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SUMMARY:From Spec Z to Zeta spectral triples [Oberseminar Global Analysis 
 and Operator Algebras]
DTSTART:20260702T070000Z
DTEND:20260702T090000Z
DTSTAMP:20260706T084400Z
UID:indico-event-1478@math-events.uni-bonn.de
DESCRIPTION:Speakers: Alain Connes (IHES Paris)\n\nIn this extended two-ho
 ur seminar (joint work with Katia Consani for the first two parts and also
  with Henri Moscovici for the last part)\, we will explore the global geom
 etric structure of the absolute curve--the one-point compactification $\\o
 verline{\\text{Spec } \\mathbb{Z}}$--and its profound implications for the
  Riemann Hypothesis.    In the first part\, we expound recent developmen
 ts demonstrating that the adele class space arises naturally as the Picard
  monoid of this compactified curve. We subsequently construct a tentative 
 global geometric structure $C=({\\text{Spec } \\mathbb{Z}})_{\\mathbb{F}_1
 }$ for ${\\text{Spec } \\mathbb{Z}}$ over $\\mathbb{F}_1$.Taking the geome
 tric points of $C$ over a perfectoid field of characteristic $p$ provides 
 a foundational compatibility check: it successfully recovers  the moduli 
 space of untilts\, yielding a precise geometric realization of the "Scholz
 e heuristic" as formulated by J. Lurie\, while illuminating the boundaries
  of local vs. global expectations at other primes.    The geometry of th
 e adele class space transitions us\, in the second half of the talk\, to t
 he spectral properties of the Riemann Zeta function. Starting from Riemann
 's explicit formulas\, we show how the action of the idele class group on 
 the adele class space gives the zeros of $L$-functions as an absorption sp
 ectrum\, with the explicit formulas emerging as a Lefschetz formula carrie
 d by the image of the curve through the Abel-Jacobi map. We then detail th
 e translation from an absorption to an emission spectrum via the semilocal
  adele class space.Finally\, we present a remarkable computational and ana
 lytic convergence: using a "zeta spectral triple" constructed from only th
 e first five finite primes ($2\, 3\, 5\, 7\, 11$)\, one can access the fir
 st fifty nontrivial zeros of the Riemann Zeta function with incredible pre
 cision. Crucially\, the spectral triple truncated to finitely many primes 
 is self-adjoint\, ensuring these approximate zeros lie strictly on the cri
 tical line. This analytic architecture ultimately reduces the proof of the
  Riemann Hypothesis to demonstrating the convergence of the approximation 
 of the lowest eigenvector of the Weil quadratic form by prolate spheroidal
  wave functions.\n\nhttps://math-events.uni-bonn.de/event/1478/
LOCATION:Lipschitzsaal (Mathezentrum)
URL:https://math-events.uni-bonn.de/event/1478/
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