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This seminar talk explores the fascinating connections between matrix spaces and graphs, building on Avi Wigderson's previous work on linear spaces of matrices. Discover how certain matrix spaces naturally emerge from graphs, tracing back to Tutte and Lovász's foundational work on perfect matchings that established links between perfect matchings and full-rank matrices. Learn about additional correspondences between graph structures and matrix space structures, including independent sets and totally-isotropic spaces, graph connectivity and orthogonal decompositions, and various isomorphism notions. Examine how graph-theoretic questions and techniques can be extended to matrix spaces in numerous cases, such as alternating paths, independence polynomials, Turán and Ramsey type questions, expanders, and threshold phenomena. The presentation also covers how these techniques and results find applications in other fields including invariant theory, group theory, quantum information, and geometry. This talk by Youming Qiao from the Institute for Advanced Study is based on joint works with Avi Wigderson, Yuval Wigderson, Gábor Ivanyos, Yinan Li, Chuanqi Zhang, and Markus Bläser.
