One Step Closer to a 'Grand Unified Theory of Math': Geometric Langlands

Mathematicians recently proved a central component of the Langlands program, an ambitious effort to develop a “grand unified theory” of mathematics. The monumental proof of the geometric Langlands conjecture totaled more than 800 pages and marked the culmination of 30 years of work by nine mathematicians. Lead author, Dennis Gaitsgory, is the recipient of the 2025 Breakthrough Prize in Mathematics. Watch Quanta’s video explainer about the full Langlands program here: https://www.youtube.com/watch?v=_bJeKUosqoY Read the companion Quanta Magazine article: https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/ PAPER - D. Gaitsgory, S. Raskin D. Arinkin, D. Beraldo, J. Campbell, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin and N. Rozenblyum (2024) "Proof of the geometric Langlands conjecture" - https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ CORRECTION 02/24/25: In the end credits, "Qaunta Math Writer" is misspelled. It should have been "Quanta Math Writer." (Thanks to Milton Lee for catching that) -------- 00:00 Introduction 01:01 What is the Langlands Programs? 01:35 Fourier theory and analysis 02:23 Fourier transform, building blocks and labels 04:37 Sheaves as building blocks 05:49 Geometric Langlands and eigensheaves 06:30 Gaitsgory and his fundamental diagram 07:01 Poincaré sheaf and the solution to conjecture --------- - VISIT our website: https://www.quantamagazine.org - LIKE us on Facebook: / quantanews - FOLLOW us Twitter: / quantamagazine Quanta Magazine is an editorially independent publication supported by the Simons Foundation: https://www.simonsfoundation.org