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Explore the local-global principle for representations of binary by quaternary quadratic forms in this advanced mathematics seminar from the Institute for Advanced Study. Delve into the fundamental question of when an integral quadratic form q in m variables can be represented by another form Q in n variables, specifically examining the challenging case where the codimension n-m equals 2. Learn about joint research with Wooyeon Kim and Pengyu Yang that establishes conditions under which local representations (over real numbers and modulo N) guarantee global representations exist. Discover how the proof employs cutting-edge techniques including recent measure rigidity results from Einsiedler and Lindenstrauss for higher-rank diagonalizable actions, combined with the determinant method. Understand the mathematical framework involving n×m-integer matrices T such that Q∘T=q, and examine the two Linnik-type splitting conditions that enable this local-global principle. This presentation addresses one of the most delicate problems in quadratic form theory, offering insights into advanced algebraic number theory and representation theory through rigorous mathematical analysis.
