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Explore the mathematical foundations of topological phases of matter through this comprehensive graduate summer school lecture series covering 21+ hours of advanced content. Delve into the intersection of topology and physics as it relates to the second quantum revolution, examining how topological insulators and superconductors have transformed our understanding beyond the quantum Hall effect. Master two well-established mathematical frameworks: topological quantum field theories (TQFTs) and modular tensor categories (MTCs) for two-dimensional systems, plus symmetry protected topological phases (SPTs) classified by generalized cohomology theories including group cohomology for bosonic systems and K-theory for fermions. Investigate emerging paradigms that challenge existing mathematical understanding, including fracton theories and quantum cellular automata (QCA), which exhibit both additive and multiplicative characteristics depending on whether they involve free or interacting systems. Learn from leading experts in mathematics and theoretical physics as they present both mathematical rigor and physical intuition across topics spanning fusion categories, modular fusion category classification, quantum cellular automata, Floquet topological phases, lattice topological phases, error correcting codes, and exotic quantum field theories. Gain insights into how topology influences phase transitions, quantum entanglement, and quantum computing while building foundational knowledge for future research in this rapidly developing field where mathematical theory is still catching up to physical discoveries.
