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，植根于代数几何、复分析、交换代数和组合学，除了其他数学领域外，还应用于计算机科学、生物学和统计物理学。近十年来，该学科取得了巨大的发展，既使该领域成为一个独立的领域，又揭示了其与纯数学和应用数学的众多分支的深厚联系。模空间是对代数几何数学对象进行分类的参数空间，例如簇、簇之间的映射和向量丛。该研究项目旨在为经典模空间及其与非阿基米德解析几何的相互作用提供新的组合视角。这个方向的计算挑战是其驱动力。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的具体计算来分析这些抽象空间。该项目研究的几个主题见证了该学科的跨学科性质，适合与研究生合作研究。热带几何提供了一个使用具体组合工具解决代数几何问题的框架：代数簇被加权、平衡的多面体复形所取代。这些物体保留了足够的有关原始品种的数据以保持有意义，同时丢弃了它们的大部分复杂性。它们的组合在很大程度上取决于我们品种的嵌入。相比之下，伯科维奇空间或估值空间独立于所有此类选择。通过选择适当的坐标，这些估值的几何和拓扑反映在多面体一侧丰富的组合属性中，包括连通性、可壳性等。因此，原始空间的复杂性变成了令人兴奋的组合问题。据说这些漂亮的嵌入所产生的热带化是忠实的。该项目旨在通过品种的初始（Groebner）退化来开发当地忠诚度证书。接近这一点时，忠诚的概念就会变成这种退化的理想特性。毫不奇怪，验证这些属性需要深入了解初始退化的组合学和几何学。因此，寻找有效的方法来构建或检测忠实度是热带几何学的核心。该项目将为几个重要的例子类别开发这样的程序：Grassmannians、曲线模、del Pezzo 曲面、Fano 方案和簇簇。

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.