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.

# Past Logic Seminar

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.'

I will describe joint work with Katrin Tent, in which we consider a profinite group equipped with a uniformly definable family of open subgroups. We show that if the family is `full’ (i.e. includes all open subgroups) then the group has NIP theory if and only if it has NTP_2 theory, if and only if it has an (open) normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups (for distinct primes p). Without the `fullness’ assumption, if the group has NIP theory then it has a prosoluble open normal subgroup of finite index.

In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves.

In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.

I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind. This proves some new special cases of Vojta's conjecture for function fields.

Van den Dries has proved the decidability of the ring of algebraic integers, the integral closure of the ring of integers in

the algebraic closure of the rationals. A well-established analogy leads to ask the same question for the ring of complex polynomials.

This turns out to go the other way, interpreting the rational field. An interesting structure on the

limit of Jacobians of all complex curves is encountered along the way.

The concept of pseudofinite dimension for ultraproducts of finite structures was introduced by Hrushovski and Wagner. In this talk, I will present joint work with D. Macpherson and C. Steinhorn in which we explored conditions on the (fine) pseudofinite dimension that guarantee simplicity or supersimplicity of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of droping of the pseudofinite dimension. Also, under a suitable assumption, it can be shown that a measure-theoretic condition is equivalent to loc

p, li { white-space: pre-wrap; } A {\em monoid} is a semigroup with identity. A {\em finitary property for monoids} is a property guaranteed to be satisfied by any finite monoid. A good example is the maximal condition on the lattice of right ideals: if a monoid satisfies this condition we say it is {\em weakly right noetherian}. A monoid $S$ may be represented via mappings of sets or, equivalently and more concretely, by {\em (right) $S$-acts}. Here an $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that

for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. I will be speaking about finitary properties for monoids arising from model theoretic considerations for $S$-acts.

Let $S$ be a monoid and let $L_S$ be the first-order language of $S$-acts, so that $L_S$ has no constant or relational symbols (other than $=$) and a unary function symbol $\rho_s$ for each $s\in S$. Clearly $\Sigma_S$ axiomatises the class of $S$-acts, where

\[\Sigma_S=\big\{ (\forall x)(x\rho_s \rho_t=x\rho_{st}):s,t\in S\big\}\cup\{ (\forall x)(x\rho_1=x)

\}.\]

Model theory tells us that $\Sigma_S$

has a model companion $\Sigma_S^*$ precisely when the class

${\mathcal E}$ of existentially closed $S$-acts is axiomatisable and

in this case, $\Sigma_S^*$ axiomatises ${\mathcal E}$. An old result of Wheeler tells us that $\Sigma_S^*$ exists if and only if for every finitely generated right congruence $\mu$ on $S$, every finitely generated $S$-subact of $S/\mu$ is finitely presented, that is, $S$ is {\em right coherent}. Interest in right coherency also arises from other considerations such as {\em purity} for $S$-acts.

Until recently, little was known about right coherent monoids and, in particular, whether free monoids are (right) coherent.

I will present some work of Gould, Hartmann and Ru\v{s}kuc in this direction: specifically we answer positively the question for free monoids.

Where $\Sigma_S^*$ exists, it is known to be

stable, and is superstable if and only if $S$ is weakly right noetherian.

By using an algebraic description of types over $\Sigma_S^*$ developed in the 1980s by Fountain and Gould,

we can show that $\Sigma_S^*$ is totally

transcendental if and only if $S$ is weakly right noetherian and $S$ is {\em ranked}. The latter condition says that every right congruence possesses a finite Cantor-Bendixon rank with respect to the {\em finite type topology}.

Our results show that there is a totally transcendental theory of $S$-acts for which Morley rank of types does not coincide with $U$-rank, contrasting with the corresponding situation for modules over a ring.