Large o-minimal structures

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The main focus of the proposed research is on finding as yet unknown examples of o-minimal structures. Essentially two types of such structures are known at present: one is obtained by adding functions belonging to certain quasianalytic classes to the field of real numbers, and the other by adding Pfaffian functions to any given o-minimal structure (e.g., the field of real numbers, or an o-minimal structure of the previous type). The investigator and his colleagues intend to extend currently known methods for proving the o-minimality of such structures to other classes of functions, such as Denjoy-Carleman classes or certain solutions of differential equations (other than Pfaffian functions). The "tameness" properties of sets definable in o-minimal structures have consequences that can be of interest to many areas of mathematics. For instance, the structures above are inspired by the various current attempts to understand the mathematics of Hilbert's 16th problem, one of the famous list of problems in mathematics proposed by German mathematician David Hilbert at the turn of the last century. The problem remains unsolved to this date, and while its solution may not have immediate applications to areas of science other than mathematics, the mathematical methods developed thus far in the attempt to understand the problem have already proved useful in the study of dynamical systems (which in turn is a major tool used in physics and engineering). The investigator and his colleagues think that their proposed reasearch will eventually lead to new insights and methods in this very active area of mathematics.