Fields in Tensor-Triangular Geometry and Applications

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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.