Lagrangians from Algebra and Combinatorics

镜像对称是三种看似无关的几何之间的联系。这种联系最早是由物理学家在弦理论中观察到的，它带来了一系列令人惊讶的预测、强大的工具和令人着迷的几何问题。第一种几何，代数几何，研究由直线、圆和椭圆等方程定义的刚性物体。自古以来，几何学家就研究代数几何中的枚举问题。从常识开始（例如，两条直线相交于多少点），数列几何的问题迅速接近我们数学能力的极限。辛几何是由运动定律定义的，就像行星绕太阳运行的轨迹一样。虽然历史上用于计算约束系统的力学，例如连接在金属棒上的钟摆，辛几何的语言已经成为数学物理作为一个整体的自然设置。第三种几何——因其与巴西的历史联系而被称为热带几何——研究最优化和最大化。这里的形状描述了线性优化中的问题，例如在工厂中有效地分配工人的工作。在小尺度上，代数几何和辛几何看起来非常不同：代数几何看起来非常有序，而辛几何经常处理混乱的场景，比如一张满是台球的桌子。然而，当一个人缩小观察大尺度的行为时，这两种几何形状都可以用热带几何形状来近似。如果代数空间和辛空间在热带上看起来相同，它们被称为“镜像空间”。在过去的二十年里，“热带复对应”的研究通过热带几何的透镜来观察代数问题，产生了令人兴奋的结果。这个提议将着眼于最近发展的“热带-拉格朗日对应”，其中辛结构（拉格朗日）是由热带数据建立的。在镜像对称范围内的研究议程包括寻找新的镜像空间，开发新的辛几何计算方法，以及研究辛几何在热带曲线中的令人兴奋的应用。该项目还将研究几何之外的接口，特别是二聚体模型（对多米诺骨牌的描述）和突变（发生在辛几何、簇代数和三角分类中的修改过程）。

Mirror symmetry is a connection between three seemingly unrelated kinds of geometry. This link, first observed by physicists in string theory, has led to an flurry of surprising predictions, powerful tools, and fascinating questions for geometers.The first kind of geometry, algebraic geometry, studies rigid objects defined by equations such as lines, circles, and ellipses. Geometers have studied enumeration problems in algebraic geometry since antiquity. Starting at common knowledge (e.g., how many points do two lines intersect at), questions in enumerative geometry rapidly approach the limits of our mathematical capability. Symplectic geometry is defined by the laws of motion, like the trajectory of a planet around the sun. While historically used to calculate the mechanics of constrained systems, such as a pendulum attached to a metal rod, the language of symplectic geometry has become the natural setting for mathematical physics as a whole.The third kind of geometry --- called tropical geometry for its historical connection to Brazil --- studies optimization and maximization. Shapes here describe the problems in linear optimization, such as efficiently assigning workers to jobs in a factory.On a small scale, algebraic and symplectic geometry look incredibly different: algebraic geometry seems very orderly, while symplectic geometry frequently handles chaotic scenarios such as a table full of billiard balls. However, when one zooms out to view the large-scale behaviour, both geometries can be approximated by tropical geometry. If an algebraic space and symplectic space tropically look the same, they are called ``mirror spaces''.In the last two decades, the study of ``Complex to Tropical correspondences'' has produced exciting results by viewing problems in algebra through the lens of tropical geometry. This proposal will look at recently developed ``Tropical to Lagrangian correspondences,'' where symplectic structures (Lagrangians) are built from tropical data.The research agenda within the scope of mirror symmetry includes finding new mirror spaces, developing new computational methods in symplectic geometry, and investigating exciting applications of symplectic geometry in tropical curves. The project will also look at interfaces outside of geometry, specifically to dimer models (a description of domino tilings) and mutations (a modification process occurring in symplectic geometry, cluster algebras, and triangulated categories).