|Math 285D: An Introduction to O-Minimality
Time and Place: MWF 3-3:50pm, Mathematical Sciences Building 5217
E-mail address: matthiasmath.ucla.edu
Office: Mathematical Sciences Building 5614
Office Phone: (310) 206-8576
Office Hours: M 2-2:50pm, 4-4:50pm, W 4-4:50pm, or by appointment.
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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 model theory
, over the last twenty years, the notion of an o-minimal structure has proven to be increasingly useful in the fields of real algebraic
and real analytic geometry. The general theory has even had applications to subjects as varied as Lie theory
, and neural networks
This course will be split into two parts. In the first half, we will
introduce the o-minimality axiom and develop its main consequences. We
will discuss basic facts about o-minimal structures: their geometric,
model-theoretic, and algebraic (valuation-theoretic) properties. The
second half of the course will be designed according to the interests
of the participants. Possibilities include the discussion of some of
the various new and mathematically interesting examples of o-minimal
structures, and methods to construct them, which have emerged in recent
years; or a further development of the valuation theory
of o-minimial structures, with applications to limit sets and tropical geometry
Prerequisities Some basic knowledge of first-order logic, model theory, and
abstract algebra should be sufficient. No knowledge of real algebraic
and analytic geometry will be assumed.
The following book will be a good companion for the first half of the course:
Lou van den Dries
, Tame Topology and O-Minimal Structures
, London Math. Soc. Lecture Note Series, vol. 248, Cambridge University Press, Cambridge (1998).
A few of the papers possibly relevant for the second half of the course are:
Lou van den Dries, Limit sets in o-minimal structures.
Jean-Philippe Rolin, Patrick Speissegger, Alex Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality
. J. Amer. Math. Soc. 16
(2003), no. 4, 751--777.
Patrick Speissegger, The Pfaffian closure of an o-minimal structure
. J. Reine Angew. Math. 508
Lou van den Dries, T-convexity and tame extensions.
II. J. Symbolic Logic 62
(1997), no. 1, 14--34.
Alex Wilkie, Model
completeness results for expansions of the ordered field of real
numbers by restricted Pfaffian functions and the exponential function
. J. Amer. Math. Soc. 9
(1996), no. 4, 1051--1094.
Lou van den Dries, Adam Lewenberg, T-convexity and tame extensions
. J. Symbolic Logic 60
(1995), no. 1, 74--102.
Lou van den Dries, Angus Macintyre, David Marker, The elementary theory of restricted analytic fields with exponentiation
. Ann. of Math. (2) 140
(1994), no. 1, 183--205.
Chris Miller, Exponentiation is hard to avoid
. Proc. Amer. Math. Soc. 122
(1994), no. 1, 257--259.
Jan Denef, Lou van den Dries, p-adic and real subanalytic sets
. Ann. of Math. (2) 128
(1988), no. 1, 79--138.
There will be an occasional problem set assigned on a semi-regular basis,
handed out in class, and also posted on this website.
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modified January 5, 2008.