Spaces of rational curves and diophantine geometry

这一建议的主要问题涉及代数非闭地域上有理曲线的有理点和空间与几何或拓扑不变量的关系。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，以及它们相对于高度的分布。在几何方面，重点是二元性，如合理性和有理连通性，投射不变量，如有效和充分因子的圆锥，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发展最快的领域之一。在理解曲线的算术方面取得了巨大的进步。这项提议的目的是促进我们对更高维度空间的理解。如果不吸收其他数学分支的思想，这些发展是不可能的：超越理论、代数拓扑学和调和分析。反过来，算术几何的进步对数学物理、动力系统和复杂分析产生了强大的影响。对算术几何实验的需要导致了强大的计算工具和软件的发展，它们现在被广泛使用，例如在密码学和数据分析中。

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.