# Seminar Day on Model theory of valued fields and resolution of singularities

On this seminar day we will have a few talks around valued fields, connecting algebra, model theory and algebraic geometry. The talks take place on **June 17th 2022** at TU Dresden in room REC/B214/H.

### Speakers

Sylvy Anscombe, IMJ-PRG, Université Paris Cité

Steven Dale Cutkosky, Department of Mathematics, University of Missouri

Philip Dittmann, Institute of Algebra, Technische Universität Dresden

Franz-Viktor Kuhlmann, Institute of Mathematics, University of Szczecin

### Schedule

13:15-14:15 | Philip Dittmann | Resolution of singularities in the model theory of valued fields |

14:30-15:30 | Steven Dale Cutkosky | On the construction of valuations and generating sequences on hypersurface singularities |

Coffee break | ||

16:00-17:00 | Franz-Viktor Kuhlmann | Deeply ramified fields and their relatives |

17:15-18:15 | Sylvy Anscombe | Cohen rings, NIP transfer, and representatives |

### Abstracts

**Sylvy Anscombe, Cohen rings, NIP transfer, and representatives **

I will speak about Cohen rings: complete Noetherian local rings A with maximal ideal pA. If the residue field k is perfect, then A is canonically isomorphic to a Witt ring over k. When k is imperfect there is still a structure theorem (classical, due to Cohen). More recently, with Franziska Jahnke, we proved a "NIP transfer result" (NIP=not the independence property): A is NIP if and only if k is NIP. In this talk I will say a little about NIP, and NIP transfer. I will also discuss some work in progress on representatives: partial maps from k to A that are sections of the residue map.

**Steven Dale Cutkosky, On the construction of valuations and generating sequences on hypersurface singularities**

Abstract

**Philip Dittmann, Resolution of singularities in the model theory of valued fields**

I will talk about various results concerning axiomatisations of the existential theory of a fixed valued field, notably of F_p((t)). These include the algorithm of Denef-Schoutens as well as recent joint work with Anscombe and Fehm. These results are conditional on certain desingularisation conjectures, which I will discuss in detail.

**Franz-Viktor Kuhlmann, Deeply ramified fields and their relatives**

I will review the main results of my work with Anna Rzepka on deeply ramified fields and their relatives. One of the latter are what we had originally called "generalized deeply ramified fields". In light of very recent developments in the literature and in our work, we have now renamed them to "roughly deeply ramified fields". The idea has appeared that in order for a valued field (K,v) to have good algebraic and model theoretic properties, one only has to ask that the core field (a notion from the theory of p-adic fields) satisfies suitable conditions. This idea had in fact led us to our introduction of generalized deeply ramified fields, which have nicely demonstrated its usefulness. As another example, a paper of Halevi and Hasson deals with the generalization of model theoretic properties from the class of algebraically maximal Kaplansky fields to the more general class of henselian fields whose core fields are algebraically maximal Kaplansky fields. If the core field of (K,v) is a Kaplansky field, then we may call (K,v) a roughly Kaplansky field, as the passage from Kaplansky field to roughly Kaplansky field is nothing but the passage from a p-divisible value group to a roughly p-divisible value group (a notion introduced by W. A. Johnson).

Similarly, (K,v) is a defectless field if and only if its core field is. Let us call (K,v) a roughly tame field if it is henselian and its core field is a tame field; again the generalization consists in replacing ``p-divisible value group'' by ``roughly p-divisible value group''. Anna Rzepka and Piotr Szewczyk have recently shown that a henselian field (K,v) is roughly tame if and only if all of its algebraic extensions are defectless fields. Note that in general, infinite algebraic extensions of defectless fields may not again be defectless fields.

We have chosen the name ``roughly deeply ramified field'' since in this case, the generalization of the notion ``deeply ramified field'' consists in replacing a certain condition on all archimedean components of the value group by the condition ``roughly p-divisible value group'', which removes any condition on the value groups of the coarsenings of v which have residue characteristic 0. We will describe a result for roughly deeply ramified field that is in analogy to the above mentioned result for roughly tame fields. Finally, I will mention a bunch of open questions about the passage from classes of valued fields to their rough relatives.