Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry

Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separ...

Arithmetic and Diophantine Geometry

Topics in Diophantine Geometry

7 Faltings’ Theorem 36 7.1 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.2 Height Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.3 Removing the Nonvanishing Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.4 Proof of Nonvanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Galois Theory and Diophantine geometry

To a large extent, the investigations to be brought up today arise from a curious inadequacy having to do with the arrow on the left. On the one hand, it is widely acknowledged that the theory of motives finds a strong source of inspiration in Diophantine geometry, inasmuch so many of the structures, conjectures, and results therein have as model the conjecture of Birch and Swinnerton-Dyer, whe...

Model Theory and Diophantine Geometry

I discuss some recent applications of model theory to diophantinetype problems in algebraic geometry. I give the required background, as well as a sketch of the proofs.