Topics suggestions for the Workshop on Valuation Theory in Positive Characteristic The two big open problems are: A) Does local uniformization hold in positive (and in a suitably adapted form) in mixed characteristic? B) What is the model theory of non-perfect henselian defectless valued fields. In particular, does F_p((t)) have a decidable theory? Here are some more detailed (and perhaps more feasible) problems. This list is put together from suggestions by FV, Jochen, Florian and Salih. 1) The structure of (minimal) purely wild extensions. Can we say more about minimal polynomials for such extensions than what is already known (minimal polynomials for minimal purely wild extensions can be chosen to be p-polynomials)? What can be said for a finite extension of a non-henselian valued field that after passing to henselizations becomes a minimal purely wild extension? What can be said about purely wild extensions in general? Literature: Kuhlmann, F.-V.: Additive Polynomials and Their Role in the Model Theory of Valued Fields, Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, held October 18-22, 2003. Lecture Notes in Logic 26 (2006), 160-203, available at http://math.usask.ca/~fvk/ADDPOL.pdf 2) What are possible improvements of the Theorem of Henselian Rationality? This problem is particularly interesting for inseparable local uniformization. 3) What can be said about the structure of valued function fields when the valuation is trivial on the ground field but not Abhyankar? This question is essential in order to avoid using extensions of the function fields for obtaining local uniformization. What can be said when the valuation is not trivial on the ground field, the extension of transcendence degree is not immediate, but valuation-algebraic? What can be said for higher transcendence degree? 4) Dependent and independent Artin-Schreier extensions in algebraic geometry. Are the two Artin-Schreier defect extensions in an example given by Cutkosky and Piltant dependent or independent? The lower one has recently been proven to be dependent. Are dependent Artin-Schreier defect extensions more harmful than independent ones? Temkin's work on inseparable local uniformization seems to indicate this. Details have to be worked out. Literature: Cutkosky, D., Piltant, O.: Ramification of valuations. Adv. Math. 183, 1-79 (2004) Kuhlmann, F.-V.: A classification of Artin Schreier defect extensions and a characterization of defectless fields, to appear in Illinois J Math, available at http://math.usask.ca/~fvk/CIASECDI.pdf. 5) What is the suitable analogue of the notions of dependent and independent Artin-Schreier extensions in mixed characteristic? What are useful characterizations of defectless fields in mixed characteristic? Literature: as for 4) 6) If a finite extension of a valued function field F within its henselization admits local uniformization, does F also admit local uniformization? In other words, can local uniformization be pulled down through extensions within the henselization? (It can be pulled down through tame extensions of henselian fields.) This question may be characteristic blind. But it is still linked with our subject, as it is essential for an alternate approach to a proof of inseparable local uniformization. In characteristic 0, a positive solution may provide an easy proof for local uniformization. 7) Is there quantifier elimination for tame fields? 8) What are the extremal valued fields in mixed characteristic? Literature: Azgin, S. - Kuhlmann, F.-V. - Pop, F.: Characterization of extremal valued fields, to appear in Proc AMS, available at http://math.usask.ca/fvk/EXTFIN.pdf 9) If F is an algebraic function field over a large, but not necessarily perfect ground field K, and if F admits a K-rational place, is K existentially closed in F? Is this problem in any way equivalent to the problem of local uniformization? (If F has a K-rational place that admits local uniformization, then K is indeed existentially closed in F.) 10) Suppose a valued field F admits a truncation closed embedding in a power series field. In positive characteristic it can happen that some of its maximal immediate extensions admit extensions of that embedding that are again truncation closed, and some don't. Do these two sorts differ when it comes to their model theory? (The first sort of fields seem to be the "better" ones.) Literature: Kuhlmann, F.-V. - Kuhlmann, S. - Fornasiero, A.: Towers of complements to valuation rings and truncation closed embeddings of valued fields, Journal of Algebra 323 (2010), 574-600, available at http://math.usask.ca/fvk/FORNAC38.pdf 11) Find a natural language in which F_p((t)) has quantifier elimination. Is the henselisation of F_p(t) an elementary substructure of F_p((t))? 12) Find a uniform first-order definition (in the language of rings, allowing parameters) for all DVR's in fields finitely generated over their prime fields (maybe allowing a predicate for "the field of constants" in the non-global case). 13) Prove or disprove: For any valuation v on a field K with henselisation neither real nor separably closed there is a (in the language of rings, allowing parameters) first-order definable valuation (ring) w on K inducing the same topology as v. This is true if v is henselian (shown by Koenigsmann) or if K has rich arithmetic structure. 14) What is the structure of the Zariski space of valuations of a field, together with its Zarisky and its patch topology. Particularly interesting is the case of all valuations/places of an algebraic function fields which are trivial/non-trivial on the base field (or induce a fixed valuation on it). How well do "nice valuations" approximate the general valuations? How well can the inertia of general valuations be approximated by inertia at "nice valuations"? It is a nice/hard/very important question whether the inertia elements in general can be approximated by inertia at "nice valuations" (The answer is YES for tame inertia, but the question is completely open for wild inertia.) This question is related to several other ones we are interested in. Literature: Kuhlmann, F.-V. - Prestel, A.: On places of algebraic function fields, J. reine angew. Math. 353 (1984), 182-195 Kuhlmann, F.-V.: On places of algebraic function fields in arbitrary characteristic, Advances in Math. 188 (2004), 399-424, available at http://math.usask.ca/~fvk/Places.dvi Pop, F.: Inertia elements versus Frobenius elements, Math. Annalen 438 (2010), 1005-1017, available at http://www.math.upenn.edu/~pop/Research/files-Res/InFr-corr.pdf 15) Are certain properties of Galois extensions preserved under taking decomposition groups. For example: Let L|K be maximal abelian, and for a valuation v of K, let K_v be the decomposition field of v in L|K. Is then L the maximal abelian extension of K_v ? (The answer is YES, if K contains all roots of unity, but the characteristic might cause problems). The same question can be asked for other "verbal" extensions of K, e.g., maximal solvable, maximal pro-p, maximal metabelian, etc. 16) Let $K$ be a valued difference field whose distinguished automorphism induces the identity on the residue field (In particular, the Kaplansky condition is not met). Is it true that $\sigma$-transcendence degree 1 immediate extensions are $\sigma$-henselian rational? (This would be the $\sigma$-analogue of henselian rationality result of F.-V. Kuhlmann). A positive result would be crucial to proving $\sigma$-algebraically maximal valued difference fields are existentially closed in all immediate extensions and consequently aiming for Ax-Kochen-Ershov type results for valued difference fields. 17) How can one define a notion for valued difference fields that would correspond to "purely wild extensions" of valued fields in positive characteristic? ($\sigma$-algebraic immediate extensions of a $\sigma$-henselian field should be purely wild). Would it be correct with this notion that minimal purely wild extensions of valued difference fields are generated by roots of $\sigma$-polynomials of the form $L(x)-a$ where $L$ is additive (e.g. L(x)=\sigma(x)-x$). This would be the analogue of Pop's result on the structure of minimal purely wild extensions. If a positive result can be obtained there is good hope to apply F.-V. Kuhlmann's "Artin-Schreier surgery" technique to solve question 1). 18) Concerning the structure of purely wild extensions, Pop's result on minimal purely wild extensions is about the only thing we know. Even worse, the proof uses Galois theory of valued fields which is highly non-trivial to adapt to the case of valued difference fields. Can one obtain an elementary proof (in the sense of finding roots of polynomials in an explicit manner) of Pop's result? Further, what can one say about the structure of purely wild extensions in general? It looks like the answer to these questions are the key to obtaining quantifier elimination for tame fields.