Metrics and Completions of Triangulated Categories

The proposed project focuses on metrics and completions of triangulated categories. The two main objectives are to exploit recent breakthroughs in the theory of metrics on triangulated categories to answer open questions in the representation theory of algebras, and to push their development to the next level.

Distance is a fundamental notion which allows us to interpret the world around us. The idea of distance applies across myriad contexts, from distance between physical objects and navigating the space we live in to more conceptual notions of distance in sets of data that provides us with enormous predictive power. Abstracting these disparate incarnations leads to the mathematical notion of a metric space. In his transformative 1973 paper, Lawvere introduced the notion of a metric on a category, by assigning to each morphism a length, and with it a way of measuring how far objects are away from each other, thus linking these fundamental concepts to the categorical world. This provides a potent formalism for simultaneously treating both the distance between objects and how they interact with one another. The proposed project tackles pressing questions relating to the theory of metrics, specifically in triangulated categories.

Triangulated categories were introduced more than half a century ago by Verdier in his thesis. With roots and a continuing key role in the fields of algebraic geometry (derived categories of coherent sheaves, motives) and algebraic topology (stable homotopy theory), triangulated categories are crucial to modern day research in a plethora of contexts beyond these subjects, such as in representation theory (derived and stable module categories), symplectic geometry (Fukaya categories), algebraic analysis (Fourier-Sato transform and microlocalisation), and mathematical physics (D-branes and homological mirror symmetry). Given their ubiquity throughout mathematics, it might initially come as a surprise that interesting methods for constructing a new triangulated category from a given one are notoriously elusive. Most recently, Neeman has succeeded in using the technology of metrics and completions to provide a way to obtain a new triangulated category from a triangulated category with a "good" metric. Considering the scarcity, and relative restrictiveness, of previously known methods for constructing a new triangulated category from a given one, the potential of this result is immense. In particular, being able to produce new triangulated categories has the potential to impact several conjectures, particularly in noncommutative motives.

The goal of the proposed project is twofold: To further the theory of metrics and completions of triangulated categories and to exploit it to advance our understanding of the representation theory of finite dimensional algebras. In light of the new and interesting way of constructing triangulated categories via metrics and completions, and the dream of an explicit computation of these at our fingertips, we use combinatorial models on the one hand, and dg enhancements on the other to provide machinery to make this become a reality. At the same time, we exploit the theory of metrics and completions to allow for a fresh approach to study the poset of t-structures, with an emphasis on determining precisely under what circumstances this poset forms lattice.