Fields in Tensor-Triangular Geometry and Applications

张量三角几何是数学的一部分，它统一了代数几何、拓扑学、表示论和动机理论中其他不同分支的几个方面。在所有这些专门领域中，出现了非常复杂的结构，这些结构不能在颗粒级上完全理解，但其“整体形状”可以通过称为谱的几何不变量来理解。这一理论的一个特点是，在这些明显非常不同的环境中，相同的不变量是有意义的，并提供了深刻的洞察力。这种多功能性提供了统一的方法，并在数学领域的不同子专业之间架起了许多桥梁。这个项目的目标是分析张量三角形几何的“基本粒子”，即最小的这种结构，以及它们是如何组装成更大的结构的。该项目将为研究生提供研究培训机会。更详细地说，这个项目要解决的主要问题是张量三角几何中的点的概念，换句话说，张量三角场，每个大的张量三角范畴都构成了张量三角场。这种张量三角场在特殊情况下已经存在，就像代数几何中交换代数的普通场，或者稳定同伦理论中的Morava K-理论。该计划的一个主要组成部分是在其他环境中引入新的技术来解决构造这种张量三角场的问题，比如表象理论，绕过所谓的圆周点的缺点，或者更雄心勃勃地在动机理论中，目前还不知道场的作用的候选者。从代数几何中(残数)场在定义等级、计算重数等方面的重要性来看，对张量三角场的深入理解有望类似地在张量三角形几何中产生许多应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Tensor triangular geometry is a part of mathematics that unifies several aspects of otherwise distinct branches of algebraic geometry, topology, representation theory, and the theory of motives. In all those specialized areas, very complicated structures emerge that cannot be completely understood at a granular level but whose "overall shape" can be understood by means of a geometric invariant, called the spectrum. One feature of this theory is that the same invariant makes sense, and provides deep insight, in every one of these apparently very different settings. This versatility provides a unified methodology and builds many bridges between different sub-specialties of the mathematical landscape. The objective of this project is to analyze the "fundamental particles" of tensor-triangular geometry, that is, the minimal such structures and how they assemble to build much larger ones. This project will provide research training opportunities for graduate students. In more detail, the main problem to be addressed in this project is the concept of "point" in tensor-triangular geometry, in other words, the tensor-triangular fields, of which every large tensor-triangulated category is constituted. Such tensor-triangular fields already exist in special cases, like the ordinary fields of commutative algebra in algebraic geometry, or the Morava K-theories in stable homotopy theory. A main component of the program is to bring new techniques to bear on the problem of constructing such tensor-triangular fields in other settings, like representation theory, circumventing the shortcomings of so-called pi-points, or more ambitiously in motivic theory, where no candidates for the role of fields are known yet. Judging from the importance of (residue) fields in algebraic geometry for defining ranks, counting multiplicities, etc., a deeper understanding of tensor-triangular fields is expected to similarly generate many applications throughout tensor-triangular geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.