Abstract:
We collect here a number of open questions concerning Diophantine equations (including Pillai's Conjectures), Diophantine approximation (featuring the $abc$ Conjecture) and transcendental number theory (with, for instance, Schanuel's Conjecture). Some questions related to Mahler's measure and Weil absolute logarithmic height are then considered (e.g., Lehmer's Problem). We also discuss Mazur's question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Key words and phrases:Diophantine problems, transcendence, linear and algebraic independence, Schanuel conjecture, four exponential conjecture, multizeta values, $abc$ conjecture, Waring problem, Diophantine approximation, continued fractions, Thue–Siegel–Roth–Schmidt, irrationality measures, Markoff spectrum, height, Lehmer problem, Mazur density conjecture, abelian varieties, special functions, function fields.
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