Lecture on Cyclotomic Fields and Iwasawa Theory -- Introduction, Research Problems, Computational Aspects and Some New Results
The content of this lecture will gravitate in the larger area described by the title. The closer framework will become evident in the first weeks of the lecture, by interaction with the attendants. For a warm up, we shall discuss computational and research aspects around a classical (from the 1950es ... ) conjecture of Ankeny, Artin and Chowla (AAC). With some luck this might exemplify ways in which a practicing number theorist can get help from the computer and symbolic computation systems. The further major topics will be connected to the conjecture of Kummer { Vandiver, and several important consequences, which are also long standing questions, but were never formulated as a conjecture.
The amount of background teaching of basics, versus actual research results will emerge from the public interaction. The basic lecture time will be two sessions of 1.30 hours { with or without break. It is possible to arrange for some additional session, on a weekly base or once every two weeks, dedicated to questions and problem solving.
If you have any questions, please send emails to Prof. Preda Mihăilescu by preda@bimsa.cn or preda@mathematik.uni-goettingen.de.
Simple questions can be discussed on WeChat, but more involved ones (which require more text answer) are preferable on mail, for ease of editing.
The amount of background teaching of basics, versus actual research results will emerge from the public interaction. The basic lecture time will be two sessions of 1.30 hours { with or without break. It is possible to arrange for some additional session, on a weekly base or once every two weeks, dedicated to questions and problem solving.
If you have any questions, please send emails to Prof. Preda Mihăilescu by preda@bimsa.cn or preda@mathematik.uni-goettingen.de.
Simple questions can be discussed on WeChat, but more involved ones (which require more text answer) are preferable on mail, for ease of editing.
Lecturer
Date
18th September ~ 4th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 14:20 - 16:05 | A3-4-101 | ZOOM B | 462 110 5973 | BIMSA |
Prerequisite
Algebra -- good knowledge, level of 2 semester lecture (must) . Basic algebraic number theory and cycltomic fields (highly wished). Class field theory, p-adic analysis (helpful)
Reference
[1] K.~Iwasawa. On $\mathbb{Z}_{\ell}$ - extensions of number fields, Ann. Math. Second Series, 98:247 -- 326, 1973.
[2] S. Lang: Cyclotomic Fields I & II, Springer 1989
[3] S. Lang: Algebraic Number Theory, Springer.
[4] R. Long: Algebraic number theory, Marcel Dekker, Monographs in Pure and Applied Mathematics 41, (1977).
[5] P. Ribenboim: $13$ Lectures on Fermat's Last Theorem, Springer Verlag (1979).
[6] P. Ribenboim: Algebraic Numbers, Springer 2000.
[7] L. Washington: Introduction to Cyclotomic Fields, Second Edition, Springer (1996), Graduate Texts in Mathematics 83.
[2] S. Lang: Cyclotomic Fields I & II, Springer 1989
[3] S. Lang: Algebraic Number Theory, Springer.
[4] R. Long: Algebraic number theory, Marcel Dekker, Monographs in Pure and Applied Mathematics 41, (1977).
[5] P. Ribenboim: $13$ Lectures on Fermat's Last Theorem, Springer Verlag (1979).
[6] P. Ribenboim: Algebraic Numbers, Springer 2000.
[7] L. Washington: Introduction to Cyclotomic Fields, Second Edition, Springer (1996), Graduate Texts in Mathematics 83.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
He studied mathematics and computer science in Zürich, receiving a PhD from ETH Zürich in 1997. His PhD thesis, titled Cyclotomy of rings and primality testing, was written under the direction of Erwin Engeler and Hendrik Lenstra. After his first studies, during close to 20 years, he worked in Zürich in the industry, first as a numerical analyst, then as a developer and consultant in IT-Security: cryptogrphy and fingerprint-identification. After 2000, during five years, he did research at the University of Paderborn, Germany, where he proved the long standing 'Conjecture of Catalan', in 2002. In 2005 he received a Wolkswagen Foundation professorship at the University of Göttingen, where he has been a professor ever since.
During the last decade he specialized in abelian Iwasawa Theory and developed new methods for the investigation of open problems in the domain.
During the last decade he specialized in abelian Iwasawa Theory and developed new methods for the investigation of open problems in the domain.