Past Logic Seminar

9 February 2017
11:00
Kobi Peterzil
Abstract

(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.

2 February 2017
17:30
Boris Zilber
Abstract

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. 

 

26 January 2017
17:30
Arno Fehm
Abstract

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.

19 January 2017
16:00
Adam Topaz
Abstract

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.

1 December 2016
17:30
Gareth Jones
Abstract

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.

24 November 2016
17:30
Alex Wilkie
Abstract

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.
 

17 November 2016
17:30
Robin Knight
Abstract

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.'

10 November 2016
17:30
Dugald Macpherson
Abstract

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.

3 November 2016
16:00
Natalia Garcia-Fritz
Abstract

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.
 

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