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 "Rigid analytic geometry I" WS24/25
The lecture has been moved to Tuesdays and Thursdays 18ct-20 in Großer
Hörsaal Wegelerstraße 10. Rigid analytic geometry will be introduced in the
classical framework of Tate and the Grauert school but additional points will
be introduced early, using the definition of van der Put. Knowledge of
topology, homological algebra and basic commutative algebra at a level
similar to the prerequisites for a basic algebraic geometry course is
required.
Lecture "Algebra II" WS24/25
The lecture is scheduled for Mondays 16:00-18:00ct and Thursdays
14:00-16:00ct in Kleiner Hörsaal Wegelerstraße 10. The theory of Dedekind
rings will be developed and applied to the theory of algebraic number
fields and their local fields. In particular, the ideal group and the ideal
class group will be introduced and the finiteness of the ideal class group
and Dirichlet's theorem about the structure of the group of units will be
shown. A good knowledge of Galois theory and the other Introduction to
Algebra topics, as well as some familiarity with Noetherian rings and
knowledge of the analysis lectures from the first year will be required.
The exercise sheets will be published on this home page:
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
- On the spaces Fspq of Triebel-Lizorkin type:
pointwise multipliers and spaces on domains, Math. Nachr. 125 (1986),
113-149.
- (with Yu. I. Manin and Yu. Tschinkel)
Rational points of bounded height on Fano varieties,
Invent. Math. 95(1989), 421-435
- (with T. Runst)
Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces.
Math. Nachr. 174 (1995), 113-149.
- Harmonic analysis in weighted L2-spaces,
Ann. Sci. École Norm. Sup. (4), 31(1998), 181-279
- (with J. Schwermer),
A decomposition of spaces of automorphic forms, and the Eisenstein
cohomology of arithmetic groups,
Math. Ann. 311(1998), 765-790.
- On the singularities of residual Eisenstein series,
Invent. Math. 138(1999), 307-317
- (With T. Kleinjung, F. Morain and T. Wirth)
Proving the primality of very large numbers with fastECPP,
in Algorithmic number theory,
Lecture Notes in Comput. Sci., 3076, 2004, pages 194-207.
- (With T. Kleinjung),
Continued fractions and lattice sieving.
In: Proceedings SHARCS 2005
- (with K. Aoki, T. Kleinjung, A. Lenstra, D. Osvik)
A kilobit special number field sieve factorization,
in Advances in cryptology. ASIACRYPT 2007,
Lecture Notes in Comput. Sci., 4833, 2007, pages 1-12.
- A topological model for some summand of the Eisenstein
cohomology of congruence subgroups, in
Eisenstein series and applications, Progr. Math., 258, 2008,
pages 27-85.