Mirror symmetry, Berkovich spaces and the Minimal Model Programme

代数几何研究可以用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，该理论提出将基本粒子数学描述为微小的振动弦。弦理论中的宇宙模型以代数几何中一种特定类型的物体为输入：所谓的Calabi-Yau变种，以数学家尤金尼奥·卡拉比和游星东的名字命名。这些天体因其特殊的对称性引起了物理学家的注意。Calabi-Yau变种产生了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，Calabi-Yau变种并不是唯一由物理模型决定的：相反，Calabi-Yau变种似乎是以镜像对形式出现的，从而产生了等价的理论。在1990年左右，在计数几何(计算生活在Calabi-Yau变种上的特殊类型的曲线)的一些壮观应用之后，代数几何学者被迫非常认真地对待这个想法。代数几何学家的主要挑战是为这些想法提供数学基础，也就是给出镜像对意味着什么的准确定义，并设计出构建这种镜像对的技术。这仍然是一个正在进行的故事，但已经取得了很大进展。这个项目涉及由康采维奇和索贝尔曼发展的镜像对称性的数学方法之一：Strominger-Yau-Zaslow(Syz)猜想的非阿基米德方法。Syz猜想是一次雄心勃勃的尝试，试图给出镜像对称性的几何解释，它在数学中非常有影响力。大约在2000年，Kontsevich和Soibelman有了开创性的见解，即人们应该能够在一个看似无关的领域找到Syz猜想预测的结构：非阿基米德几何，最初设计用于解决数论问题的几何和分析的一个分支。在过去的几年里，我意识到了康采维奇和索贝尔曼提议的一个重要部分，在画面中引入了一个新的成分：二次几何中的最小模型程序(MMP)。该计划是过去40年来代数几何领域最成功的发展之一；2018年，剑桥数学家考彻·比卡尔因其对数学几何的贡献而获得菲尔兹奖(数学界最负盛名的奖项)。它的目的是对代数几何中出现的所有几何对象进行分类。我发现，人们可以使用非阿基米德几何作为一本词典，在镜像对称领域和基质理论之间来回传递问题和结果，从而为同时研究这两个领域提供了新的工具。这个项目的目标是进一步探索镜面对称性、非阿基米德几何和双曲面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性的非阿基米德方法中的一些核心猜想，并开发新的工具来理解基质金属氧化物。

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.