Title:

Varieties of tameness phenomena in expansions of the real field

Abstract:

What does it mean for a first-order expansion of the field of real
numbers to be tame, or well behaved? In recent years, much attention has
been paid by model theorists and real-analytic geometers to the
o-minimal setting: expansions of the real field in which every definable
set has finitely many connected components. But there are expansions of
the real field that are tame in some well-defined sense, yet define sets
with infinitely many connected components. Moreover, there are different
kinds of tameness that can arise. The analysis of these structures often
requires (at least, on the face of it) a mixture of model-theoretic,
analytic-geometric and descriptive set-theoretic techniques. Underlying
all this is an idea that first-order definability, in combination with
the field structure, can be used as a tool for determining how
complicated are given sets of real numbers.

I will give a (fairly self-contained) survey of some known results and
discuss some open questions.