Equiangular lines, spherical two-distance sets, and spectral graph theory
Yufei Zhao (Massachusetts Institute of Technology)
Abstract: Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. The answer is expressed in terms of spectral radii of graphs.
Generalizing to spherical two-distance sets, we conjecturally relate the problem to a certain eigenvalue problem for signed graphs, and solve it in a number of cases.
A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.
Joint work with Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang (all MIT)
combinatorics
Audience: researchers in the topic
( paper )
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