 
Summary: UIC Model Theory Seminar, April 13, 2004
Expansions of ominimal structures
by trajectories of definable planar vector fields
Chris Miller,
The Ohio State University,
Columbus, OH
An expansion of the real field is said to be ominimal if every definable
set has finitely many connected components. Such structures are a natural
setting for studying "tame" objects of realanalytic geometry such as non
oscillatory trajectories of realanalytic planar vector fields. It turns out that
even some infinitely spiralling trajectories of such vector fields have a reason
ably wellbehaved model theory; this motivates the notion of dminimality,
a generalization of ominimality that allows for some definable sets to have
infinitely many connected components. The following trichotomy illustrates
why we are interested in this notion.
Let U R2 be open and F : U R2 be real analytic such that the origin
0 is an elementary singularity of F (i.e., f1(0) = {0} and the linear part
of F at 0 has a nonzero eigenvalue). Let g: (a, b) R2 be a solution to
y = F(y) such that g(t) 0 as t a+. Then, after possibly shrinking b,
exactly one of the following holds for the expansion M of the real field by
