Universität Wien
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250053 VO Model Theory of Valued Fields (2021S)

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

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

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

Information

Aims, contents and method of the course

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.

Assessment and permitted materials

Minimum requirements and assessment criteria

Examination topics

Reading list

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.

Association in the course directory

MLOV;

Last modified: Fr 12.05.2023 00:21