Ibrahim Mohammed (Leeds)

Weak o-minimality

A structure expanding a dense linear order is weakly o-minimal if any definable set in one variable is a finite union of points and convex sets. This is a more general condition compared to regular o-minimality, where we require these sets to be a finite union of points and intervals.

As a result, weak o-minimality doesn't have all the useful properties that o-minimality has, for example it's not preserved under elementary equivalence, nor can we show that any definable function is piecewise monotone and continuous.

In this talk I'll highlight some of the nice properties o-minimal structures have, give examples of weakly o-minimal structures which break those properties, and then show how we can salvage weaker versions of the properties in a weakly o-minimal setting.

Most of this talk will be based on a paper by Dugald Macpherson, David Marker and Charles Steinhorn called "Weakly O-minimal structures and real closed fields".

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