CIRM 上的霍奇理论与 o-极小性
Hodge theory and o-minimality at CIRM
AI总结：
本文综述了 o-极小性与 Zilber-Pink 猜想的关系，重点介绍 Ax-Schanuel 定理、霍奇轨迹分布及周期映射像的代数性和拟射影性等近期进展。
摘要：
我们讨论 o-极小性与所谓 Zilber-Pink 猜想之间的关系。自 Pila 和 Zannier 的工作以来，o-极小几何中的代数化定理对丢番图几何产生了深远影响（尤其是在阿贝尔簇和 Shimura 簇中特殊点的研究上）。我们首先聚焦于函数超越性，讨论各种近期且引人注目的 Ax-Schanuel 定理，以及 Zilber-Pink 猜想中相关的几何部分。借助这些工具，我们将研究任意霍奇结构变分的霍奇轨迹的分布（典型/非典型二分法），并介绍一些近期应用。最后，我们将描述周期映射像的代数性和拟射影性。
英文摘要：
We discuss the relationship between o-minimality and the so called Zilber-Pink conjecture. Since the work of Pila and Zannier, algebraization theorems in o-minimal geometry had profound impacts in Diophantine geometry (most notably on the study of special points in abelian and Shimura varieties). We will first focus on functional transcendence, discussing various recent and spectacular Ax-Schanuel theorems, and the related geometric part of Zilber-Pink. Armed with these tools, we will study the distribution of the Hodge locus of an arbitrary variation of Hodge structures (the typical/atypical dichotomy) and present some recent applications. We will conclude by describing the algebraicity and quasiprojectivity of images of period maps.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
