Spaces of rational curves and diophantine geometry

这个建议的主要问题涉及理性点和空间的理性曲线代数簇代数非封闭的地面领域的关系几何或拓扑不变量。在算术方面，人们感兴趣的是有理点的存在，它们在各种拓扑中的密度，以及它们关于高度的分布。在几何方面，重点是双有理性质，如理性和理性连通性，投影不变量，如有效和充足因子的圆锥，以及几何对应。在过去的几十年里，算术几何已经成为最令人兴奋和快速增长的领域之一。在理解曲线的算术方面已经有了巨大的进步。这个提议的目标是推进我们对高维空间的理解。这些发展将不可能没有同化的想法从其他分支的数学：超越理论，代数拓扑学和谐波分析。作为回报，算术几何的进步对数学物理、动力系统、复分析产生了强烈的影响。对算术几何实验的需求导致了强大的计算工具和软件的发展，这些工具和软件现在被广泛使用，例如，密码学和数据分析

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.