Fundamentals and applications of tropical schemes

热带几何是热带半环上的几何，其中乘法用加法代替，加法用最小代替。这会将代数簇(多项式方程的解)从多面体几何转化为称为热带簇的对象。目标是将关于原始品种的代数几何问题转化为关于热带品种的更容易处理的问题。自20年前成立以来，热带几何作为一个领域迅速扩展，在数学内外都有应用(包括经济学、机器学习、系统发生学、最优化和统计学)。然而，大多数应用程序并没有充分利用现代代数几何的优势。这是因为这些工具的热带版本仍在开发中。这个项目的主要目标是继续发展热带格式理论，并将其应用于代数几何中的问题。方案理论在20世纪60年代由Grothendieck和合作者的工作引入到代数几何中，并支撑着所有现代代数几何。通过将其整合到热带几何中，我们将极大地扩展该领域的覆盖范围和应用。潜在的应用既包括经典代数几何中的公开问题，也包括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.