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SUMMARY:Pavel Kurasov (Stockholm University)
DTSTART:20200601T150000Z
DTEND:20200601T155000Z
DTSTAMP:20260416T065120Z
UID:IML-SMR/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/IML-SMR/6/">
 The additive structure of the spectra of quantum graphs and discrete measu
 res</a>\nby Pavel Kurasov (Stockholm University) as part of Scattering\, m
 icrolocal analysis and renormalisation\n\n\nAbstract\nIt is proven that th
 e spectrum of the Laplacian on a metric graph $\\Gamma$ contains arithmeti
 c sequences if and only if the graph has a loop -- an edge connected to on
 e vertex\nby both end points. Moreover the length of the longest possible 
 arithmetic subsequence\nis estimated using the corresponding discrete grap
 h $G$.\n \nOur main tool is diophantine analysis\, specifically ''Lang's $
 G_m $ Conjecture'' concerning the intersection of the division group of a 
 finitely generated subgroup of $(\\mathbb C^*)^N$ with a subvariety of  $(
 \\mathbb C^*)^N$.\n On our way we prove recent Colin de Verdière's Conjec
 ture concerning structure\n of polynomials associated with metric graphs.\
 n\n\n \n The trace formula connecting spectra of standard Laplacians on me
 tric graphs to the\n sets of periodic orbits allows us to construct  a lar
 ge family of exotic crystalline measures\, studied recently\n by Y. Meyer.
  Crystalline measures are discrete measures with Fourier transform being a
  discrete measure as well.\n Our analysis in the first part imply that con
 structed measures are\n not just combinations of Poisson summation formula
 e.\n\n\n\n\nThis is a joint work with Peter Sarnak.\n
LOCATION:https://researchseminars.org/talk/IML-SMR/6/
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