Polyhedral Fan Structures in Toric and Tropical Geometry

格多面体是介于组合数学和代数几何之间的对象。对它们的三角剖分、粗剖分、混合剖分和其他分解的研究是由这些领域之间的相互作用以及在数论、优化、统计、数学物理和算法生物学中的应用所激发的。要解决这一领域的基本开放问题，需要将联合收割机理论洞察力与算法独创性和计算机实验结合起来。在这个建议中涉及的具体议题包括以下内容：格多面体（特别是拟阵多面体）的幺模三角剖分，光滑性和正规性之间的关系的复曲面品种，组合和几何解释的h-多项式，和对称格多面体。与1489优先计划中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。

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.