Math 285D: An Introduction to O-Minimality |

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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 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, economics, 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.

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.

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. Preprint (2004).

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 (1999), 189--211.

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.

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. Preprint (2004).

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 (1999), 189--211.

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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