MIT-Harvard-MSR Combinatorics Seminar: Equiangular lines and large multiplicity of fixed second eigenvalue
SEMINARS, HARVARD-MIT COMBINATORICS
Speaker:
Carl Schildkraut - MIT
Given a fixed angle alpha and growing dimension n, what is the maximum number of lines in n dimensions, all pairs of which meet at the same angle alpha? In 2019, Jiang, Tidor, Yao, Zhang, and Zhao determined this to be n + o(n) for "most" angles alpha, and determined the answer within O(1) for the others; the main technical portion was a sublinear upper bound on the multiplicity of the second-largest eigenvalue of bounded degree graphs. We present two constructions of bounded degree graphs with second-largest eigenvalue of large multiplicity. The first gives multiplicity about n^(1/2) using group-theoretic techniques. The second gives multiplicity only log log n, but allows precise control on the value of the second eigenvalue. This corresponds to families of n + log log n equiangular lines with the same fixed angle alpha. For some values of alpha, this answers a question of Jiang and Polyanskii, as well as Jiang, Tidor, Yao, Zhang, and Zhao, in the negative. Partially based on joint work with Milan Haiman, Shengtong Zhang, and Yufei Zhao.
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http://math.mit.edu/seminars/c
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