The higher algebra of spaces of quantum systems

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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.