O-minimality

This is the title of a talk I gave at the CMU Graduate Student Seminar on Feb. 19, 2014.

Abstract

Imagine living in a world where every set of real is a finite union of intervals and points. How much easier would life then be ?

The talk will make this question precise by introducing the notion of o-minimality. I will show that in this universe, many dreams come true. For example, every real-valued function is piecewise continuous, and every subset of the plane can be decomposed into finitely many very simple sets, the so-called cells.

As an application, I will outline a proof of the following result of Khovanskiǐ: For any k and n, there exists C such that if p is a real polynomial with n variables, k non-zero terms (but unbounded degree), and finitely many real roots, then it has fewer than C such roots.

References

• Lou van den Dries. Tame Topology and O-minimal Structures, Cambridge University Press (1998).
• Lou van den Dries. Remarks on Tarski's problem concerning (R,+,⋅,exp), Logic Colloquium '82, North-Holland (1984), pp. 97–121.
• A. G. Khovanskiǐ. Fewnomials, Princeton University Press (1991).