Remi Jaoui, Department of Pure Mathematics, University of Waterloo
"Qualitative probability theory and types"
Following Tao's blog post "Qualitative probability theory, types and the group chunk and group configuration theorems'', we will define a qualitative version of probability theory, suitable to relate the probabilistic notion of independence with the model-theoretic one.
In this setting, qualitative measures are finitely additive functions from a Boolean algebra $(M,\mathcal B)$ to the set with three elements ${0,I,1}$. In fact, qualitative measures simply provide a different way of speaking about filters --- and therefore, partial types --- of the Boolean algebra $(M,\mathcal B)$.
A typical feature of model-theoretic structures $\mathcal M$ is that the Boolean algebra $\mathcal D( M \times \cdots \times M)$ of definable sets in some product of $M$ is often larger than the product Boolean algebra $\mathcal D(M) \times \cdots \times \mathcal D(M)$. Taking this aspect into account, one has to work carefully to define qualitative measures on a product from qualitative measures on its factors.
Once this has been settled, we will apply the "probabilistic way of thinking'' to definable sets in the theory of algebraically closed fields. In other words, we will study qualitative random variables with values in definable sets.
MC 5403