# Model Theory Seminar

### School of Mathematics, University of Leeds

This week's seminar will be in Roger Stevens Lecture Theatre 7 and will be on Tuesday 11th March at 2pm. We will meet before the seminar at about 1.40pm in the common room for tea and coffee.

The seminar is organised by Will Anscombe.

### Spring Term 2014

Week 9, Tuesday 11th March
Will Anscombe
Supersimple generalised measurable structures

I will discuss some recent (and ongoing) work (with Dugald Macpherson and Daniel Wood) about generalised measurable structures. A generalised measurable structure is a structure $\mathcal{M}$, a semiring $R$, and a function $h$ with domain the set of definable sets in $\mathcal{M}$ and codomain $R$, which all satisfy a short list of axioms. The intuition is that $h(X)$ is the `size' of a definable set $X$. Generalised measurable structures can be obtained as ultraproducts of multidimensional asymptotic classes (MACs). In this context, $h(X)$ is a function which, when evaluated in a structure $\mathcal{M}$ from the MAC, gives an approximation to the cardinality of $X(\mathcal{M})$. If we impose the assumption that the semiring $R$ is a polynomial ring in finitely many variables, then our generalised measurable structure must be supersimple. It's also possible to give a description of $D$-rank.