Polyhedral Fan Structures in Toric and Tropical Geometry

环面和热带几何中的多面体扇形结构

• 批准号: 171126520
• 负责人: Professor Dr. Michael Joswig
• 金额: --
• 依托单位: Max-Planck-Institut für Mathematik in den Naturwissenschaften (MIS)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171126520?language=en
• 关键词: Polyhedral; Fan; Structures; Toric; Tropical

Lattice polytopes are objects at a junction between combinatorics and algebraic geometry. The study of their triangulations, coarsest subdivisions, mixed subdivisions, and other decompositions is motivated by the mutual interaction between these fields as well as by applications in number theory, optimization, statistics, mathematical physics, and algorithmic biology. Attacking fundamental open problems in this area requires to combine theoretical insight with algorithmic ingenuity and computer experiments. Specific topics addressed in this proposal include the following: unimodular triangulations of lattice polytopes (in particular, matroid polytopes), the relationship between smoothness and normality of a toric variety, combinatorial and geometric interpretations of h∗-polynomials, and symmetric lattice polytopes. Interaction with other researchers in the Priority Program 1489 provides a unique opportunity which makes it likely to make substantial progress.

点阵多面体是组合学和代数几何的结合点。这些领域之间的相互作用以及在数论、优化、统计学、数学物理和算法生物学中的应用，激发了对它们的三角剖分、最粗细分、混合细分和其他分解的研究。解决这一领域的基本开放问题需要将理论洞察力与算法独创性和计算机实验结合起来。本提案中涉及的具体主题包括：晶格多面体的单模三角化（特别是拟阵多面体），环型的光滑性和正态性之间的关系，h *多项式的组合和几何解释，以及对称晶格多面体。与1489优先计划的其他研究人员的互动提供了一个独特的机会，使其有可能取得实质性进展。