The differential field of transseries extends the field of real Laurent series and occurs in various contexts: asymptotic expansions, analytic vector fields, and o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field and report on our efforts to understand its elementary theory.

We study the Vapnik–Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm{U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincaré biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.

We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\overline{\mathbb{Q}}$. This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension $1$, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over $\overline{\mathbb{Q}}$. We also derive the Grothendieck period conjecture for cycles of codimension $1$ in abelian varieties over $\overline{\mathbb{Q}}$ from a classical transcendence theorem à la Schneider–Lang.

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.

Let $\mathcal{M}$ be an o-minimal expansion of a real closed field $R$. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of $\langle R^n,+\rangle$ contains a definable generic set and therefore admits a lattice.

We attempt to formulate issues around modularity and Zilber’s trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].

This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries and make some new observations on the strong types over those. In particular, we show that in a simple theory $Ltp\equiv stp$ in real context implies that for hyperimaginary context.

We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary class, answering a question of Kesälä and Baldwin.

We use hyperbolic towers to answer some model-theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$ but that there is a finitely generated model which omits $p_0^{\left(2\right)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

In a recent paper I established an analogue of the Lindemann–Weierstrass part of Ax–Schanuel for the elliptic modular function. Here I extend this to include its first and second derivatives. A generalization is given that includes exponential and Weierstrass elliptic functions as well.

We initiate an account of Shelah’s notion of “strong dependence” in terms of generically stable measures, proving a measure analogue (for NIP theories) of the fact that a stable theory $T$ is “strongly dependent” if and only if all (finitary) types have almost finite weight.

In this article we consider some questions raised by F. Benoist, E. Bouscaren, and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mathbb{Z}$, then there are no infinitely $p$-divisible points of order a power of $p$.

In this paper, we prove that a pseudo-exponential field has continuum many nonisomorphic countable real closed exponential subfields, each with an order-preserving exponential map which is surjective onto the nonnegative elements. Indeed, this is true of any algebraically closed exponential field satisfying Schanuel’s conjecture.

Working in an o-minimal expansion of the real field, we investigate when a germ (around zero, say) of a complex analytic function has a definable analytic continuation to its Mittag–Leffler star.

As an application we show that any algebro-logarithmic function that is complex analytic in a neighborhood of the origin in $\mathbb{C}$ has an analytic continuation to all but finitely many points in $\mathbb{C}$.