Tropical geometry: combinatorics, topology, and algorithms

热带几何是代数几何的多面体阴影，自然地处于几何组合学和代数几何的交叉点。 它在枚举几何、镜像对称、计算代数、最优化、代数统计和计算生物学中有着广泛的应用。热带方法的优势在于热带对象本质上是组合的，并且计算在组合对象上比在代数几何对象上走得更远。 对热带几何中组合结构的更好理解导致了枚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，这些结构也自然地出现在离散几何和组合代数中，例如图、多面体的细分和三角剖分、纤维多面体、拟阵理论、系统发育树的空间和单项式理想的细胞分辨率，仅举几例。 该项目旨在了解组合和拓扑结构，并进一步发展新的算法在三个主要方向：热带品种，热带曲线和热带半代数集。热带代数簇是分段线性的，所以热带化取代困难的代数计算与更容易的多面体计算。 热带方法可用于开发算法和软件，以解决计算交换代数中的经典问题。 虽然热带品种已经存在多年，但很少有家族具有同源性。 该项目包括热带品种的自然家族的组合拓扑学的研究，例如完全交叉的热带化，行列式品种，格拉斯曼和结果。 一个困难是缺乏例子来检查插图和发展直觉。PI建议建立一个图书馆的例子，为上述家庭的品种和完整的分类时，可行的。 热带曲线是在图论和电网络理论中自然出现的度量图。 它们是简单的组合对象，但它们足够强大，可以证明关于经典代数曲线的新定理。 该建议旨在更好地理解热带曲线的投影嵌入和分支。 应用热带几何的半代数集和优化是一个有前途的，但未开发的方向。 本计画旨在发展热带凸性的新演算法，并了解热带半代数集合的组合学。该项目的许多部分适合学生参与和跨学科合作。 将为离散几何和计算代数开发研究工具和软件。 计算方法可能是有用的其他领域，如半代数优化，代数统计，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.