Universität Wien
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250136 VO Topics in Model Theory (2023S)

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

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

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

Information

Aims, contents and method of the course

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.

Assessment and permitted materials

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

Minimum requirements and assessment criteria

Examination topics

Reading list

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.

Association in the course directory

MLOV

Last modified: Tu 03.10.2023 14:28