Universität Wien
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250126 VO Topics in Model Theory (2025W)

Constructing O-minimal Structures

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 02.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 07.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 09.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 14.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 16.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 21.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 23.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 28.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 30.10. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 04.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 06.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 11.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 13.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 18.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 20.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 25.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 27.11. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 02.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 04.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 09.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 11.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 16.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 18.12. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 08.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 13.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 15.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 20.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Donnerstag 22.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01
  • Dienstag 27.01. 13:15 - 14:45 Seminarraum 10, Kolingasse 14-16, OG01

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

O-minimality is a property of ordered structures which yields results generalizing the classical finiteness theorems long known to hold for semialgebraic and subanalytic sets, such as the existence of cell decompositions and Whitney stratifications. This leads to a development of a kind of "tame topology" (envisaged by Grothendieck). Although originating in mathematical logic (model theory), the notion of an o-minimal structure
has proven to be useful in real algebraic and real analytic geometry, and the general theory has even had applications to subjects as varied as Lie theory, economics, and neural networks.

Most recently, o-minimality has also found surprising uses in diophantine geometry, among other things being fundamental to the (unconditional) proof of the André-Oort Conjecture (2021). An important role in these developments is played by an o-minimal structure denoted by R_{an,exp} (the ordered field of real numbers expanded by restricted analytic functions and the exponential function), because it defines all elementary functions (with suitable necessary restrictions on periodic ones such as sine and cosine).

The construction of o-minimal structures often uses ideas from elimination theory and resolution of singularities. In this course we will focus on such methods, using R_{an,exp} as our guiding example. We will introduce the o-minimality axiom and its main consequences, and then give a complete proof of the o-minimality of R_{an,exp}, using techniques of van den Dries, Rolin, Speissegger, Wilkie. Time permitting, we will also touch on further topics.

I will try to make the course accessible to students with varying backgrounds, and hence only assume a basic knowledge of analysis, algebra, and logic (on the bachelor level). Feel free to ask me if you are in doubt about your level of preparation.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Final exam at the end of the semester.

Mindestanforderungen und Beurteilungsmaßstab

Prüfungsstoff

Literatur

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

J. Denef, L. van den Dries, p-adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), no. 1, 79–138.

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

L. van den Dries, A. Macintyre, D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205.

L. van den Dries, P. Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513–565.

J.-P. Rolin, P. Speissegger, A. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777.

Zuordnung im Vorlesungsverzeichnis

MLOV; ML2; MEL

Letzte Änderung: Mi 17.12.2025 11:27