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Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $\Ff_p^n \to \Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where $\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character., Comment: Version 3: various local changes, with some material reorganized for clarity. The main mathematical difference is in section 5, where the connection to Duke-Friedlander-Iwaniec is considerably tightened

We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration., 69 pages

We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjectures from [24]. We introduce $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. We study the relationship between definability of an action of a definable group on a compact space, weakly almost periodic actions, and stability. We conclude that for any group $G$ definable in a sufficiently saturated structure, every definable action of $G$ on a compact space supports a $G$-invariant probability measure. This gives negative solutions to some questions and conjectures from [22] and [24]. We give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language for which the $\bigvee$-definable group $H:=\langle X \rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact "model" exists for each approximate subgroup does not work in general., Version 4 contains the material in Sections 2, 3, and 5 of version 1. Following the advice of editors and referees we have divided version 1 into two papers, version 4 being the first. The second paper is entitled "On first order amenability"

In a series of imaginary conversations, a philosopher of physics and a mathematical logician discuss the investigation of a mathematical relational universe and the description of the world by physical theory, as metaphors for each other. They take for granted that basic events can be observed directly, as atomic formulas, and ask about the ability to interpret deeper underlying structures. This depends on the logic used. They are interested specifically in the role of classical probability logic in this endeavor. Within the model they adopt, as a case study, probability logic is highly competent in discovering and describing spatial structures; where a very general notion of space is taken to correspond to the logical notion of monadic, both with natural relativistic corrections. However, this logic can go no further in itself, quite unlike the unbounded abilities of first-order logic. In some cases new first-order structures emerge at the point of failure of probability logic.

We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure preserving bijections., Local corrections

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort $\Gamma$) if every type over a sufficiently rich base structure can be viewed as part of a $\Gamma$-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from $\Gamma$, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF., Comment: 48 pages. Minor corrections and improvements following a referee report

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

This book deals with non-archimedean tame topology and stably dominated types. It considers o-minimality as an analogy and reduces questions over valued fields to the o-minimal setting. A fundamental tool, imported from stability theory, is the notion of a definable type, which plays a number of roles, starting from the definition of a point of the fundamental spaces. One of the roles of definable types is to be a substitute for the classical notion of a sequence, especially in situations where one is willing to refine to a subsequence. To each algebraic variety V over a valued field K, the book associates in a canonical way a projective limit unit vector V of spaces, which is the stable completion of V. In case the value group is ℝ, the results presented in this book relate to similar tameness theorems for Berkovich spaces.

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

This chapter presents various applications to the topology of classical Berkovich spaces. It deduces from the main theorem several new results on the topology of V(superscript an) which were not known previously in such a level of generality. In particular, it shows that V(superscript an) admits a strong deformation retraction to a subspace homeomorphic to a finite simplicial complex and that V(superscript an) is locally contractible. The chapter also proves the existence of strong retractions to skeleta for analytifications of definable subsets of quasi-projective varieties and goes on to prove finiteness of homotopy types in families in a strong sense and a result on homotopy equivalence of upper level sets of definable functions. Finally, it describes an injection in the opposite direction (over an algebraically closed field) which in general provides an identification between points of Berkovich analytifications and Galois orbits of stably dominated points.

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

This chapter introduces the theory ACV²F of iterated places and describes some algebraic criteria for v- and g-continuity. It considers the theory ACV²F of triples (K₂,K₁,K₀) of fields with surjective, non-injective places rᵢⱼ : Kᵢ → Kⱼ for i > j, r₂₀ = r₁₀ ° r₂₁, such that K₂ is algebraically closed. The chapter shows that the family of g-open sets is definable in definable families. It also presents some applications of the continuity criteria and concludes by proving that for each definable set of definable functions V → Wsuperscript Number Sign the subset of those that are g-continuous is definable.

This chapter proves the iso-definability of unit vector C when C is a curve using Riemann-Roch. Recall that a pro-definable set is called iso-definable if it is isomorphic, as a pro-definable set, to a definable set. If C is an algebraic curve defined over a valued field F, then unit vector C is an iso-definable set. The topology on unit vector C is definably generated, that is, generated by a definable family of (iso)-definable subsets. In other words, there is a definable family giving a pre-basis of the topology. The chapter explains how definable types on C correspond to germs of paths on unit vector C. It also constructs the retraction on skeleta for curves. A key result is the finiteness of forward-branching points.

This chapter focuses on the properties of strongly stably dominated types over valued fields bases. In this setting, strong stability corresponds to a strong form of the Abhyankar property for valuations: the transcendence degrees of the extension coincide with those of the residue field extension. The chapter proves a Bertini type result and shows that the strongly stable points form a strict ind-definable subset Vsuperscript Number Sign of unit vector V. It then proves a rigidity statement for iso-definable Γ‎-internal subsets of maximal o-minimal dimension of unit vector V, namely that they cannot be deformed by any homotopy leaving appropriate functions invariant. The chapter also describes the closure of iso-definable Γ‎-internal sets in Vsuperscript Number Sign and proves that Vsuperscript Number Sign is exactly the union of all skeleta.

This chapter introduces the space unit vector V of stably dominated types on a definable set V. It first endows unit vector V with a canonical structure of a (strict) pro-definable set before providing some examples of stably dominated types. It then endows unit vector V with the structure of a definable topological space, and the properties of this definable topology are discussed. It also examines the canonical embedding of V in unit vector V as the set of simple points. An essential feature in the approach used in this chapter is the existence of a canonical extension for a definable function on V to unit vector V. This is considered in the next section where continuity criteria are given. The chapter concludes by describing basic notions of (generalized) paths and homotopies, along with good metrics, Zariski topology, and schematic distance.

This chapter describes the notion of definable compactness for subsets of unit vector V. One of the main results is Theorem 4.2.20, which establishes the equivalence between being definably compact and being closed and bounded. The chapter gives a general definition of definable compactness that may be useful when the definable topology has enough definable types. The o-minimal formulation regarding limits of curves is replaced by limits of definable types. The chapter relates definable compactness to being closed and bounded and shows that the expected properties hold. In particular, the image of a definably compact set under a continuous definable map is definably compact.

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable homotopy category of definable spaces over the o-minimal Γ‎. The chapter introduces three categories that can be viewed as ind-pro definable and admit natural functors to the category TOP of topological spaces with continuous maps. The discussion is often limited to the subcategory consisting of A-definable objects and morphisms. The morphisms are factored out by (strong) homotopy equivalence. The chapter presents the proof of the equivalence of categories before concluding with remarks on homotopies over imaginary base sets.

Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an of any d-dimensional quasi-projective scheme V over K embeds in R^{2d+1}. If, moreover, the value group of K is dense in R_{>0} and V is a curve, then we describe the homeomorphism type of V^an by using the theory of local dendrites., 15 pages

We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef., Comment: 15pp