The Combinatorial Game Theory of Well-Tempered Scoring Games

We consider the class of "well-tempered" integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal-play partizan games. We show that the monoid of well-tempered scoring games modulo indistinguishability is cancellative but not a group, and we describe its structure in terms of the group of normal-play partizan games. We also classify Boolean-valued well-tempered scoring games, showing that there are exactly seventy, up to equivalence.Comment: 60 pages, 21 figure

Finite burden in multivalued algebraically closed fields

We prove that an expansion of an algebraically closed field by $n$ arbitrary valuation rings is NTP${}_2$, and in fact has finite burden. It fails to be NIP, however, unless the valuation rings form a chain. Moreover, the incomplete theory of algebraically closed fields with $n$ valuation rings is decidable.Comment: 40 page

Dp-finite fields VI: the dp-finite Shelah conjecture

We prove the dp-finite case of the Shelah conjecture on NIP fields. If K is a dp-finite field, then K admits a non-trivial definable henselian valuation ring, unless K is finite, real closed, or algebraically closed. As a consequence, the conjectural classification of dp-finite fields holds. Additionally, dp-finite valued fields are henselian. Lastly, if K is an unstable dp-finite expansion of a field, then K admits a unique definable V-topology.Comment: 26 page

Dp-finite fields V: topological fields of finite weight

We prove that unstable dp-finite fields admit definable V-topologies. As a consequence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof of the classification of dp-finite fields of positive characteristic. For $n \ge 1$, we define a local class of "$W_n$-topological fields", generalizing V-topological fields. A $W_1$-topology is the same thing as a V-topology, and a $W_n$-topology is some higher-rank analogue. If $K$ is an unstable dp-finite field, then the canonical topology is a definable $W_n$-topology for $n = \operatorname{dp-rk}(K)$. Every $W_n$-topology has between 1 and $n$ coarsenings that are V-topologies. If the given $W_n$-topology is definable in some structure, then so are the V-topological coarsenings.Comment: Preliminary draft, comments welcom

On the proof of elimination of imaginaries in algebraically closed valued fields

We give a simplified proof of elimination of imaginaries (in the geometric sorts) in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.Comment: New and improved coding for finite sets (section 4.3

A pathological o-minimal quotient

We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in an o-minimal structure is interdefinable over an independent set of parameters with a tuple of real elements. This can be interpreted as saying that interpretable sets look "locally" like definable sets, in a sense which can be made precise.Comment: Withdrawn; incorporated into arXiv:1911.1007

Circular Planar Resistor Networks with Nonlinear and Signed Conductors

We consider the inverse boundary value problem in the case of discrete electrical networks containing nonlinear (non-ohmic) resistors. Generalizing work of Curtis, Ingerman, Morrow, Colin de Verdiere, Gitler, and Vertigan, we characterize the circular planar graphs for which the inverse boundary value problem has a solution in this generalized non-linear setting. The answer is the same as in the linear setting. Our method of proof never requires that the resistors behave in a continuous or monotone fashion; this allows us to recover signed conductances in many cases. We apply this to the problem of recovery in graphs that are not circular planar. We also use our results to make a frivolous knot-theoretic statement, and to slightly generalize a fact proved by Lam and Pylyavskyy about factorization schemes in their electrical linear group.Comment: 52 pages, 36 figure

Dp-finite fields I: infinitesimals and positive characteristic

We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a field, then either K has finite Morley rank or K has a non-trivial Aut(K/A)-invariant valuation ring for a small set A. In the positive characteristic case, we can even demand that the valuation ring is henselian. Using this, we classify the positive characteristic dp-finite pure fields.Comment: 89 pages; changed title, corrected typos, fixed some references, improved formatting, added references to sequel

Forking and Dividing in Fields with Several Orderings and Valuations

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.Comment: 41 pages; dissertation chapte

A criterion for uniform finiteness in the imaginary sorts

Let $T$ be a theory. If $T$ eliminates $\exists^\infty$, it need not follow that $T^{eq}$ eliminates $\exists^\infty$, as shown by the example of the $p$-adics. We give a criterion to determine whether $T^{eq}$ eliminates $\exists^\infty$. Specifically, we show that $T^{eq}$ eliminates $\exists^\infty$ if and only if $\exists^\infty$ is eliminated on all interpretable sets of "unary imaginaries." This criterion can be applied in cases where a full description of $T^{eq}$ is unknown. As an application, we show that $T^{eq}$ eliminates $\exists^\infty$ when $T$ is a C-minimal expansion of ACVF.Comment: 6 page