We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {em core} $mathcal{J}$ associated to
such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $mathcal{G}$. The automorphism groups of models of the theory are related with $mathcal{G}$, not in general via a homomorphism, but by a {em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({mathbb{R}})$ or $SL_n({mathbb{Q}}_p)$.
We identify a canonical structure J associated to any first-order theory, the {it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits
a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupinski, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $beth_5$ to $beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the Kechris-Pestov-Todorv cevic correspondence. We define the Ramsey property for a first order theory, and show - as a simple application of the construction applied to an auxiliary theory - that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraisse classes, first by Kechris-Pestov-Todorv cevic for expansions by orderings, then by Melleray, Nguyen Van The, Tsankov and Zucker for more general expansions.
Motivated by Emmanuel Kowalskis exponential sums over definable sets in finite fields, we generalize Axs theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weils 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.
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{h
hm} 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.
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X ^{-1} X |/ |X| bounded is close to a
finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromovs proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the
theory of pseudo-finite fields we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.
We develop a motivic integration version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and obtain relations among the
motivic Fourier transforms of a test function at different completions. We use these to prove, in a special case, a motivic version of a theorem of Deligne-Kazhdan-Vigneras.
We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then
the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of generically stable types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.
This second part of the paper strengthens the descent theory described in the first part to rational maps, arbitrary base fields, and dynamics given by correspondences. We obtain in particular a decomposition of any difference field extension into a
tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the dynamical Northcott Theorem 1.11 of Part I in sharp form.
