Spaces of rational curves and diophantine geometry

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The main questions of this proposal concern rational points and spaces of rational curves on algebraic varieties over algebraically nonclosed ground fields in relation to geometric or topological invariants. On the arithmetic side, one is interested in existence of rational points,their density in various topologies, and their distribution with respect to heights. On the geometric side,the focus is on birational properties, such as rationality and rational connectedness, on projective invariants, such as the cones of effective and ample divisors, and on geometric correspondences. In the last decades, arithmetic geometry has become one of the most exciting and rapidly growing fields. There has been tremendous progress in understanding the arithmetic of curves. The goal of this proposal is to advance our understanding of higher-dimensional spaces. These developments would not be possible without the assimilation of ideas from other branches of mathematics: transcendence theory, algebraic topology, and harmonic analysis. In return, advances in arithmetic geometry have had strong impact in mathematical physics, dynamical systems, complex analysis. The need for experimentation in arithmetic geometry has lead to the development of powerful computational tools and software, which are now widely used, e.g., in cryptography and data analysis.