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- Banff, AB
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The main objectives of the proposed conference on Diophantine approximation and algebraic curves will be the study of rational and integral solutions to Diophantine equations and inequalities and the connection with algebraic curves. Since early last century and even before, Diophantine approximation has played a large role in the study of solutions to Diophantine equations, a very old and influential topic in number theory. Thue's famous theorem was subsequently refined and expanded upon, culminating in Roth's celebrated result and his winning of the Fields medal. Shortly after that, Baker's ``effective" methods (earning him a Fields medal) were added into the mix. Concurrently with all this, the more systematic and algebraic development of the theory of curves (as opposed to the more ad hoc methods employed previously) was championed by the likes of Artin, Chevalley and Weil. A great achievement here was Falting's famous proof of Mordell's conjecture (yet another Fields medal for these areas).
Clearly these topics have intrigued mathematicians for a very long time. The techniques applied have been varied, but machinery originating here has also found use in a wide variety of fields. To mention just one example, it was noted over a century ago that the theories being developed over the rational number field applied equally well to fields of transcendence degree one over a finite field ("function fields"). Now curves defined over finite fields and their corresponding function fields are a cornerstone of modern computer coding theory.
During the proposed conference, experts in the areas of linear forms in logarithms, heights, the subspace theorem, the connections between Diophantine approximation and Nevanlinna theory, and others will come together with those in elliptic curves, abelian varieties and other closely related subjects in algebraic geometry. It is hoped that new light may be shed and insight gained into questions such as the existence of elliptic curves of large rank, the complete solution to certain families of equations, and deeper connections between the approximation of algebraic numbers and algebraic properties of curves and surfaces.
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