We give a description of the spectra of $\hat{\mathbb Z}$ and of the

finite adeles using ultraproducts. In describing the prime ideals and the

localizations, ultrapowers of the group $\mathbb Z$ and ultraproducts of

rings of $p$-adic integers are used.

# Past Logic Seminar

Using some new results in probabilistic recognition of black box groups

(Joint work with Sukru Yalcinkaya), I will discuss some emerging

structures that beg for a model-theoretic analysis.

In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion. (Joint work with Tobias Kaiser)

(joint work with Sergei Starchenko)

Let p:C^n ->A be the covering map of a complex abelian variety and let X be an algebraic variety of C^n, or more generally a definable set in an o-minimal expansion of the real field. Ullmo and Yafaev investigated the topological closure of p(X) in A in the above two settings and conjectured that the frontier of p(X) can be described, when X is algebraic as finitely many cosets of real sub tori of A, They proved the conjecture when dim X=1. They make a similar conjecture for X definable in an o-minimal structure.

In recent work we show that the above conjecture fails as stated, and prove a modified version, describing the frontier of p(X) as finitely many families of cosets of subtori. We prove a similar result when X is a definable set in an o-minimal structure and p:R^n-> T is the covering map of a real torus. The proofs use model theory of o-minimal structures as well as algebraically closed valued fields.

I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction/Schanuel-type conjecture analysis. Then I am able to formulate very precise axioms that such a field must satisfy. The main theorem then states that under the axioms the structure has the desired algebraic properties.

The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimansional' valuation theory and the validity of these statements is an open problem to be discussed.

In joint work with Sylvy Anscombe we had found an abstract

valuation theoretic condition characterizing those fields F for which

the power series ring F is existentially 0-definable in its

quotient field F((t)). In this talk I will report on recent joint work

with Sylvy Anscombe and Philip Dittmann in which the study of this

condition leads us to some beautiful results on the border of number

theory and model theory. In particular, I will suggest and apply a

p-adic analogue of Lagrange's Four Squares Theorem.

There are several conjectures in the literature suggesting that absolute Galois groups of fields tend to be "as free as possible," given their "almost-abelian" data.

This can be made precise in various ways, one of which is via the notion of "1-formality" arising in analogy with the concept in rational homotopy theory.

In this talk, I will discuss several examples which illustrate this phenomenon, as well as some surprising diophantine consequences.

This discussion will also include some recent joint work with Guillot, Mináč, Tân and Wittenberg, concerning the vanishing of mod-2 4-fold Massey products in the Galois cohomology of number fields.

After giving some motivation, I will discuss work in progress with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result.

After mentioning, by way of motivation (mine at least), some diophantine questions concerning

sets definable in the restricted analytic, exponential field $\R_{an, exp}$, I discuss the

problem of extending a given $\R_{an, exp}$-definable function $f:(a, \infty) \to \R$ to

a holomorphic function $\hat f : \{z \in \C : Re(z) > b \} \to \C$ (for some $b > a$).

In particular, I give a necessary and sufficient condition on $f$ for such an $\hat f$ to exist and be

$\R_{an, exp}$-definable.

One of many overlaps between logic and topology is duality: Stone duality links Boolean algebras with zero-dimensional compact Hausdorff spaces, and gives a useful topological way of describing certain phenomena in first order logic; and there are generalisations that allow one to study infinitary logics also. We will look at a couple of ways in which this duality theory is useful.'