Symplectic Cohomology, log Calabi-Yau Varieties, and equivariant Lagrangian Submanifolds

该项目研究几何的不同领域之间的联系，这些领域受到理论物理学思想的启发。理论物理学中的一个重要思想是物理理论之间的二元性概念。这种二重性的一个结果是，描述这些理论的数学模型之间存在着深刻的联系。这些联系使我们能够从另一个角度看待一个模型中的数学问题，从而得出新的结果。在这个项目中，数学模型一方面来自代数几何（由多项式方程定义的集合的几何）和表示论（对称性的研究），另一方面来自辛几何（经典力学相空间的几何）。与它们相关的对偶性被称为同调镜像对称。这个项目的重点是挖掘这种联系，以获得对二元性每一方产生的结构的新见解。该项目的一部分研究如何在辛几何中出现对称性，以及这种新的观点如何能够深入了解表示论。另一部分研究辛几何中的动力学与代数几何中最基本的对象之一，即函数之间的关系。该项目还支持在这个快速发展的研究领域培养研究生和早期职业数学家。更广泛地说，这个项目符合数学和物理之间持续的相互作用，几个世纪以来，导致了理论的进步，使我们这个时代的变革技术成为可能。这个项目的组织原则是对数卡-丘簇的同调镜像对称（簇作为紧化中的反规范因子的补充而出现）。我们认为这种品种的代数和辛双方的对应关系。在辛方面，我们考虑的主要结构是辛上同调，这是一种代数结构，由辛流形上某些哈密顿流的周期轨道构建而成（因此与动力学有关）。该项目的核心是将这种结构与镜像代数簇上的函数和向量场联系起来。与表示论的联系是通过考虑半单代数群G在代数方面的旗簇而出现的。这些不是对数卡-丘，但它们包含了开子集，该项目的一部分是为了更好地理解如何在两种情况之间传递（这涉及到考虑辛侧的势函数）。旗簇的对称性，即群G和它的李代数，也应该出现在辛侧。李代数的自然家园是辛上同调，而群作用本身表现在这个李代数对等变拉格朗日子流形的Floer上同调的作用上，这些子流形是代数几何中等变向量丛的对应物。最终，人们期望得到G在拉格朗日Floer上同调群中的表示。这一努力的回报是，这些Floer上同调群有一个杰出的基础，这个项目试图了解这个基础是如何与Lusztig，Mirkovic-Vilonen等人的表示理论中的各种已知的典型基础相关的。在处理这些问题时，该项目使用了Strominger-Yau-Zaslow方法的思想来实现镜像对称，由Gross-Siebert和Gross-Hacking-Keel开发，以及PI在以前的工作中开发的技术在日志Calabi-Yau表面（复杂的二维）的情况下。

This project studies connections between different areas of geometry that are inspired by ideas from theoretical physics. An important idea in theoretical physics is the notion of a duality between physical theories. A consequence of such a duality is a deep connection between the mathematical models that describe those theories. These connections allow us to look at a mathematical question in one model from another perspective, and thus derive new results. For this project, the mathematical models come from algebraic geometry (the geometry of sets defined by polynomial equations) and representation theory (the study of symmetry) on the one hand, and symplectic geometry (the geometry of the phase spaces of classical mechanics) on the other. The duality relating them is known as homological mirror symmetry. The focus of this project is to mine this connection for new insights into structures arising on each side of the duality. One part of the project studies how symmetries arise in symplectic geometry, and how this new perspective can give insights into representation theory. Another part studies the relationship between dynamics in symplectic geometry and one of the most basic objects in algebraic geometry, namely functions. The project also supports the training of graduate and early-career mathematicians in this rapidly-developing area of research. More broadly, this project fits into the ongoing interaction between mathematics and physics that has, over the centuries, led to theoretical advances that have enabled the transformative technologies of our time.The organizing principle for this project is homological mirror symmetry for log Calabi-Yau varieties (varieties arising as the complement of an anticanonical divisor in a compactification). We consider such varieties both for the algebraic and the symplectic sides of the correspondence. On the symplectic side, the main structure we consider is symplectic cohomology, an algebraic structure that is built out of periodic orbits of certain Hamiltonian flows on a symplectic manifold (hence the connection to dynamics). The heart of the project is to relate this structure to functions and vector fields on the mirror algebraic variety. The connection to representation theory appears by considering the flag variety of a semisimple algebraic group G on the algebraic side. These are not log Calabi-Yau, but they contain open subsets which are, and part of the project is to understand better how to pass between the two situations (this involves considering a potential function on the symplectic side). The symmetries of the flag variety, namely the group G and its Lie algebra, should appear in the symplectic side as well. The natural home for the Lie algebra is symplectic cohomology, and the group action itself is manifested in the action of this Lie algebra on the Floer cohomology of equivariant Lagrangian submanifolds, which are the counterpart of equivariant vector bundles in algebraic geometry. Ultimately, one expects to obtain representations of G in the Lagrangian Floer cohomology groups. The pay-off for this effort is that these Floer cohomology groups come with a distinguished basis, and this project seeks to understand how that basis is related to the various known canonical bases in representation theory of Lusztig, Mirkovic-Vilonen, and others. In approaching these problems, the project uses ideas from the Strominger-Yau-Zaslow approach to mirror symmetry, as developed by Gross-Siebert and Gross-Hacking-Keel, as well as techniques developed by the PI in previous work on the case of log Calabi-Yau surfaces (complex dimension two).