BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Pavel Etingof (MIT Mathematics) DTSTART:20211111T130000Z DTEND:20211111T150000Z DTSTAMP:20260423T005805Z UID:AGNTISTA/43 DESCRIPTION:Title: Hecke operators over local fields and an analytic approach to the geomet ric Langlands correspondence\nby Pavel Etingof (MIT Mathematics) as pa rt of Algebraic Geometry and Number Theory seminar - ISTA\n\n\nAbstract\nI will review an analytic approach to the geometric Langlands correspondenc e\, following my work with E. Frenkel and D. Kazhdan\,\narXiv:1908.09677\, arXiv:2103.01509\, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing wor ks of a number of mathematicians and mathematical physicists\, Kontsevich\ , Langlands\, Teschner\, and Gaiotto-Witten. One of the goals of this appr oach is to understand single-valued real analytic eigenfunctions of the qu antum Hitchin integrable system. The main method of studying these functio ns is realizing them as the eigenbasis for certain compact normal commutin g integral operators the Hilbert space of L2 half-densities on the (comple x points of) the moduli space Bun_G of principal G-bundles on a smooth pro jective curve X\, possibly with parabolic points. These operators actually make sense over any local field\, and over non-archimedian fields are a r eplacement for the quantum Hitchin system. We conjecture them to be compac t and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). \nI will first discuss the simplest non-trivial example of Hecke operators over local fields\, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles B un_G^{ss} is P^1\, and the situation is relatively well understood\; over C it is the theory of single-valued eigenfunctions of the Lame operator wi th coupling parameter -1/2 (previously studied by Beukers and later in a m ore functional-analytic sense in our work with Frenkel and Kazhdan). I wil l consider the corresponding spectral theory and then explain its generali zation to N>4 points and conjecturally to higher genus curves.\n LOCATION:https://researchseminars.org/talk/AGNTISTA/43/ END:VEVENT END:VCALENDAR