A Non-Archimedean Approach to Mirror Symmetry

镜像对称是数学中最神秘的二元性之一。它起源于理论物理，并对数学的各个领域产生了巨大的影响，特别是微分几何和代数几何。研究对象是被称为Calabi-Yau流形的几何形状，它们在不同的数学和物理背景下自然出现。镜像对称性预言，给定任何Calabi-Yau流形，都存在一个镜像流形，使得两者之间的几何关系列表不断增长，涉及到原本互不相关的深刻而精细的不变量。尽管这一学科不断取得进展，但镜像流形的全部存在范围和镜像对称的基本数学机制至今仍未解决。这个项目基于非阿基米德几何的最新发展，为镜面对称性的研究带来了一种新的方法。我们的目标是构思和追求一个成熟的非阿基米德镜像对称理论，这将导致现有方法无法获得的新结果，以及超出当前范围的应用。该奖项还将支持研究生和本科生。通过对Calabi-Yau流形的非阿基米德Syz环面上边界的解析曲线的计数，研究镜像的存在性问题。除了最小模型程序外，还将使用非阿基米德Monge-Ampère方程来探索这种纤颤的新构造。非阿基米德曲线的Gromov紧致性定理需要建立在广义尾部条件的基础上，而广义尾部条件依赖于目标空间的新的解析运算。曲线计数的虚拟基本类的构造将依赖于我们以前在派生的非阿基米德几何上的工作。PI将证明从曲线计数获得的镜像代数的各种性质，包括结合性、收敛半径和奇异性估计。局部镜像代数有望利用跨越墙的公式粘合在一起，形成全局镜像多样性。一个长期的目标是表明镜面结构是一种对合，是镜面二元性的最佳表现。PI还致力于镜像对称性之外的应用，特别是在表示理论中的团簇代数和双曲几何中的KSBA稳定对的模。PI期待着未来与格罗斯-西伯特计划基于对数几何的卓有成效的互动。这一奖项反映了NSF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。

Mirror symmetry is one of the most mysterious dualities in mathematics. It originated from theoretical physics, and generated great impact on various areas of mathematics, in particular differential geometry and algebraic geometry. The objects of study are geometric shapes called Calabi-Yau manifolds that appear naturally in different contexts of mathematics and physics. Mirror symmetry predicts that given any Calabi-Yau manifold, there exists a mirror manifold, such that an ever-growing list of geometric relations hold between the two, involving deep and elaborate invariants that are otherwise unrelated. Despite the continual progress in the subject, both the full extent of existence of mirror manifolds and the underlying mathematical mechanism of mirror symmetry remain unsolved today. This project brings a new approach to the study of mirror symmetry, based on latest developments from non-archimedean geometry. The goal is to conceive and pursue a full-fledged theory of non-archimedean mirror symmetry, which will lead to new results unattainable from existing methods, as well as applications beyond the current scope. This award will also support graduate and undergraduate students. The mirror existence problem will be studied via the enumeration of analytic curves with boundaries in non-archimedean SYZ torus fibrations of Calabi-Yau manifolds. New constructions of such fibrations will be explored using non-archimedean Monge-Ampère equations in addition to the minimal model program. An analog of the Gromov compactness theorem for non-archimedean curves with boundaries needs to be established, based on generalized tail conditions, which depend on novel analytic surgeries of the target spaces. The construction of virtual fundamental classes for the curve counts will rely on our previous works on derived non-archimedean geometry. PI will prove various properties of the mirror algebra obtained from the curve counts, including associativity, radius of convergence and singularity estimates. The local mirror algebras are expected to glue together to form the global mirror variety using wall-crossing formulas. A long-term goal is to show that the mirror construction is an involution, the best exhibition of mirror duality. PI also aims for applications outside mirror symmetry, in particular towards cluster algebras in representation theory and the moduli of KSBA stable pairs in birational geometry. PI expects fruitful future interactions with the Gross-Siebert program based on logarithmic geometry.This 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.