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Metrics and Completions of Triangulated Categories

三角类别的指标和完成情况

基本信息

批准号：
EP/V038672/1
负责人：
Sira Gratz
金额：
$ 12.38万
依托单位：
Aarhus University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV038672%2F1
关键词：
Metrics Completions Triangulated Categories

项目摘要

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.

拟议项目的重点是衡量标准和完成三角分类。两个主要目标是利用最近的突破理论的度量的三角范畴回答开放的问题表示理论的代数，并推动其发展到一个新的水平。距离是一个基本概念，它使我们能够解释我们周围的世界。距离的概念适用于无数的情境，从物理对象之间的距离和导航我们生活的空间到数据集中的距离概念，为我们提供了巨大的预测能力。抽象这些不同的化身导致了度量空间的数学概念。在他1973年的论文中，Lawvere引入了范畴度量的概念，通过给每个态射分配一个长度，以及一种测量对象彼此距离的方法，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主义。拟议的项目解决紧迫的问题有关的理论度量，特别是在triangulated categories.Triangulated类别介绍了超过半个世纪前Verdier在他的论文。在代数几何学领域中，（导出范畴的凝聚层，动机）和代数拓扑（稳定同伦理论），三角范畴是至关重要的现代研究在大量的背景下超出这些主题，如在表示论（导出和稳定模范畴），辛几何（福谷范畴），代数分析（傅里叶-佐藤变换和微局部化），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性，最初可能会感到惊讶的是，从一个给定的三角范畴构造一个新的三角范畴的有趣方法是出了名的难以捉摸。最近，尼曼已经成功地使用技术的指标和完善，以提供一种方法来获得一个新的三角形类别从一个“好”的度量。考虑到以前已知的方法的稀缺性和相对的限制性，从一个给定的构建一个新的三角范畴，这个结果的潜力是巨大的。特别是，能够产生新的三角范畴有可能影响几个代数，特别是在noncommutative motives.The拟议项目的目标是双重的：进一步的度量和三角范畴的完成理论，并利用它来推进我们的理解有限维代数的表示理论。鉴于通过度量和完备化构建三角分类的新的有趣的方法，以及我们触手可及的显式计算的梦想，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制。与此同时，我们利用度量和完备化理论，允许一个新的方法来研究偏序集的t-结构，重点是精确地确定在什么情况下，这个偏序集形成格。