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Friday, April 21, 2017

**Abstract:** Since the work of G\"odel we know that the theory of the ring $\mathbb Z$ of integers is very complicated. Using the coding techniques introduced by him, every finitely generated commutative ring can be interpreted in $\mathbb Z$ and therefore has a theory which is no more complicated than that of $\mathbb Z$. It has also been long known that conversely, every infinite finitely generated commutative ring interprets the integers, and hence its theory is at least as complex as that of $\mathbb Z$. However, this mutual interpretability does not fully describe the class of definable sets in such rings. The correct point of view is provided by the concept of bi-interpretability, an equivalence relation on the class of first-order structures which captures what it means for two structures to essentially have the same categories of definable sets and maps. We characterize algebraically those finitely generated rings which are bi-interpretable with $\mathbb Z$. (Joint work with Anatole Kh\'elif, Eudes Naziazeno, and Thomas Scanlon.)

Friday, May 12, 2017