# 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

3110^{63}+63^{3110}
and

8656^{2929}+2929^{8656}
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 "Weil I" SS24

The lecture is scheduled for Tuesdays and Fridays 14-16ct in room 0.007
Endenicher Allee 60. As there will be no lectures on April 8 or 9 the first
lecture will be on Friday April 12.
The proof of the Weil conjectures from Deligne's Weil I paper will be
presented. Knowledge of etale cohomology, from the basic definitions to
proper base change, is a prerequisite. The other needed results from etale
cohomology will be stated mostly without proofs. If sufficient time remains
at the end of the lecture, some but certainly not all of these omitted proofs
may be given.

## Lecture "Real algebraic geometry 2"

The lecture is scheduled for Tuesdays 8:00-10:00 and Fridays 10:00-12:00 and
the exercises for Mondays 14-16ct, all in room N0.007 Endenicher Allee 60.
As there are no lectures or exercises in Endenicher Allee 60 on April 8 or
9, the first lecture will be on Friday April 12.
Sheaf cohomology of the real spectrum will be discussed. In particular, it
will be shown that the cohomology of locally constant sheaves has the
properties expexted by the comparison with algebraic topology. While some
basic familiarity with algebraic topology makes it easier to follow the
lecture and understand the motivation for these considerations, it is not a
prerequisite. However, a knowledge of real closed fields, of the real
spectrum and the general theory of spectral spaces is necessary. Previous
knowledge of the machinery of sheaf cohomology as a derived functor of the
functor of global sections will simplify things, but a crash course on this
topic can be given in the exercises if needed.

- Exercise sheet 1, due Friday April 19.
- Exercise sheet 2, due Friday April 26.
- Exercise sheet 3, due Friday May 3.
- Exercise sheet 4, due Friday May 10.
- Exercise sheet 5, due Friday May 17.
- Exercise sheet 6, due Friday May 31.
- Exercise sheet 7, due Friday June 7.
- Exercise sheet 8, due Friday June 14.
- Exercise sheet 9, due Friday June 21.
- Exercise sheet 10, due Friday June 28.

## Seminar "Geometrische Konstruktionen und transzendente Zahlen."

Das Seminar fand 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 F**^{s}_{pq} 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 L**_{2}-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.