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The Tenfold Way

The importance of the tenfold way in physics was only recognized in this century. Simply put, it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when the topologist C. T. C. Wall classified the associative real super division algebras and found ten of them. The three 'purely even' examples were already familiar: the real numbers, complex numbers and quaternions. The rest become important when we classify representations of groups on super Hilbert spaces. In this talk I explain how the tenfold way arises naturally in quantum physics, and its connection to Clifford algebras. I gave this talk on February 6, 2023 in Nichol Furey's seminar Algebra, Particles and Quantum Theory. For my slides and a web page on the tenfold way, go to: http://math.ucr.edu/home/baez/tenfold/