 # Mathematical Sciences Research Institute

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# Seminar

DDC - Valuation Theory: Beyond V-topologies October 07, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) William Johnson (University of California, Berkeley)
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

#### Beyond V-Topologies

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

A V-topology is a field topology induced by a valuation ring or an absolute value.  V-topologies play an important role in number theory and algebra.  In the analysis of dp-finite fields, a more general class of finite weight'' field topologies naturally arises.  Weight-1 topologies are equivalent to V-topologies.  A topology generated by $n$ distinct V-topologies is a weight-$n$ topology, but there are other unexpected examples of weight-$n$ topologies when $n > 1$.  On the algebraic side, there is a class of weight n'' rings generalizing valuation rings.  From this, we get some new examples of dp-finite integral domains.  We will discuss the basic theory, some open questions, and how this notion arises in the context of model-theoretic classification problems.