The next five installments in the Model Theory Seminar will feature a series of talks:

• Wednesday, November 16 (slides)
• Wednesday, November 23 (slides)
• Wednesday, November 30 (slides: see pdf below)
• Wednesday, December 7 (slides)
• Wednesday, December 14  (slides)

I will introduce the notion of "tame field" and describe what is known and is not known about the model theory of tame fields. Further, I will introduce the notion of "defect" of finite extensions of valued fields and indicate why it is important for the model theory of valued fields with residue fields of positive characteristic. Tame fields are defectless fields, i.e., all of their finite extensions have only trivial defect.

Tame fields are the closest we have come to the open problem of the decidability of formal Laurent series fields over finite fields, such as \(F_p((t))\), where \(F_p\) is the field with \(p\) elements; yet tame fields are perfect, while \(F_p((t))\) is not. We are still lacking a complete recursive axiomatization for \(F_p((t))\) with the \(t\)-adic valuation. It is known that a simple adaptation of the axiom system that works for \(p\)-adically closed fields is not complete.

In the year 2003 Yuri Ershov introduced the notion of “extremal valued field” and proved that \(F_p((t))\) is extremal, but this was wrong as the definition of the notion was incorrect. In joint work with Salih Azgin and Florian Pop, we corrected the definition, and thereby the proof, and partially characterized extremal valued fields. This work was then continued together with Sylvy Anscombe. It has remained an open problem whether adding the (very handy) axiom system expressing extremality to the aforementioned adapted system will render it complete.

Another important question is whether we can push things beyond the class of defectless fields. In joint work with Anna Rzepka I have studied the valuation theory of "deeply ramified fields", which admit only certain less harmful defects. Perfectoid fields belong to this (elementary) class of valued fields, as well as all perfect valued fields of positive characteristic, such as the perfect hull of \(F_p((t))\). Thus the theory of deeply ramified fields may offer some clues for the open problem of the decidability of the latter.

In passing, I will also sketch the connection of the model theory of valued fields with the open problem of resolution of singularities in positive characteristic.