BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Pavel Etingof (MIT) DTSTART:20210913T190000Z DTEND:20210913T200000Z DTSTAMP:20260423T021444Z UID:WHCGP/45 DESCRIPTION:Title: H ecke operators over local fields and an analytic approach to the geometric Langlands correspondence\nby Pavel Etingof (MIT) as part of Western H emisphere colloquium on geometry and physics\n\n\nAbstract\nI will review an analytic approach to the geometric Langlands correspondence\, following my work with E. Frenkel and D. Kazhdan\, arXiv:1908.09677\, arXiv:2103.01 509\, arXiv:2106.05243. This approach was developed by us in the last coup le of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists\, Kontsevich\, Langlands\, Teschner\, and Gaiotto-Witten. One of the goals of this approach is to und erstand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizin g them as the eigenbasis for certain compact normal commuting integral ope rators the Hilbert space of L2 half-densities on the (complex points of) t he moduli space Bun_G of principal G-bundles on a smooth projective curve X\, possibly with parabolic points. These operators actually make sense ov er any local field\, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove th is conjecture in the genus zero case (with parabolic points) for G=PGL(2). I 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 poi nts. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1 \, and the situation is relatively well understood\; over C it is the theo ry of single-valued eigenfunctions of the Lame operator with coupling para meter -1/2 (previously studied by Beukers and later in a more functional-a nalytic sense in our work with Frenkel and Kazhdan). I will consider the c orresponding spectral theory and then explain its generalization to N>4 po ints and conjecturally to higher genus curves.\n LOCATION:https://researchseminars.org/talk/WHCGP/45/ END:VEVENT END:VCALENDAR