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SUMMARY:A bridge built using differential equations in automorphic forms -
  Kim Klinger-Logan (Rutgers\, The State University of New Jersey)
DTSTART:20220823T150000Z
DTEND:20220823T160000Z
UID:TALK177221@talks.cam.ac.uk
DESCRIPTION:In this talk we will attempt to build our metaphircal bridge b
 y making a connection between zeros and special values of L-functions and 
 scattering amplitudes. The connection is best seen through solutions to di
 fferential equations of the form $$(\\Delta - \\lambda ) f = S$$ on $SL_2(
 \\mathbb{Z})\\backslash\\mathfrak{H}$ for $\\Delta=y^2(\\partial_x^2+\\par
 tial_y^2)$ and $S \\in H^{-infty}(SL_2(\\mathbb{Z})\\backslash\\mathfrak{H
 })\\cup M$ where $M$ is the space of moderate growth functions. Recently\,
  Bombieri and Garrett (following work of Hass\, Hejhal\, and Colin de Verd
 iere) laid out the possibly connection with eigenvalue solutions to equati
 ons of this form with zeros of L-functions.&nbsp\; On the other hand\, phy
 sicists such as Green\, Kwan\, Russo\, Vanhove found that eigenfunction so
 lutions to equations of this form give coefficients of the 4-graviton scat
 tering amlitude. We will elaborate on these connections and discuss some r
 ecent work on finiding solutions for such equations. This work is in colla
 boration with Ksenia Fedosova\, Stephen D. Miller\, and Danylo Radchenko.
LOCATION:Seminar Room 1\, Newton Institute
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