Fundamentals and applications of tropical schemes

热带几何是热带半环上的几何，其中乘法被加法取代，加法被最小值取代。这将代数簇（多项式方程的解）转化为多面体几何中的对象，称为热带簇。我们的目标是将原来的代数几何问题转化为更容易处理的热带几何问题。自二十年前成立以来，热带几何作为一个领域迅速扩大，在数学内外都有应用（包括经济学，机器学习，系统发育学，优化和统计学）。然而，大多数应用并没有充分利用现代代数几何的全部力量。这是因为这些工具的热带版本仍在开发中。该项目的主要目标是继续开发热带方案理论，并将其应用于代数几何问题。格形理论在1960年代由格罗滕迪克及其合作者的工作引入代数几何，并支撑了所有现代代数几何。通过将其整合到热带几何中，我们将大大扩展该领域的范围和应用。潜在的应用包括经典代数几何中的开放问题，以及代数几何中基本模空间的热带版本。

Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. This turns an algebraic variety (the solution to polynomial equations) into an object from polyhedral geometry called a tropical variety. The goal is to translate algebraic geometry questions about the original variety into more tractable questions about the tropical variety.Since its inception two decades ago, tropical geometry has rapidly expanded as a field, with applications both inside and outside mathematics (including to Economics, Machine Learning, Phylogenetics, Optimization and Statistics). Most applications, however, do not exploit the full strength of modern algebraic geometry. This is because tropical versions of these tools are still being developed.The main goal of this project is to continue the development of tropical scheme theory, and apply this to problems in algebraic geometry. Scheme theory was introduced to algebraic geometry in the 1960s by the work of Grothendieck and collaborators, and underpins all of modern algebraic geometry. By integrating this into tropical geometry we will greatly expand the reach and application of the field. Potential applications include both open problems in classical algebraic geometry, and also tropical versions of fundamental moduli spaces in algebraic geometry.