Jeff.LagariasAffiliation: University of Michigan Title Of Talk: Complex equiangular lines and orders of real quadratic fields Abstract: This talk surveys the problem of the existence of maximal sets of $d^2$ complex equiangular lines in $\CC^d$. Conjecturally such sets exist in all dimensions $d$. This is currently proved in a finite number of dimensions, now up to dimension $50$. This pure geometric problem, which is also important in quantum information theory, was studied by physicists, who found it to have a surprising connection with number theory. The known constructions involve algebraic numbers in abelian extensions of the real quadratic field $\QQ(\sqrt{(d+1)(d-3)})$, for $d \ge 4$. Gene Kopp (LSU) made a connection of the algebraic numbers in this problem with the Stark conjectures for real quadratic fields, published in 2019. This talk describes this important connection and conjectural phenomenology of these configurations, with specific ray class fields of orders, associated to over-orders of the quadratic order of discriminant $(d+1)(d-3)$.
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Last update made Tue Mar 10 21:28:04 CDT 2026.
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