Matroids in tropical geometry

热带几何是用最小和加法运算分别代替实数的加法和乘法运算得到的几何。热带数学已经在许多不同的背景下进行了研究，但与代数几何的深刻联系只是在最近几十年才建立起来的。这一发展导致了许多不同领域的应用，如枚举代数几何、镜像对称、优化和计算生物学。拟阵是数学对象，在数学中抽象了许多不同的独立性概念。它们在热带几何中是必不可少的，因为它们在经典数学中扮演着与线性子空间相同的角色。热带几何与拟阵理论之间的联系既深又强，对这两个领域都非常有益。最近，PI和他的合作者在热带几何中引入了两个新概念：热带理想和热带CSM类，这两个概念对该领域非常有用。热带理想作为代数和组合对象，跟踪定义热带品种的方程。热带CSM类是携带关于任何光滑热带品种的组合和拓扑信息的热带对象。拟阵在这两个物体的构造中都是必不可少的。这个项目的目的是通过推动这两个新的热带概念：热带理想和热带CSM类的研究，继续发展拟阵理论和热带几何之间的紧密联系。研究这些有希望的对象将进一步推动热带几何的发展，为许多应用打开大门，如希尔伯特方案的热带研究，热带几何中可实现性问题的更深入探索，以及枚举代数几何问题的新方法。

Tropical geometry is the geometry obtained when the operations of addition and multiplication on the real numbers are replaced by the operations of minimum and addition, respectively. Tropical mathematics have been studied in many different contexts, but a deep connection to algebraic geometry has only been established in the last few decades. This development has led to numerous applications in many different areas, such as enumerative algebraic geometry, mirror symmetry, optimisation, and computational biology.Matroids are mathematical objects that abstract many different notions of independence throughout mathematics. They are essential in tropical geometry, as they play the same role as linear subspaces in classical mathematics. The connection between tropical geometry and matroid theory is deep and strong, and has been very beneficial to both fields. Recently, the PI and his collaborators have introduced two new notions in tropical geometry that promise to be very useful for the field: tropical ideals and tropical CSM classes. Tropical ideals serve as algebraic and combinatorial objects that keep track of the equations that define a tropical variety. Tropical CSM classes are tropical objects that carry combinatorial and topological information about any smooth tropical variety. Matroids are essential in the construction of both of these objects.The aim of this project is to continue to develop the strong connections between matroid theory and tropical geometry, by pushing the study of these two novel tropical notions: tropical ideals and tropical CSM classes. Investigating these promising objects will push the reach of tropical geometry further, opening the door to numerous applications such as a tropical study of Hilbert schemes, a deeper exploration of realisability questions in tropical geometry, and new approaches to enumerative algebro-geometric problems.