Lattice Polytopes with a View Toward Algebraic Geometry

PI追求晶格多面体的系统研究，重点是在相邻领域的应用，特别是在复曲面几何。第一个项目的重点是代数几何和几何中出现的Ehrhart多项式的研究，它计数的整数倍的晶格多面体的晶格点的数量的关系。最近，一个Ehrhart-theoretical不变量（h*-多项式的次数）开辟了一种精细的方法来研究没有内部格点的格多面体，PI研究了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，并探索了格多面体领域之外的可能的推广。第二个项目的目标是提高我们对自反和Gorenstein多面体的理解，这些多面体在镜像对称Calabi-Yau簇族的Batyrev-Borisov构造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。这项研究的一个重要部分也涉及到获得分类结果，以检查结构和搜索的counterexamples.The理论的格多面体位于交叉的代数，凸和离散几何，优化和几何的号码。格多面体的定义非常简单：它是格中许多点的凸船体。由于它们的基本性质，这些凸几何对象以各种伪装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土壤。最突出的是，格多面体提供了一个明确的，组合的方法来高维代数簇，称为环面簇。这种相互作用已被证明是成功的代数几何以及多面体组合，并在其他领域，特别是在弦理论意想不到的应用。PI研究了从这些不同观点出发的关于格多面体的开放性问题。格多面体的魅力还在于这样一个事实，即许多问题可以制定一个基本的方式，非常适合计算方法，使该地区吸引学生。该项目的一个组成部分是完成与Christian Haase和Andreas Paffenholz一起写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.