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Abstracts for Conference on Interactions of model theory with number theory and algebraic geometry

Alternatively have a look at the program.

On special sets and their geometry

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Speaker: 
Boris Zilber
Date: 
Mon, 11/06/2012 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will discuss a very general definition of a special subvariety inspired by recent works on those for Shimura varieties by Pila, Ullmo, Yafaev and others. The main theorem states that the general special subvarieties are exactly closed irreducible subsets of a certain $1$-based Zariski structure, so are essentially classifiable. This allows to formulate a general form of a Schanuel-type conjecture and, correspondingly, a Diophantine conjecture on atypical intersections.

Pairs of algebraically closed fields

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Speaker: 
Françoise Delon
Date: 
Mon, 11/06/2012 - 11:10 - 12:10
Location: 
MPIM Lecture Hall

A pair of algebraically closed fields consists of an algebraically closed field enriched with an additional unary predicate interpreted as an algebraically closed subfield. Proper pairs of algebraically closed fields are known to be omega-stable of Morley Rank omega. We propose a language in which they eliminate quantifiers, and which has the advantage of adapting to some expansions such as pairs of algebraically closed valued fields.

Abelian integrals and categoricity

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Speaker: 
Martin Bays
Date: 
Mon, 11/06/2012 - 14:50 - 15:50
Location: 
MPIM Lecture Hall

An integral of the form $\int w(z)dz$ where $w(z)$ is an algebraic function of $z$, such as $\int\frac{dz}{\sqrt{z^{3}-1}}$, is known as an Abelian integral. Once represented as a path integral of a rational differential form on a complex algebraic curve, it can be seen as a multifunction of the endpoints whose value depends on (the homology class of) the path along which the integral is taken.

Geometric stability theory methods in algebraic dynamics

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Speaker: 
Alice Medvedev
Date: 
Mon, 11/06/2012 - 16:30 - 17:30
Location: 
MPIM Lecture Hall

Certain definable sets in difference fields, namely $\sigma$-varieties, are a natural generalization of algebraic dynamical systems, so some arithmetic questions about algebraic dynamical systems can be addressed with the model theory of difference fields. Ideas around the Zilber Trichotomy Principle, such as minimality, orthogonality, and disintegratedness (triviality) are especially useful.

Pell's Equation over Polynomial Rings

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Speaker: 
David Masser
Date: 
Tue, 12/06/2012 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

It is classical that for any positive integer $d$ not a perfect square there are integers $x$ and $y\neq 0$ with $x^{2}-dy^{2}=1$. The analogous assertion for $D,\ X,\ Y\neq 0$ in $\mathrm{C}[\mathrm{t}]$ with $X^{2}-DY^{2}=1$ clearly requires that the degree of $D$ be even, and it is easy to see that it holds for all quadratic $D$. However for quartic $D$ the problem changes character; this can be seen from that fact that the set of complex $\lambda$ such that $X,\ Y$ exist for $D=t^{4}+t +\lambda$ is infinite but ``scarce'' ; for example it contains at most finitely many rationals.

Families of abelian varieties with many isogenous fibres

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Speaker: 
Martin Orr
Date: 
Tue, 12/06/2012 - 11:10 - 12:10
Location: 
MPIM Lecture Hall

Let $V$ be a subvariety of the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. Suppose that a Zariski dense set of points of $V$ lie in a single Hecke orbit; in other words they correspond to abelian varieties from a single polarised isogeny class. The Zilber-Pink conjecture predicts that $V$ is weakly special. We will prove this when $\dim V=1$ using the Pila-Zannier method and the Masser-Wustholz isogeny theorem.

Non-Archimedean Approximations by Special Points

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Speaker: 
Philipp Habegger
Date: 
Tue, 12/06/2012 - 14:50 - 15:50
Location: 
MPIM Lecture Hall

Tate and Voloch proved that a linear form in roots of unity is either zero or p-adically bounded from below by a positive constant that is independent of the roots of unity. They also conjectured that a subvariety of a semi-abelian variety does not admit arbitrarily good p-adic approximations by torsion points. Buium and Mattuck made progress and Scanlon gave a full proof using work of Chatzidakis and Hrushovski on the model theory of difference fields.

Axiomatizing exponentiation

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Speaker: 
Jonathan Kirby
Date: 
Tue, 12/06/2012 - 16:30 - 17:30
Location: 
MPIM Lecture Hall

Zilber gave non-first order axioms for an exponential field B with good model-theoretic properties which is conjecturally isomorphic to complex exponentiation. I will explain how the diophantine conjecture CIT allows us to axiomatize the complete first-order theory Th(B), and how the existence of an axiomatization of Th(B) along similar lines implies CIT. This is joint work with Boris Zilber.

Berkovich spaces, polytopes and model theory

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Speaker: 
Antoine Ducros
Date: 
Wed, 13/06/2012 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

Let $X$ be a Berkovich analytic space and let $f= (f_{1},\ \ldots,\ f_{n})$ be a family of invertible holomorphic functions on it. We will explain how model theory can help to recover two results which were originally proven using de Jong's alterations, and which tell the following.

New transfer principles between $\mathbb{Q}_p$ and $\mathbb{F}_p(t)$

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Speaker: 
Immanuel Halupczok
Date: 
Wed, 13/06/2012 - 11:10 - 12:10
Location: 
MPIM Lecture Hall

The classical transfer principle by Ax-Kochen-Ershov states that any first order sentence holds in $\mathbb{Q}_{p}$ if and only if it holds in $\mathbb{F}_{p}(t)$ for large $p$. Motivic integration provides a framework in which this transfer principle can be generalized to ``sentences'' speaking about the measure of definable sets and integrals of certain functions. In my talk, I will explain such transfer principles.

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