Abstract: (Joint work with Sylvy Anscombe) We consider four properties

of a field K related to the existence of (definable) henselian

valuations on K and on elementarily equivalent fields and study the

implications between them. Surprisingly, the full pictures look very

different in equicharacteristic and mixed characteristic.

# Past Logic Seminar

If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame. We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases. Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing: in this case, we are looking at the so-called "generic multiverse".

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations.

Let M be an algebraic curve over an algebraically closed field and let

$(M, ...)$ be a strongly minimal non-locally modular structure with

basic relations definable in the full Zariski language on $M$. In this

talk I will present the proof of the fact that $(M, ...)$ interprets

an algebraically closed field.

We consider the decision problem of determining whether an exponential

polynomial has a real zero. This is motivated by reachability questions

for continuous-time linear dynamical systems, where exponential

polynomials naturally arise as solutions of linear differential equations.

The decidability of the Zero Problem is open in general and our results

concern restricted versions. We show decidability of a bounded

variant---asking for a zero in a given bounded interval---subject to

Schanuel's conjecture. In the unbounded case, we obtain partial

decidability results, using Baker's Theorem on linear forms in logarithms

as a key tool. We show also that decidability of the Zero Problem in full

generality would entail powerful new effectiveness results concerning

Diophantine approximation of algebraic numbers.

This is joint work with Ventsislav Chonev and Joel Ouaknine.

In contrast to the Artin-Schreier Theorem, its p-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of p-adic fields. We plan to give a characterization of p-adic p-Henselian valuations in an essentially finite way. This relates to the Z/p metabelian form of the birational p-adic Grothendieck section conjecture.

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short. Since then the consistency of NF has been a problem that remains open today. But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have solved the problem by showing that TST^+ is indeed consistent. But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.