Lattice Polytopes with a View Toward Algebraic Geometry

PI致力于晶格多面体的系统研究，重点是在邻近领域的应用，特别是在环面几何中的应用。第一个项目关注于在研究Ehrhart多项式时产生的猜想及其与代数几何和数几何的关系，Ehrhart多项式计算晶格多面体的整数倍中的晶格点的数量。最近，Ehrhart理论不变量(h*-多项式的次数)开辟了一条研究没有内格点的格多面体的新途径。PI研究了它与其他不变量的关系，如A-判别式的次数、极化环面簇的谱值或NEF值，并探索了格多面体以外的可能的推广。第二个项目的目标是加强我们对反身性多面体和Gorenstein多面体的理解，这些多面体在构造镜像对称的Calabi-Yau变种家族的Batyrev-Borisv中起着至关重要的作用。这里，一个特别感兴趣的不变量是Gorenstein多面体的弦E-多项式。格多面体理论是代数几何、凸几何和离散几何、最优化理论和数字几何的交集。格多面体的定义非常简单：它是一个格子中有限多个点的凸壳。由于它们的基本性质，这些凸几何对象在整个纯数学和应用数学中以各种伪装无处不在，它们为跨学科研究提供了肥沃的土壤。最突出的是，格多面体为高维代数簇提供了一种显式的组合方法，称为环簇。这种相互作用已被证明在代数几何和多面体组合数学中是成功的，并在其他领域有意想不到的应用，特别是在弦论中。PI从这些不同的观点出发，研究关于晶格多面体的开放问题。格子多面体的魅力还在于，许多问题都可以用初等的方法来表示，并且非常适合计算方法，这使得这个领域对学生很有吸引力。这个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面体的书，这将使学生尽可能容易地接触到当前的研究主题。

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.