A first-order theory is Noetherian with respect to the collection of formulae F if every definable set is a Boolean combination of instances of formulae in F and the topology whose subbasis of closed sets is the collection of instances of arbitrary formulae in F is Noetherian. Noetherianity is a strengthening of equationality, which itself implies stability. We show the Noetherianity of the theory of proper pairs of algebraically closed fields in any characteristic.

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers and of Nikolov-Pyber, on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. Furthermore, we bound the number of solutions to certain equations in subsets of small tripling in groups. In particular, we show the existence of lower bounds on the number of arithmetic progressions of length 3 for subsets of small doubling without involutions in arbitrary abelian groups.

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories of fields with operators.

We introduce a relaxation of stability, called robust stability, which is insensitive to perturbations by subsets of Loeb measure 0 in a pseudofinite group. We show that robust stability satisfies a stationarity principle for measure independent elements. We apply this principle to deduce the existence of squares and L-shapes in dense subsets of Cartesian squares of pseudofinite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian squares of finite groups.

We provide elementary model-theoretic proofs to some existing results on sumset phenomena and IP sets, motivated by Hrushovski's work on the stabilizer theorem.

A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in abelian groups.

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability yet so far only two examples of non-equational stable theories are known. We construct non-equational ω-stable theories by a suitable colouring of the free pseudospace, based on Hrushovski and Srour's original example.

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of three theories of fields: the theory of proper extension of algebraically closed fields of some fixed characteristic, the theory of differentially closed fields in positive characteristic and the theory of separably closed fields of infinite imperfection degree. As a by-product, we obtain relative elimination of imaginaries for the the theory of separably closed fields of infinite imperfection degree, after adding sorts for the canonical parameters of equations.

In this short note, we mimic the proof of the simplicity of the theory ACFA of generic difference fields in order to provide a criterion, valid for certain theories of pure fields and fields equipped with operators, which shows that a complete theory is simple whenever its definable and algebraic closures are controlled by an underlying stable theory.

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of H-structures of a trivial theory of rank 1.

We give an alternative proof, valid in all characteristics, of a result of Lascar characterising the bounded automorphisms of an algebraically closed field. We generalise this method to various fields equipped with operators.

We study the model theory of countable right-angled buildings with infinite residues. For every Coxeter graph we obtain a complete theory with a natural axiomatisation, which is ω-stable and equational. Furthermore, we provide sharp lower and upper bounds for its degree of ampleness, computed exclusively in terms of the associated Coxeter graph. This generalises and provides an alternative treatment of the free pseudospace.

We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of a Cartesian power of the subgroup of coloured elements by an algebraic group. In particular, a simple group in a coloured field is algebraic.

We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a subgroup of a product of groups definable in the reducts. In the relatively CM-trivial case, which contains certain Hrushovski amalgamations (the fusion of two strongly minimal sets or the expansions of a field by a predicate), every definable group allows a homomorphism with virtually central kernel into a product of groups definable in the reducts.

In this short paper, we will provide a characterisation of interpretable groups in a beautiful pair (K, E) of algebraically closed fields: every interpretable group is, up to isogeny, the extension of the subgroup of E-rational points of an algebraic group by an interpretable group which is the quotient of an algebraic group by the E-rational points of an algebraic subgroup.

The ample hierarchy of geometries of stables theories is strict. We generalise the construction of the free pseudospace to higher dimensions and show that the n-dimensional free pseudospace is ω-stable n-ample yet not (n+1)-ample. In particular, the free pseudospace is not 3-ample. A thorough study of forking is conducted and an explicit description of canonical bases is exhibited.

We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family.

Elliptic curves over a supersimple field with exactly one extension of degree 2 have s-generic rational points.

We collapse Poizat's red fields in characteristic 0 to obtain a differentially closed field of rank ω ⋅ 2 equipped with a definable additive subgroup of commensurable rank. We obtain by using the logarithmic derivative a green multiplicative subgroup which cannot be of finite rank.

We construct a bad field in characteristic zero. That is, an algebraically closed field with a notion of dimension analogous to Zariski dimension, equipped with an infinite proper multiplicative subgroup of dimension one, such that the field itself has dimension 2. This answers a longstanding open question by Zilber.

We apply Hrushovski-Fraïssé's amalgamation procedure to obtain a theory of fields of prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1.

We exhibit a simplified exposition of Hrushovski's fusion of two strongly minimal theories over a trivial geometry.

Given two countable strongly minimal theories with the DMP, whose common theory is the theory of vector spaces over a fixed finite field, we show that the union of the two theories has a strongly minimal completion.

We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank $p-1$, extracting the main ideas for the construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given and ranks are computed.

We consider fields with an abstract notion of dimension as stated by Pillay and Poizat in their paper of 1995. We prove that for every finite extension L of K and for every finite Galois extension L₁ of L , the Brauer group of L₁ over L is finite, as well as the first cohomology group of L₁ over L with coefficients in some algebraic group G.

In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case.

It is proved that, if F is an infinite field with characteristic different from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic modulus, then C has a generic F-rational point. The notion of genericity here is in the sense of the supersimple field F.