Metrics and Completions of Triangulated Categories

拟议的项目侧重于三角分类的度量和完成。两个主要目标是利用三角分类度量理论的最新突破来回答代数表示理论中的开放问题，并将其发展推向下一个水平。距离是一个基本概念，它使我们能够理解我们周围的世界。距离的概念适用于各种情况，从物理对象之间的距离到我们所处空间的导航，再到为我们提供巨大预测能力的数据集中的距离概念性概念。将这些不同的化身抽象出来，就产生了度量空间的数学概念。在他1973年的一篇具有变革意义的论文中，Lawvere引入了范畴度规的概念，通过给每个态射分配一个长度，以及一种测量物体彼此之间距离的方法，从而将这些基本概念与范畴世界联系起来。这为同时处理对象之间的距离以及它们如何相互作用提供了一个强有力的形式。拟议的项目解决了与度量理论有关的紧迫问题，特别是在三角分类中。三角分类是半个多世纪前由Verdier在他的论文中提出的。在代数几何（相干束的派生范畴，动机）和代数拓扑（稳定同伦理论）领域中具有根和持续的关键作用，三角化范畴对现代研究在这些主题之外的大量背景下至关重要，例如在表示理论（派生和稳定的模范畴），辛几何（深谷范畴），代数分析（傅立叶-佐托变换和微定位），以及数学物理（d膜和同调镜像对称）。考虑到它们在数学中的普遍存在，最初可能会感到惊讶的是，从一个给定的分类中构造一个新的三角分类的有趣方法是出了名的难以捉摸。最近，Neeman成功地使用了指标和完井技术，提供了一种从具有“良好”指标的三角分类中获得新三角分类的方法。考虑到以前已知的从给定类别构建新三角分类的方法的稀缺性和相对限制性，这一结果的潜力是巨大的。特别是，能够产生新的三角分类有可能影响一些猜想，特别是在非交换动机方面。提出的项目的目标是双重的：进一步测度和三角化范畴的补全理论，并利用它来推进我们对有限维代数的表示理论的理解。鉴于通过度量和补全构建三角分类的新而有趣的方法，以及在我们的指尖上对这些进行显式计算的梦想，我们一方面使用组合模型，另一方面使用dg增强来提供使其成为现实的机器。同时，我们利用度量和补全理论，以允许一种新的方法来研究t结构的偏序集，重点是精确地确定在什么情况下这个偏序集形成格。

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.