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Explore the homotopical foundations of quantum spin systems in this 51-minute lecture by Markus Pflaum from the University of Colorado Boulder. Delivered on December 4, 2023, as part of the Fields Institute's Workshop on Operator Algebras and Applications: Non-Commutative Geometry, this talk delves into advanced concepts in quantum physics and mathematics. Gain insights into the intersection of operator algebras, non-commutative geometry, and quantum spin systems. Suitable for researchers and advanced students in mathematics and theoretical physics, this lecture contributes to the broader Thematic Program on Operator Algebras and Applications.