Mirror symmetry, Berkovich spaces and the Minimal Model Programme

代数几何学研究可以通过多项式方程定义的几何对象的形状，导致数论，代数和几何之间的深刻互动。自20世纪90年代以来，物理学通过镜像对称理论对代数几何产生了重大而令人惊讶的影响，镜像对称理论起源于弦理论，弦理论是一种将基本粒子数学描述为微小振动弦的提议。弦理论中的宇宙模型以代数几何中的一种特定类型的对象作为输入：所谓的卡拉比-丘簇，以数学家尤金尼奥·卡拉比和丘成桐的名字命名。这些物体因其特殊的对称性而吸引了物理学家的注意。卡-丘簇负责弦理论中用来解释粒子基本性质的6个“隐藏维度”。物理学家很快意识到，卡-丘变体并不是由物理模型唯一决定的：相反，卡-丘变体似乎是以“镜像对”的形式出现的，从而产生了等价的理论。在1990年前后，代数几何学家们不得不非常认真地对待这个想法，因为他们在枚举几何中有了一些引人注目的应用（计算卡-丘变种上的特殊类型曲线）。代数几何学家面临的主要挑战是为这些想法提供数学基础，也就是说，给出镜像对的确切定义，并设计出构造这种镜像对的技术。这仍然是一个持续的故事，但已经取得了很大进展。这个项目关注的是由Kontsevich和Soibelman开发的镜像对称的数学方法之一：Strominger-Yau-Zaslow（SYZ）猜想的非阿基米德方法。SYZ猜想是一个雄心勃勃的尝试，试图给出镜像对称的几何解释，它在数学中非常有影响力。大约在2000年，孔采维奇和索贝尔曼有了突破性的见解，人们应该能够在一个看似无关的领域找到SYZ猜想所预测的结构：非阿基米德几何，一个几何和分析的分支，最初是为了解决数论问题而设计的。在过去的几年里，我已经实现了一个重要组成部分Kontsevich和Soibelman的建议，通过引入一个新的成分到图片：最小模型计划（MMP）在双有理几何。该计划是过去40年来代数几何最成功的发展之一; 2018年，剑桥数学家Caucher Birkar因其对MMP的贡献而获得菲尔兹奖（数学界最负盛名的奖项）。MMP的目的是对代数几何中出现的所有几何对象进行分类。我已经发现，可以使用非阿基米德几何作为一个字典之间的镜像对称和MMP领域来回转移的问题和结果，从而提供了新的工具，同时研究这两个领域。这个项目的目标是进一步利用镜像对称，非阿基米德几何和双有理几何之间的相互作用。通过这种方式，我的目标是证明镜像对称的非阿基米德方法中的一些中心结构，并开发新的工具来理解MMP。

Algebraic geometry studies the shapes of geometric objects that can be defined by means of polynomials equations, leading to profound interactions between number theory, algebra and geometry. Since the 1990s, physics has exerted a major and surprising influence on algebraic geometry through the theory of mirror symmetry, which grew out of string theory, a proposal to describe fundamental particles mathematically as tiny vibrating strings. Models of the universe in string theory take as input a specific type of object in algebraic geometry: a so-called Calabi-Yau variety, named after the mathematicians Eugenio Calabi and Shing-Tung Yau. These objects attracted the attention of physicists because of their special symmetry properties. The Calabi-Yau variety is responsible for the 6 "hidden dimensions" that are postulated in string theory to explain the fundamental properties of particles. Physicists soon realized that the Calabi-Yau variety is not uniquely determined by the physical model: rather, Calabi-Yau varieties seemed to come in "mirror pairs" giving rise to equivalent theories. Algebraic geometers were forced to take this idea very seriously after some spectacular applications to enumerative geometry (counting special types of curves living on Calabi-Yau varieties) around 1990. The main challenge for algebraic geometers was to provide mathematical foundations for these ideas, that is, give an exact definition of what it means to be a mirror pair, and devise techniques to construct such pairs. This is still an ongoing story, but much progress has been made. This project is concerned with one of the mathematical approaches to mirror symmetry, developed by Kontsevich and Soibelman: the non-archimedean approach to the Strominger-Yau-Zaslow (SYZ) conjecture. The SYZ conjecture is an ambitious attempt to give a geometric explanation of mirror symmetry, and it has been very influential in mathematics. Around 2000, Kontsevich and Soibelman had the groundbreaking insight that one should be able to find the structures predicted by the SYZ conjecture in a seemingly unrelated field: non-archimedean geometry, a branch of geometry and analysis that was originally designed to solve problems in number theory. In the last few years, I have realized an important part of Kontsevich and Soibelman's proposal, by introducing a new ingredient into the picture: the minimal model programme (MMP) in birational geometry. This programme is one of the most successful developments in algebraic geometry in the last 40 years; in 2018, the Cambridge mathematician Caucher Birkar received the Fields medal (the most prestigious award in mathematics) for his contributions to the MMP. The aim of the MMP is to classify all the geometric objects that arise in algebraic geometry. I have discovered that one can use non-archimedean geometry as a dictionary to transfer questions and results back and forth between the field of mirror symmetry and the MMP, thus providing new tools to study both fields simultaneously. The goal of this project is to further exploit these interactions between mirror symmetry, non-archimedean geometry, and birational geometry. In this way, I aim to prove some of the central conjectures in the non-archimedean approach to mirror symmetry, and to develop new tools to understand the MMP.