The higher algebra of spaces of quantum systems

（量子）物质的相态分类是一个古老的问题，它在物理学和数学中不断产生新的结果。近年来，来自拓扑学的强大工具——对抽象空间的数学研究，其中没有距离和角度的自然刚性几何——对该领域产生了深远的影响（包括导致2016年诺贝尔物理学奖的工作）。我的研究项目增加了量子物质系统空间的一个显著特征：这些空间表现出丰富的高等代数结构，并再现了不仅在物理学中而且在纯数学中都感兴趣的高等代数对象。尽管高等代数和量子系统之间的联系早在20世纪后期就已被认识到，但精确地表述这些联系所需的数学直到最近才发展起来。特别是，我最近的工作已经回答了许多关于“量子系统中存在的高等代数是什么？”的定义问题。这打开了无数新的研究途径，许多围绕着后续问题“量子系统中存在的高等代数是什么？”不是定义，而是例子的细节。其中一些例子现在已经足够具体，一个年轻的研究生都可以计算它们，并在这个过程中学习物理和数学之间的字典：例如，描述某些3+1d固态系统的低能量行为的例子。其他的例子现在刚刚出现在地平线上，仍然模糊，但值得研究，因为它们将阐明某些“特殊”数学对象的神秘特征，这些对象似乎只是由于巧合而存在，但迫切需要解释。就其本质而言，这个研究项目借鉴了数学和物理学的思想。许多本科生同时追求这两个领域，但在研究生院被迫选择一门学科。从事我的研究项目的学生将成为“双语”：既精通高等代数又精通理论物理。这种双语能力非常有价值，因为它有助于为所有科学带来背景和思想的多样性。虽然我从数学和物理中汲取灵感，但我首先是一个数学家，所以我的主要目标是利用物理学的思想来探索新的数学。当来自数学的想法导致新的物理应用时，这也是令人兴奋的，我的研究计划将通过提供构建和分析量子物质新相的工具来继续这样做。

The classification of phases of (quantum) matter is an ancient question that continues to lead to new results in both physics and mathematics. In recent years, powerful tools from topology - the mathematical study of abstract spaces in which there is no natural rigid geometry of distances and angles - has had a profound impact on the field (including work leading to the 2016 Nobel Prize in Physics). My research program adds to this picture a remarkable feature of spaces of quantum matter systems: these spaces exhibit rich higher algebraic structure, and reproduce higher algebraic objects that are of interest not just in physics but also in pure mathematics. Although connections between higher algebra and quantum systems have been recognized since the late 20th century, the mathematics needed to formulate these connections precisely has only recently been developed. In particular, my recent work has answered many of the definitional questions concerning "What is the higher algebra present in a quantum system?". This opens myriad new avenues of research, many around the follow-up question "What are the higher algebras present in quantum systems?": not the definition, but the details of examples. Some of these examples are now sufficiently concrete that a young graduate student can compute them, and in the process learn the dictionary between physics and mathematics: for instance, examples that describe the low-energy behaviour of certain solid-state systems in 3+1d. Other examples are now just coming into view on the horizon, still hazy but worth pursuing, because they would illuminate mysterious features of certain "exceptional" mathematical objects, which seem to exist only because of a coincidence but cry out for an explanation. By its very nature, this research program draws on ideas from both mathematics and physics. Many undergraduate students pursue both fields, but then are forced to choose a discipline in graduate school. Students pursuing projects from my research program will instead be "bilingual": fluent in both higher algebra and theoretical physics. Such bilingualism can be extremely valuable, because it helps bring a diversity of backgrounds and ideas to all Sciences. While I draw from both mathematics and physics, I am first a mathematician, and so my primary goal is to use ideas from physics to explore new mathematics. It is also thrilling when ideas from mathematics lead to new physical applications, and my research program will continue to do so by providing tools for constructing and analyzing novel phases of quantum matter.