Title: On definable quotients in valued fields and related structures
Abstract: The question of determining the shape of “definable” quotients—equivalently, of constructing moduli spaces for definable sets—has been investigated in various classes of henselian fields over the past 20 years, beginning with the foundational work of Haskell–Hrushovski–Macpherson on algebraically closed valued fields (ACVF). In the first part of the talk I will present a general result obtained in joint work with Rideau-Kikuchi for (enriched) henselian valued fields of equicharacteristic zero: definable quotients can then be described via the moduli spaces of definable modules (over the valuation ring) , quotients of the value group and quotients of certain residue-field vector spaces. This gives a positive answer to a question posed by Hrushovski in the early 2000. Later I will present applications of this result from a two folded perspective: the model theoretic implications and the geometric ones. This is joint work with Rideau-Kikuchi and Cubides, and builds in joint work with P.Simon. This work aims to study further topological geometric spaces like the real analyfication of a semialgebraic set as an analogue of the berkovich analifyication of a variety in ACVF.
If time permits, I will present current work in progress with E. Hurshovski on the classification of definable quotients in the ring of algebraic integers. This builds on previous work of Hrushovski and Derakshan to classify definable quotients in the ring of adeles, together with work of van den Dries that builds on Rumely's principle and permits to consider the algebraic integers as a booleanization of ACVF_{0} relative to ACF_{0} (the theory of algebraically closed fields).