25/11/09 16:02:39.89 QrKJGO9s.net
つづき
In the case of the geometric Langlands conjecture, the proposed one-to-one correspondence is between two very different sets of objects, analogous to these spectrums and waves: on the spectrum side are abstract algebraic objects called representations of the fundamental group, which capture information about the kinds of loop that can wrap around certain complex surfaces; on the “wave” side are sheaves, which, loosely speaking, are rules assigning vector spaces to points on a surface. Gaitsgory has dedicated much of the last 30 years to the geometric Langlands conjecture. In 2013 he wrote an outline of the steps required for a proof, and after more than a decade of intensive research in 2024 he and his colleagues published the full proof, comprising over 800 pages spread over 5 papers. This is a monumental advance, expected to have deep implications in other areas of mathematics too, including number theory, algebraic geometry and mathematical physics.
2025 New Horizons in Mathematics Prize
Sam Raskin, Yale University
For contributions to the geometric Langlands program, including the theory of the Whittaker model and the proof of the geometric Langlands conjecture in characteristic 0.
URLﾘﾝｸ(people.mpim-bonn.mpg.de)
Proof of the geometric Langlands conjecture
This page contains five papers, the combined content of which constitutes the proof of the (categorical, unramified) geometric Langlands conjecture.
This is a collaborative project of D. Arinkin, D. Beraldo, J. Campbell, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin and N. Rozenblyum.
Papers:
GLC I: Construction of the functor
GLC II: Kac-Moody localization and the FLE
GLC III: Compatibility with parabolic induction
GLC IV: Ambidexterity
GLC V: The multiplicity one theorem
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