Talks

Jean-Paul Allouche (Paris)
Thue, combinatorics on words, and the ubiquitous Thue-Morse sequence
Abstract Two seminal papers of Thue (1906 and 1912), somehow marginal in his mathematical works, gave rise to a very active branch of discrete mathematics, the combinatorics on words. In particular a binary sequence studied by Thue (now called the Thue-Morse sequence) shows up - sometimes unexpectedly - in many mathematical fields, as well as in other fields.

Mike Bennett (Vancouver)
Rational approximation to algebraic numbers of small height revisited
Abstract We consider effective methods for approximating roots of rational numbers, originating in work of Thue. New estimates for non-archimedean valuations of Pade approximations enable us sharpen earlier results of Siegel and of Evertse on binomial Thue inequalities. This is joint work with Greg Martin and Kevin O'Bryant.

Daniel Bertrand (Paris)
Manin kernels, algebraic independence, and diophantine equations over function fields
Abstract The first and third themes of this talk are well-known to be related. Here, I will report on another type of link, based on various versions of a theorem of Y. André on algebraic independence of abelian integrals over function fields. I will then discuss their role in a joint work with D. Masser, A. Pillay, U. Zannier, on the relative Manin-Mumford conjecture for families of semi-abelian surfaces. Finally, I will mention an application to Pell's equations over polynomial rings, in the style of D. Masser's talk, but with a non-square free discriminant (and with a French accent).

Frits Beukers (Utrecht)
Geodesic continued fractions and LLL
Abstract In the lecture we briefly explain the idea of higher dimensional geodesic continued fractions and more briefly, the LLL-algorithm. The main goal of this lecture is to present a proposal to combine the best of these two ideas into a higher dimensional continued fraction algorithm.

Yann Bugeaud (Strasbourg)
Transcendence measures for continued fractions with low complexity
Abstract We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville's inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler's exponent w₂ and Koksma's exponent w₂*.

Pietro Corvaja (Udine)
Thue's equations in several variables
Abstract In the article of Thue we are celebrating, it is proved that certain diophantine equations of the form F(x,y)=c, where F is a homogeneous form and c a constant, have only finitely many integral solutions. A main point in Thue's argument is the reducibility (over the algebraic numbers) of homogeneous forms in two variables. In a joint research with Zannier (related also to works of Evertse and Ferretti), we consider similar equations in several variables, namely equations of the form

Jan-Hendrik Evertse (Leiden)
Reduction of binary forms of given discriminant and root separation of irreducible polynomials
Abstract

Catherine Goldstein (Paris)
Axel Thue in context
Abstract Axel Thue's works, in particular his celebrated paper, "Über Annäherungswerte algebraischer Zahlen", are usually perceived as solitary gems in the mathematics of their time. This talk will sketch out Axel Thue's biography and discuss some features of his work in the context of Thue's mathematical formation, as well as that of Diophantine analysis and approximation at the turn of the XXth century.

Kálmán Győry (Debrecen)
Effective results for Thue equations over finitely generated domains (joint work with A. Bérczes and J.H. Evertse)
Abstract In the first part of the talk, I give a brief survey of effective finiteness results concerning Thue equations over number fields. In the second part, a new, general effective finiteness theorem will be presented for Thue equations over arbitrary finitely generated domains over Z which may contain transcendental elements, too. Further, a bound will be formulated for the size of the solutions which is explicitly given in terms of each parameter involved. Similar results will be mentioned for some other classical equations as well. Finally, the method of proof will be outlined.

Philipp Habegger (Darmstadt)
Unlikely Intersections and o-Minimal Structures
Abstract Conjectures on unlikely intersections generalize various problems in diophantine geometry such as the Manin-Mumford Conjecture. They originated in work of Bombieri-Masser-Zannier, Pink, and Zilber. Using a point counting result of Pila-Wilkie, Zannier devised a new strategy to prove the Manin-Mumford Conjecture. After providing some background information I will explain how to generalize this approach to get new result on higher dimensional unlikely intersections in abelian varieties defined over number fields. This is joint work in progress with Jonathan Pila.

Aaron Levin (East Lansing)
Integral points of bounded degree on curves
Abstract Let C be an affine curve over a number field k. If there exists a finite morphism of degree d from C to G_m=P¹\{0,∞}, then by pulling back units, C will contain infinitely many S-integral points of degree d over k (for some finite S). We prove a converse to this statement, completely classifying affine curves with infinitely many integral points of degree d.

David Masser (Basel)
Relative Manin-Mumford and Pell's Equation over polynomial rings
Abstract

Tanguy Rivoal (Grenoble)
Values of G-functions
Abstract I will present a few structural properties of the set of all possible values taken by G-functions (in Siegel's sense). I will obtain certain consequences concerning the set of real numbers having rational approximations generated by G-functions. This is a joint work with Stéphane Fischler.

Damien Roy (Ottawa)
On Schmidt and Summerer parametric geometry of numbers
Abstract In a series of recent papers, W.M. Schmidt and L. Summerer develop a remarkable theory of parametric geometry of numbers which enables them to recover many results about simultaneous rational approximation to families of Q-linearly independent real numbers, or about the dual problem of forming small linear integer combinations of such numbers. They recover classical results of Khintchine and Jarnik as well as more recent results by Bugeaud and Laurent. They also find many new results of Diophantine approximation.
Their theory provides constraints on the behavior of the successive minima of a natural family of one parameter convex bodies attached to a given n-tuple of real numbers, in terms of this varying parameter. In this talk, we are interested in the converse problem of constructing n-tuples of numbers for which the corresponding successive minima obey given behavior. We will present the general theory of Schmidt and Summerer, and report on recent progress concerning the above problem.

Wolfgang Schmidt (Boulder)
Standard and uniform approximation exponents
Abstract Given reals x₁,...,x_n, these exponents are the supremum of the numbers u such that the relations

Martin Sombra (Barcelona)
Successive minima and distribution of small points on toric varieties
Abstract Given a toric metrized line bundle on a toric variety over an number field, I will present a formula for the successive minima of the associated height function. From this, we will be able to deduce a toric version of Zhang's theorem of successive minima, comparing the quotient of the χ-arithmetic volume and the degree with the successive minima of the height function.
As a consequence, we obtain criteria for the equidistribution of the Galois orbits of points of small height in a toric variety.
This is joint work with J. I. Burgos Gil, P. Philippon, and J. Rivera-Letelier.

Paul Vojta (Berkley)
Toric geometry and the lemmas of Dyson and Roth
Abstract In 1989, I proved a Dyson lemma for products of two smooth projective curves of arbitrary genus. In 1995, M. Nakamaye extended this to a result for a product of an arbitrary number of smooth projective curves of arbitrary genus, in a formulation involving an additional "perturbation divisor". In 1998, he also found an example in which a hoped-for Dyson lemma is false without such a perturbation divisor. This talk will present work in progress on eliminating the perturbation divisor by using a different definition of "volume" at the points under consideration. The proof involves toric and toroidal geometry, and this is reflected in the statement as well.
I will also describe nascent work on applying methods to give a stronger version of Roth's lemma.

Michel Waldschmidt (Paris)
Families of Diophantine Thue equations with only finitely many nontrivial solutions
Abstract We consider infinite families of Thue Diophantine equations F_λ(x,y)=m in two unknowns x,y depending on one parameter λ. Each form F_λ(X,Y)∈ Z[X,Y] is irreducible of degree ≥3, and m is a fixed nonzero integer. We are interested with families for which the set of solutions (λ,x,y) with xy≠0 is finite. Up to some years ago, very few families were exhibited, but we will describe many new such families obtained in a joint work with Claude Levesque. The lecture will start with a survey of a few results on Thue equations.

Umberto Zannier (Pisa)
Diophantine problems with S-units and perfect powers with few nonzero digits
Abstract As a central example in this talk, we shall discuss a recent result (obtained jointly with P. Corvaja) asserting the finiteness of odd perfect powers with four nonzero binary digits. The investigation was motivated mainly by a question of D. Ghioca and T. Scanlon in the context of the Dynamical Mordell-Lang Conjecture. We shall frame this kind of issue in the more general context of S-unit equations, and shall recall several related results.