Let $T$ be an $\aleph_0$-categorical theory in first order logic, $M$ its countable model, and $G(T) = \mathrm{Aut}(M)$ with the topology of pointwise convergence. A classical result, due to Coquand (and appearing in a paper by Ahlbrandt and Ziegler) asserts that two aleph0-categorical theories $T$ and $T'$ are bi-interpretable if and only if $G(T)$ and $G(T')$ are isomorphic as topological groups. Moreover, one can reconstruct from $G(T)$ a theory $T'$ that is bi-interpretable with $T$. I will discuss a generalisation of this result to arbitrary complete theories in a countable language, at the cost of replacing the topological group $G(T)$ with a topological groupoid. Time permitting, I will also discuss what happens in continuous logic : I. Coquand's result for $aleph_0$-categorical theories generlises to continuous logic (B.-Kaïchouh). II. The groupoid approach works under an additional hypothesis (which holds for all classical theories and all $\aleph_0$-categorical theories, as well as for a few more). III. In the general case we may run into trouble when we try to choose witnesses (i.e., introduce Skolem functions, in some sense). Some very recent work (still in progress) may allow us to overcome this as well.

https://us06web.zoom.us/j/87929792112?pwd=WWNnTkEwOFQwRDdOVXNaK1UzZHBFZz09
Meeting ID: 879 2979 2112
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