The right angle to look at orthogonal sets

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developped. Moreover, a version of Schlichting's Theorem for hyperdefinable families of commensurable subgroups is shown

Stable domination and independence in algebraically closed valued fields

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries

Imaginaries and definable types in algebraically closed valued fields

The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point

Non-archimedean tame topology and stably dominated types

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it. In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$. When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values. The space $\hat {V}$ is obtained by defining a topology on the pro-definable set of stably dominated types on $V$. The key result is the construction of a pro-definable strong retraction of $\hat {V}$ to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.Comment: Final versio

Groups definable in two orthogonal sorts

This work can be thought of as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable normal subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover

Algebraic relations between solutions of Painlev\'e equations

We calculate model theoretic ranks of Painlev\'e equations in this article, showing in particular, that any equation in any of the Painlev\'e families has Morley rank one, extending results of Nagloo and Pillay (2011). We show that the type of the generic solution of any equation in the second Painlev\'e family is geometrically trivial, extending a result of Nagloo (2015). We also establish the orthogonality of various pairs of equations in the Painlev\'e families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlev\'e family come from classically studied B{\"a}cklund transformations. For instance, we show that if at least one of $\alpha, \beta$ is transcendental, then $P_{II} (\alpha)$ is nonorthogonal to $P_{II} ( \beta )$ if and only if $\alpha+ \beta \in \mathbb Z$ or $\alpha - \beta \in \mathbb Z$. Our results have concrete interpretations in terms of characterizing the algebraic relations between solutions of Painlev\'e equations. We give similar results for orthogonality relations between equations in different Painlev\'e families, and formulate some general questions which extend conjectures of Nagloo and Pillay (2011) on transcendence and algebraic independence of solutions to Painlev\'e equations. We also apply our analysis of ranks to establish some orthogonality results for pairs of Painlev\'e equations from different families. For instance, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012).Comment: This manuscript replaces and greatly expands a portion of arXiv:1608.0475