Lagrangians from Algebra and Combinatorics

镜像对称是三种看似无关的几何之间的联系。物理学家在弦理论中首次发现了这种联系，并由此产生了一系列令人惊讶的预言、强大的工具和令几何学家着迷的问题。第一种几何是代数几何，它研究由直线、圆和椭圆等方程定义的刚性物体。自古以来，几何学家就在代数几何中研究计数问题。从常识开始（例如，两条直线相交于多少点），计数几何中的问题很快接近我们数学能力的极限。辛几何是由运动定律定义的，就像行星绕太阳的轨道一样。虽然历史上用于计算约束系统的力学，如一个钟摆连接到一个金属杆，辛几何的语言已成为自然的设置为数学物理作为一个整体。第三种几何-所谓的热带几何的历史连接到巴西-研究最优化和最大化。在这里，我们描述线性优化中的问题，比如有效地分配工人到工厂的工作岗位。在小范围内，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情况，比如一张桌子上摆满了台球。然而，当人们缩小以查看大尺度行为时，两种几何形状都可以近似为热带几何形状。如果一个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，通过热带几何的透镜观察代数问题，对”复到热带对应“的研究产生了令人兴奋的结果。这项建议将着眼于最近开发的“热带拉格朗日对应”，其中辛结构（拉格朗日）是建立在热带数据。镜像对称范围内的研究议程包括寻找新的镜像空间，发展新的计算方法辛几何，并调查辛几何在热带曲线令人兴奋的应用。该项目还将研究几何之外的界面，特别是二聚体模型（描述多米诺骨牌）和突变（发生在辛几何，簇代数和三角范畴中的修改过程）。

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).