Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

Earlier in June while staying in Leipzig, I realized that the Gaiotto correspondence, established by my collaborators and  me solving his conjecture [DFKMMN, JDG (2021)], happens to be a part of the Geometric Langlands Correspondence for the "oper" case. The surprise here is that while our proof requires analysis of solving nonlinear PDEs, GLC is purely algebraic  and number theoretic, written in the language of Artin stacks. The NLPDE appears as a translation of "stability conditions"  in algebraic geometry of moduli theory. The Artin stacks are a  far more general object than Deligne-Mumford stacks, the latter  is just orbifolds.    In this talk I will start with the historical background of  isomonodromic deformation theory of ODEs (McCoy et al), and then  define the concept of opers. Gaiotto map is introduced from his  4D N=2 SUSY YM, which translates into a nonlinear PDE. Then  I'll illustrate how it fits into the GLC in its original framework  of Beilinson-Drinfeld (1990s), and also in the final solution of  Gaitsgory and his team (2025).