Lattice Polytopes with a View Toward Algebraic Geometry

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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.