Combinatorial and Tropical Degenerations of Classical Moduli Spaces

热带几何是数学中一个年轻且快速发展的领域，它植根于代数几何、复分析、交换代数和组合数学，在计算机科学、生物学和统计物理学以及其他数学领域都有应用。近十年来，这一学科取得了巨大的发展，既确立了这一领域本身的地位，又揭示了它与纯数学和应用数学众多分支的深刻联系。模空间是对代数几何数学对象进行分类的参数空间，例如变种，变种之间的映射和向量丛。本研究计画旨在提供一个新的组合观点，探讨古典模空间及其与非阿基米德解析几何的关系。在这个方向上的计算挑战是它的驱动力。主要目标是加深对这些物体的理解，并开发新技术，通过热带几何中的具体计算来分析这些抽象空间。在这个项目中调查的几个主题见证了学科的跨学科性质，适合与研究生合作研究。热带几何提供了一个框架，使用具体的组合工具解决代数几何问题：代数簇被取代的加权，平衡多面体复形。这些对象只保留了足够的关于原始变种的数据，以保持有意义，同时丢弃了它们的大部分复杂性。它们的组合学很大程度上依赖于我们的变种的嵌入。与此相反，伯科维奇空间或估值空间独立于所有这些选择。通过选择适当的坐标，这些估值的几何和拓扑结构反映在丰富的组合性质的多面体一侧，包括连通性，shellability等，因此，原来的空间的复杂性变成令人兴奋的组合问题。由这些漂亮的嵌入产生的热带化被认为是忠实的。该项目旨在通过品种的初始（Groebner）退化来开发当地忠实度证书。在这一点附近，忠诚的概念就变成了这种堕落的可取属性。毫不奇怪，验证这些性质需要对初始退化的组合学和几何学有深刻的理解。因此，需要找到有效的方法来构建或检测忠实性是热带几何的核心。该项目将为几个重要类别的例子开发这样的程序：格拉斯曼，曲线的模量，德尔佩佐曲面，法诺计划和集群品种。

Tropical geometry is a young and rapidly growing area in mathematics, rooted in algebraic geometry, complex analysis, commutative algebra, and combinatorics, with applications in computer science, biology, and statistical physics, in addition to other areas of mathematics. The recent decade has seen tremendous development in the subject that both established the field as an area in its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. Moduli spaces are spaces of parameters that classify algebro-geometric mathematical objects, such as varieties, maps between varieties, and vector bundles. This research project aims at providing a new combinatorial perspective on classical moduli spaces and their interplay with nonarchimedean analytic geometry. Computational challenges in this direction are its driving force. The primary goal is to deepen understanding of these objects and develop new techniques to analyze such abstract spaces through concrete computations in tropical geometry. Several topics investigated in this project witness the interdisciplinary nature of the subject and are suitable for research in collaboration with graduate students.Tropical geometry provides a framework for solving algebro-geometric problems using concrete combinatorial tools: algebraic varieties are replaced by weighted, balanced polyhedral complexes. These objects preserve just enough data about the original varieties to remain meaningful, while discarding much of their complexity. Their combinatorics depends strongly on the embeddings of our varieties. By contrast, the Berkovich space or the space of valuations is independent of all such choices. By choosing appropriate coordinates, the geometry and topology of these valuations are reflected in rich combinatorial properties on the polyhedral side, including connectedness, shellability, etc. Thus, the intricacies of the original space turn into exciting combinatorial problems. Tropicalizations resulting from these nice embeddings are said to be faithful. The project aims at developing certificates for local faithfulness by means of initial (Groebner) degenerations of the variety. The notion of faithfulness near that point then turns into desirable properties of this degeneration. Not surprisingly, validating these properties requires a deep understanding of the combinatorics and geometry of the initial degenerations. Thus, the need to find effective methods for constructing or detecting faithfulness is at the core of tropical geometry. The project will develop such program for several important classes of examples: Grassmannians, moduli of curves, del Pezzo surfaces, Fano schemes and cluster varieties.