Thematic Month at CIRM in Complex Geometry

该奖项为美国数学家参加将于2019年1月28日至3月1日在法国卢米市CIRM举行的题为“主题月：复杂几何”的会议提供部分支持。会议的主要主题是几何，它是现代数学的基石之一，有着广泛的应用，从弦论到密码学。通过在几何研究的各个活跃领域(包括算术、代数和复杂微分几何)之间新发现的深刻联系，每一门学科都见证了意想不到的突破性进展。这次会议旨在促进这些不同领域的研究人员之间的互动，以促进这些发展。这一奖项支持的活动包括一个大师班(为期一周)和四个国际会议，每个会议为期一周：Kaehler几何中的奇异度量(第2周)、旋转几何和霍奇理论(第3周)、完整曲线、有理曲线和折叠(第4周)以及球商曲面和格子(第5周)。在过去的几年里，我们在理解复代数簇的几何方面取得了重要进展，更广泛地说，是Kaehler簇。这次会议的目的是促进在各个活跃的研究领域聚集具有国际地位的专家。大师班的目的是介绍将在整个会议中使用的技术和理论。本课程主要包括三门课程：Hodge理论、K3曲面和流形上的特殊度量。在第二周，目标是研究各种几何问题，其中奇异度量理论发挥着重要作用。这包括以下主题：奇异Kaehler-Einstein簇及其模，复几何中的正性和推广的Yau-Tian-Donaldson猜想。第三周的目的是研究代数几何中的各种方法，以期在出生几何和模空间中的应用。以下主题将具有特别重要的意义：霍奇理论和高维簇的模。第四周的目标是聚集不同领域的专家，研究复杂变种中的代数曲线和超越曲线的几何。主题包括：喷流空间和叶层、特殊变种和内瓦林纳理论。尽管人们一直在寻找球商曲面的几何结构，但用显式方程得到的例子却很少。最近，Cartwright和Steger为这种构造引入了新的算法，从而完成了伪射影平面的分类。最后一周的重点将是对这项基本工作及其应用的详细分析。会议网页：https://conferences.cirm-math.fr/2060.htmlThis奖反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The award provides partial support for the participation of U.S.-based Mathematicians in a conference in Pure Mathematics titled "Thematic Month: Complex Geometry", to be held at CIRM, Luminy (France) from January 28 to March 01, 2019. The main theme of the conference is Geometry; one of the cornerstones of modern Mathematics with broad applications, ranging from String Theory to Cryptography. Through newly discovered, deep connections between various active areas of research in Geometry, including Arithmetic, Algebraic and Complex Differential Geometry, each subject has witnessed unexpected and groundbreaking advances. The conference aims to facilitate interactions among researchers in these diverse fields to further these developments. Such activities have proved to be of unparalleled importance for geometers in these interconnected areas, specially for those in early stages of their careers.The event supported by this award is composed of a master class (1 week long) and 4 international conferences, each one week long: Singular Metrics in Kaehler Geometry (week 2), Birational Geometry and Hodge Theory (week 3), Entire Curves, Rational Curves and Foliations (week 4), and Ball Quotient Surfaces and Lattices (week 5). The last couple of years have been witness to important progress in our understanding of the geometry of complex algebraic varieties, and more generally Kaehler varieties. The aim of this conference is to facilitate the gathering of experts of international stature in various active areas of research. The aim of the Master Class is the introduction of techniques and theories that will be used throughout the conference. This will mainly consist of three courses: Hodge theory, K3 surfaces and special metrics on manifolds. During the second week, the goal is to study various geometric problems where the theory of singular metrics play an important role. This includes the following topics: Singular Kaehler-Einstein varieties and their moduli, Positivity in Complex Geometry and Generalized Yau-Tian-Donaldson Conjecture. The aim of the third week is to investigate various methods in Algebraic Geometry with a view towards applications in Birational Geometry and Moduli spaces. The following topics will be of particular importance: Hodge theory and Moduli of higher dimensional varieties. The goal of the fourth week is to gather specialists in different fields working on the geometry of algebraic and transcendental curves in complex varieties. Topics include: jet spaces and foliations, Special Varieties and Nevanlinna theory. Despite an intensive search for finding a geometric construction for ball quotient surfaces, very few examples have been obtained with explicit equations. Recently Cartwright and Steger have introduced new algorithms for such constructions, leading to the completion of the classification of fake projective planes. The focus of the final week will be a detailed analysis of this fundamental work and its applications. Webpage for the conference: https://conferences.cirm-math.fr/2060.htmlThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.