Universität Wien

250136 VO Topics in Model Theory (2023S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

  • Donnerstag 02.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 07.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 09.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 14.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 16.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 21.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 23.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 28.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 30.03. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 18.04. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 20.04. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 25.04. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 27.04. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 02.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 04.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 09.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 11.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 16.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 23.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 25.05. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 01.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 06.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 13.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 15.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 20.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 22.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 27.06. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

Hardy fields have their origin in the 19th century, with du Bois-Reymond's "orders of infinity''. Later, G. H. Hardy made sense of du Bois-Reymond's original ideas, and focused on logarithmic-exponential functions (LE-functions, for short): these are the real-valued functions in one variable defined on neighborhoods of infinity that are obtained from constants and the identity function by algebraic operations, exponentiation and taking logarithms. The asymptotic behavior of non-oscillating real-valued solutions of algebraic differential equations can often be described in terms of LE-functions. Hardy proved that the germs at infinity of LE-functions make up an ordered differential field: every LE-function, ultimately, has constant sign, is differentiable, and its derivative is again an LE-function. Bourbaki then took this result as the defining feature of a Hardy field: an ordered differential field of germs of real-valued differentiable functions defined on neighborhoods of infinity on the real line.

The modern theory of Hardy fields was mostly developed by Rosenlicht and Boshernitzan. Recently, Hardy fields have gained prominence in model theory and its applications to real analytic geometry and dynamical systems, via o-minimal structures on the real field. They have also found applications in ergodic theory and computer algebra.

In this course, after an introduction to the basics of Hardy fields, I plan to prove the classical extension theorems for Hardy fields, followed by a self-contained proof of Miller's growth dichotomy theorem for o-minimal structures. In the remainder of the semester I will explore the elementary theory of maximal Hardy fields.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Based on a written take-home exam (use of your lecture notes will be permitted). Possible dates announced in the first lecture.

Mindestanforderungen und Beurteilungsmaßstab

Prüfungsstoff

Literatur

I will follow my own notes. Here are some sources that I will rely on:

M. Aschenbrenner, L. van den Dries, Asymptotic differential algebra, in: O. Costin, M. D. Kruskal, A. Macintyre (eds.), Analyzable Functions and Applications, pp. 49–85, Contemp. Math., vol. 373, Amer. Math. Soc., Providence, RI, 2005.

L. van den Dries, Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Note Series, vol. 248, Cambridge University Press, Cambridge (1998).

C. Miller, Exponentiation is hard to avoid, Proc. Am. Math. Soc. 122 (1994), 257–259.
----, Basics of o-minimality and Hardy fields, in: C. Miller et al. (eds.), Lecture Notes on O-minimal Structures and Real Analytic Geometry, pp. 43–69, Fields Institute Communications, vol. 62, Springer, New York, 2012.

M. Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), 297–311.

Zuordnung im Vorlesungsverzeichnis

MLOV

Letzte Änderung: Di 03.10.2023 14:28