Lattice Polytopes with a View Toward Algebraic Geometry

从代数几何的角度看晶格多面体

• 批准号: 1203162
• 负责人: John Duncan
• 金额: $ 9.04万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1203162&HistoricalAwards=false
• 关键词: Lattice; Polytopes; View; Toward; Algebraic

The PI pursues the systematic study of lattice polytopes with an emphasis on applications in neighboring areas, in particular, in toric geometry. The first project focuses on conjectures and relations to algebraic geometry and geometry of numbers that arise in the study of Ehrhart polynomials, which count the number of lattice points in integer multiples of lattice polytopes. Recently, an Ehrhart-theoretic invariant (the degree of the h*-polynomial) has opened up a refined way of looking at lattice polytopes without interior lattice points.The PI investigates the relations to other invariants such as the degree of the A-discriminant, the spectral value or the nef value of a polarized toric variety and explores possible generalizations beyond the realm of lattice polytopes. The goal of the second project is to enhance our understanding of reflexive and Gorenstein polytopes that play a crucial role in the Batyrev-Borisov construction of families of mirror-symmetric Calabi-Yau varieties. Here, one invariant of specific interest is the stringy E-polynomial of a Gorenstein polytope. A significant part of this research is also concerned with obtaining classification results in order to check conjectures and to search for counterexamples.The theory of lattice polytopes lies at the intersection of algebraic, convex and discrete geometry, optimization and the geometry of numbers. The definition of a lattice polytope is extraordinarily simple: it is the convex hull of finitely many points in a lattice.Because of their elementary nature, these convex-geometric objects are ubiquitous in various disguises throughout pure and applied mathematics, and they provide fertile ground for interdisciplinary research. Most prominently, lattice polytopes provide an explicit, combinatorial approach to higher-dimensional algebraic varieties, called toric varieties. This interaction has proven to be successful for algebraic geometry as well as for polyhedral combinatorics and has unexpected applications in other areas, notably in string theory. The PI studies open questions on lattice polytopes motivated from these different viewpoints. The fascination of lattice polytopes lies also in the fact that many problems can be formulated in an elementary way and are well suited for computational approaches which makes the area attractive to students. One component of this project is to finish writing a book on lattice polytopes with Christian Haase and Andreas Paffenholz that will make it as easy as possible for students to get into contact with current research topics.

PI 致力于晶格多面体的系统研究，重点是邻近领域的应用，特别是在环面几何中。第一个项目重点关注埃尔哈特多项式研究中出现的猜想以及与代数几何和数字几何的关系，该多项式计算格子多面体整数倍中的格点数量。最近，Ehrhart 理论不变量（h* 多项式的次数）开辟了一种在没有内部格点的情况下观察晶格多面体的精确方法。PI 研究了与其他不变量的关系，例如 A 判别式的次数、偏振环面簇的光谱值或 nef 值，并探索了超出晶格多面体领域的可能推广。第二个项目的目标是增强我们对反身多胞体和 Gorenstein 多胞体的理解，它们在镜像对称 Calabi-Yau 品种家族的 Batyrev-Borisov 构建中发挥着至关重要的作用。这里，一个特别令人感兴趣的不变量是 Gorenstein 多面体的弦 E 多项式。这项研究的一个重要部分还涉及获得分类结果，以检查猜想并寻找反例。 格子多胞体理论位于代数、凸和离散几何、最优化和数几何的交叉点。晶格多胞体的定义非常简单：它是晶格中有限多个点的凸包。由于其基本性质，这些凸几何对象以各种形式在纯数学和应用数学中无处不在，它们为跨学科研究提供了肥沃的基础。最突出的是，晶格多面体为高维代数簇（称为环面簇）提供了一种明确的组合方法。这种相互作用已被证明对于代数几何和多面体组合学是成功的，并且在其他领域，特别是在弦理论中具有意想不到的应用。 PI 研究从这些不同的观点出发，提出了关于晶格多胞体的开放性问题。晶格多胞体的魅力还在于许多问题可以用基本的方式表达，并且非常适合计算方法，这使得该领域对学生有吸引力。该项目的一个组成部分是与 Christian Haase 和 Andreas Paffenholz 一起完成一本关于晶格多胞体的书，这将使学生尽可能容易地接触当前的研究主题。