### Overview

An expansion of the real field (in the sense of model theory) is called
o-minimal if every definable set has finitely many connected components.
Such structures are a natural setting for studying tame aspects of real
analytic geometry, such as non-oscillatory trajectories of planar vector
fields. Recently, it was discovered that even some infinitely spiralling
trajectories of such vector fields have a reasonably 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 striking trichotomy illustrates
why we are interested in d-minimality:

Let F be the germ at the origin of a real analytic planar vector field,
and assume that the origin is an elementary singularity of F.

Let g:(0,a) --> R^2 be a trajectory of F such that g(t) --> 0
as t --> 0, and let R_g be the expansion of the real field by g((0,a)).

Then exactly one of the following three cases hold:

(i) R_g is o-minimal;

(ii) R_g is d-minimal and not o-minimal;

(iii) the set of all integers is definable in R_g.

The aim of the workshop is to present the proof of this trichotomy,
and then to work jointly on the following related questions:

(1) Does the trichotomy extend to non-elementary singularities?

(2) Does the trichotomy extend to other classes of planar vector fields,
such as vector fields definable in (certain) o-minimal structures?

(3) Are there good criteria to decide which of the cases (i), (ii) and
(iii) apply to a given vector field?

The workshop is intended to be for a small group of particpants so
that considerable time can be spent on discussion (rather than presentations).
In this way, we hope to be able to come up with new ideas that might
lead to results eventually published by various participants.

For further information please contact gensci@fields.utoronto.ca