# Mathematical Sciences Research Institute

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# Seminar

DDC - Valuation Theory: Dp-Minimal Rings December 16, 2020 (09:00 AM PST - 10:00 AM PST)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) Yatir Halevi (Ben Gurion University of the Negev)
Description

Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

#### Dp-Minimal Rings

Abstract/Media

Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

Dp-minimal fields were classified up to elementary equivalence by Will Johnson. One of the next logical steps is to classify dp-minimal integral domains.

After introducing the main model theoretic definitions we will review (and prove) some algebraic consequences of dp-minimality for integral domains. Dp-minimal integral domains are close to being valuation rings, but not always. They are local, divided in the sense of Akiba, and every localisation at a non-maximal prime ideal is a valuation ring.

Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.

If time permits, we will also discuss the comparability of definable domains inside a dp-minimal field.

Joint with Christian d'Elbée