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UIC Model Theory Seminar, April 13, 2004 Expansions of o-minimal structures
 

Summary: UIC Model Theory Seminar, April 13, 2004
Expansions of o-minimal 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 o-minimal if every definable
set has finitely many connected components. Such structures are a natural
setting for studying "tame" objects of real-analytic geometry such as non-
oscillatory trajectories of real-analytic planar vector fields. It turns out that
even some infinitely spiralling trajectories of such vector fields have a reason-
ably well-behaved model theory; this motivates the notion of d-minimality,
a generalization of o-minimality 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., f-1(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

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics