 and Thursdays 14:00 in Kleiner Hörsaal Wegelerstraße 10. An introduction to real closed fields, spectral spaces and the real spectrum, Positivstellensätze and Hilbert's 17th problem will be given. A good knowlodge of the material typically presented in the basic Galois theory lectures, as well as a little bit of knowledge of general topology, will be needed. Apart from this the lecture will be self contained. Exercises are on Fridays 14:00 0.008 in Mathematikzentrum Endenicher Allee 60. Admission to the exam depends on successful participation in the exercises. The detailed conditions for this will be explained in the first lecture.

Lecture Advanced Topics in Algebra (Etale Cohomology) WS25/26

The lectures are currently scheduled for Mondays 18:00 Großer Hörsaal Wegelerstraße 10 and Thursdays 16:00 Kleiner Hörsaal in the same building. It is intended to give an introduction to Etale Cohomology leading at least to a complete proof of proper base change. Detailed proofs will be given, although Artin approximation will be used in the proof of proper base change but not proved. Also, a lot of material about flat morphisms and abelian schemes will be used without proof. The etale cohomology part of the proofs will however be relatively complete. I intend to complement this in the next term by an introduction to the proof of the Weil conjecture where the opposite approach will be taken: First the remaining necessary prerequisites from etale cohomology will be formulated without proof and then the Weil conjectures shown. After this some of the omitted proofs from etale cohomology will be given, as far as the remaining time allows. Very good knowledge of Algebraic Geometry I/II is an absolute prerequisite, as is some familiarity with abelian categories and homological algebra. As was already said some additional material will be needed. This means that you may have to look up the formulation of these results (eg, on flat morphisms) in the literature. This lecture is thus very demanding.

Seminar "Geometrische Konstruktionen und transzendente Zahlen."

Das Seminar fand erstmalig im Sommersemester 2016 für Studenten des zweiten Semesters statt. Um einen guten Anschluß an die Vorlesung "Lineare Algebra I" sicherzustellen, diente ein von mir selbst verfaßter Text als Grundlage des Seminares. Dieser soll hier weiterhin zur Verfüfung gestellt werden.

Sprechstunden

In der vorlesungsfreien Zeit sind die Sprechstunden nach Vereinbarung.

Vorlesungen "Mathematik für Physiker I-III"

Die Javascript-Programme zu den Anwesenheitsübungen dieser Vorlesungen, die ich zwischen 2008 und 2011 gehalten habe, sind weiterhin online:

Selected Publications