Foundations and Applications of Tropical Geometry

热带几何是热带半环上的几何，热带半环是通常的实数，用最小代替加法，用加法代替乘法。虽然热带半环在最优化和计算机科学中的应用已经有几十年了，但它与代数几何的联系直到本世纪初才首次出现。这门学科在过去的十年里迅速扩展。经典的代数几何研究多项式方程的解的集合的几何，称为簇。60年代，随着Grothendieck的方案的引入，代数几何的基础发生了巨大的变化。然而，在热带几何学的第一个十年里，只考虑了热带版本的变种。在过去的几年里，这种情况发生了变化，从Giansiracusas的工作开始，以及PI和Rincon的工作，他们共同将图式理论引入热带几何。这个项目的主要目的是进一步发展热带方案的理论，并将这一技术应用于代数几何中的问题。第一个目标是发展更多的基本交换代数和新的热带方案理论的代数几何。然后，这将被用来构造代数几何中重要的模空间的热带版本，从希尔伯特方案开始，并用它来解决关于希尔伯特方案的基本开放问题。它还将被用来解决热带几何中的可实现性问题，这些问题适用于双曲面几何。

Tropical geometry is geometry over the tropical semiring, which is the usual real numbers with addition replaced by minimum, and multiplication replaced by addition. While the tropical semiring has had applications in optimization and computer science for several decades, the connection to algebraic geometry was first made only at the start of this century. The subject has expanded rapidly in the past decade.Classically algebraic geometry studies the geometry of the sets of solutions to polynomial equations, called varieties. The fundamentals of algebraic geometry changed dramatically in the 60s with Grothendieck's introduction of schemes. In the first decade of tropical geometry, however, only tropical versions of varieties were considered. This has changed in the past few years, beginning with the work of the Giansiracusas, and of the PI with Rincon, which together introduced a scheme theory into tropical geometry. This allows much more of the power of modern algebraic geometry to be used in tropical geometry.The primary aim of this project is to further develop the theory of tropical schemes, and apply this technology to problems in algebraic geometry.The first goal is to develop more of the basic commutative algebra and algebraic geometry of the new theory of tropical schemes. This will then be used to construct tropical versions of important moduli spaces in algebraic geometry, starting with the Hilbert scheme, and use this to address fundamental open questions about the Hilbert scheme. It will also be used to address realizability questions in tropical geometry, which have applications to birational geometry.