3:00–4:00 pm
Eckhart Hall Room 202 5734 S University Ave Chicago, IL 60637
When is a Mathematical Object Well-Behaved?
In this talk we will come at this question from two
different angles: first, from the viewpoint of model theory, a subject
in which for nearly half a century the notion of stability has played
a central role in describing tame behaviour; secondly, from the
perspective of combinatorics, where so-called regularity
decompositions have enjoyed a similar level of prominence in a range
of finitary settings, with remarkable applications.
In recent years, these two fundamental notions have been shown to
interact in interesting ways. In particular, it has been shown that
mathematical objects that are stable in the model-theoretic sense
admit particularly well-behaved regularity decompositions. In this
talk we will explore this fruitful interplay in the context of both
finite graphs and subsets of abelian groups.
To the extent that time permits, I will go on to describe recent joint
work with Caroline Terry (The Ohio State University), in which we
develop a higher-arity generalisation of stability that implies (and
in some cases characterises) the existence of particularly pleasant
higher-order regularity decompositions.