The higher algebra of spaces of quantum systems

(量子)物质的相分类是一个古老的问题，它继续在物理学和数学中带来新的结果。近年来，拓扑学的强大工具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)。拓扑学是对抽象空间的数学研究，其中没有关于距离和角度的自然刚性几何。我的研究项目为这幅画增添了量子物质系统空间的一个显著特征：这些空间展示了丰富的高级代数结构，并再现了不仅在物理上感兴趣，而且在纯数学中也感兴趣的更高代数对象。虽然高等代数和量子系统之间的联系早在20世纪末就已经被认识到了，但精确地表述这些联系所需的数学直到最近才得到发展。特别是，我最近的工作回答了许多定义问题，这些问题涉及到“量子系统中存在的更高代数是什么？”这开辟了无数新的研究途径，其中许多都是围绕着后续的问题“量子系统中存在哪些更高的代数？”：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，一个年轻的研究生就可以计算它们，并在这个过程中学习物理和数学之间的词典：例如，描述某些固态系统在3+1D中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，仍然是模糊的，但值得追寻，因为它们将揭示某些“特殊的”数学对象的神秘特征，这些对象似乎只是因为巧合而存在，但迫切需要一个解释。就其本质而言，这项研究计划借鉴了数学和物理学的思想。许多本科生同时攻读这两个领域，但随后被迫在研究生院选择一门学科。在我的研究项目中学习项目的学生将会是“双语”的：在高等代数和理论物理方面都很流利。这种双语能力非常有价值，因为它有助于将不同的背景和想法带给所有的科学。虽然我同时从数学和物理中吸取教训，但我首先是一名数学家，因此我的首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，也是令人兴奋的，我的研究计划将继续这样做，为构建和分析量子物质的新相提供工具。

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.