Tropical geometry: combinatorics, topology, and algorithms

热带几何是代数几何的多面体阴影，自然位于几何组合学和代数几何的交叉点上。它在计数几何、镜像对称、计算代数、最优化、代数统计和计算生物学等领域有着广泛的应用。热带方法的优势来自于这样一个事实，即热带天体本质上是组合的，而且对组合天体的计算可以比对代数几何对象的计算走得更远。对热带几何中组合结构的更好理解导致了列举几何和计算代数中的新算法和公式。此外，热带几何对象具有丰富的组合结构，这些结构也是离散几何和组合代数中自然产生的，如图、多面体的细分和三角剖分、纤维多面体、拟阵理论、系统发育树空间和单项理想的细胞分辨率，仅举几例。该项目旨在了解组合和拓扑结构，并在三个主要方向上进一步开发新的算法：热带簇、热带曲线和热带半代数集。代数簇的热带化是分段线性的，因此热带化用更容易的多面体计算取代了困难的代数计算。热带方法可以用来开发算法和软件来解决计算交换代数中的经典问题。尽管热带变种已经存在多年，但它们中很少有家族具有已知的同源性。该项目包括研究热带变种自然族的组合拓扑学，如完全交集、行列式变种、Grassmannians和结果的热带化。一个困难是缺乏检验猜想和发展直觉的例子。公社建议为上述品种家族建立一个样本库，并在可行的情况下完成分类。热带曲线是在图论和电力网络理论中自然产生的公制图。它们是简单的组合对象，但它们足以证明关于经典代数曲线的新定理。该提案旨在更好地理解热带曲线的射影嵌入和分支。热带几何在半代数集和最优化中的应用是一个很有前途但还未被探索的方向。这个项目的目的是开发热带凸性的新算法，并了解热带半代数集的组合学。这个项目的许多部分都适合学生参与和跨学科合作。将开发离散几何和计算代数的研究工具和软件。这些计算方法可用于其他领域，如半代数最优化、代数统计和计算生物学。

Tropical geometry is a polyhedral shadow of algebraic geometry and naturally lies in the intersection of geometric combinatorics and algebraic geometry. It has a wide range of applications in enumerative geometry, mirror symmetry, computational algebra, optimization, algebraic statistics, and computational biology. The strengths of tropical methods come from the fact that the tropical objects are intrinsically combinatorial, and computations can go farther on combinatorial objects than on algebro-geometric objects. Better understanding of combinatorial structures in tropical geometry has led to new algorithms and formulas in enumerative geometry and computational algebra. Moreover, tropical geometric objects have rich combinatorial structures that also arise naturally in discrete geometry and combinatorial algebra, such as graphs, subdivisions and triangulations of polytopes, fiber polytopes, matroid theory, space of phylogenetic trees, and cellular resolutions of monomial ideals, just to name a few. The project aims to understand combinatorial and topological structures and further the development of new algorithms in three main directions: tropical varieties, tropical curves, and tropical semialgebraic sets.Tropicalizations of algebraic varieties are piecewise linear, so tropicalization replaces difficult algebraic computations with easier polyhedral computations. Tropical methods can be used to develop algorithms and software for solving classical problems in computational commutative algebra. Although tropical varieties have been around for years, very few families of them have known homology. This project includes a study of the combinatorial topology of natural families of tropical varieties such as tropicalizations of complete intersections, determinantal varieties, Grassmannians, and resultants. A difficulty is the lack of examples to check conjectures and develop intuitions on. The PI proposes to build a library of examples for the aforementioned families of varieties and complete classifications when feasible. Tropical curves are metric graphs that naturally arise in graph theory and electrical network theory. They are simple combinatorial objects, yet they are powerful enough for proving new theorems about classical algebraic curves. The proposal aims at a better understanding of projective embeddings and ramifications of tropical curves. Application of tropical geometry to semialgebraic sets and optimization is a promising but under-explored direction. This project aims to develop new algorithms for tropical convexity and understand the combinatorics of tropical semialgebraic sets. Many parts of this project are suitable for involvement of students and for interdisciplinary collaborations. Research tools and software will be developed for discrete geometry and computational algebra. The computational methods may be useful in other areas such as semialgebraic optimization, algebraic statistics, and computational biology.