Universität Wien

250053 VO Model Theory of Valued Fields (2021S)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

  • Dienstag 02.03. 10:30 - 11:15 Digital
  • Donnerstag 04.03. 10:30 - 11:15 Digital
  • Dienstag 09.03. 10:30 - 11:15 Digital
  • Donnerstag 11.03. 10:30 - 11:15 Digital
  • Dienstag 16.03. 10:30 - 11:15 Digital
  • Donnerstag 18.03. 10:30 - 11:15 Digital
  • Dienstag 23.03. 10:30 - 11:15 Digital
  • Donnerstag 25.03. 10:30 - 11:15 Digital
  • Dienstag 13.04. 10:30 - 11:15 Digital
  • Donnerstag 15.04. 10:30 - 11:15 Digital
  • Dienstag 20.04. 10:30 - 11:15 Digital
  • Donnerstag 22.04. 10:30 - 11:15 Digital
  • Dienstag 27.04. 10:30 - 11:15 Digital
  • Donnerstag 29.04. 10:30 - 11:15 Digital
  • Dienstag 04.05. 10:30 - 11:15 Digital
  • Donnerstag 06.05. 10:30 - 11:15 Digital
  • Dienstag 11.05. 10:30 - 11:15 Digital
  • Dienstag 18.05. 10:30 - 11:15 Digital
  • Donnerstag 20.05. 10:30 - 11:15 Digital
  • Donnerstag 27.05. 10:30 - 11:15 Digital
  • Dienstag 01.06. 10:30 - 11:15 Digital
  • Dienstag 08.06. 10:30 - 11:15 Digital
  • Donnerstag 10.06. 10:30 - 11:15 Digital
  • Dienstag 15.06. 10:30 - 11:15 Digital
  • Donnerstag 17.06. 10:30 - 11:15 Digital
  • Dienstag 22.06. 10:30 - 11:15 Digital
  • Donnerstag 24.06. 10:30 - 11:15 Digital
  • Dienstag 29.06. 10:30 - 11:15 Digital

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

The concept of a valuation, which arose early in the 20th century to better understand the then-newfangled concept of p-adic number, now plays an important role not only in number theory but also in commutative algebra and various flavors of geometry: algebraic, semialgebraic, rigid analytic, tropical, to name a few. The general theory of (Krull) valuations is not usually part of the university curriculum in algebra, but is crucial for applications of model theory to these fields.

The aim of this course is to provide a thorough introduction to valued fields, with a particular emphasis on their model theory, leading up to the classical results by Ax-Kochen-Ershov and Macintyre on the model theoretic properties of the p-adics. This should prepare students to be able to study more recent developments, such as motivic integration or the model-theoretic investigation of fields with additional structure.

I will try to make the course accessible to students with varying backgrounds, and hence only assume a basic knowledge of algebra (groups, rings, fields) and logic (on the bachelor level). If in doubt about your preparation, ask me.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Mindestanforderungen und Beurteilungsmaßstab

Prüfungsstoff

Literatur

I’ll follow my own notes, but some useful references for this class are:

M. Aschenbrenner, L. van den Dries, J. van der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, Annals of Mathematics Studies, vol. 195, Princeton University Press, Princeton, NJ, 2017. (Chapters 1, 2, 3.)

L. van den Dries, Lectures on the model theory of valued fields, in: D. Macpherson, C. Toffalori (eds.), Model Theory in Algebra, Analysis and Arithmetic, pp. 55–157, Lecture Notes in Mathematics, vol. 2111, Springer, Heidelberg, 2014.

A. J. Engler, A. Prestel, Valued Fields, Springer Monographs in Mathematics, Springer- Verlag, Berlin, 2005.

A. Prestel, P. Roquette, Formally p-adic Fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, Berlin, 1984.

P. Ribenboim, The Theory of Classical Valuations, Springer-Verlag, 1999.

Zuordnung im Vorlesungsverzeichnis

MLOV;

Letzte Änderung: Fr 12.05.2023 00:21