Monumental proof settles geometric Langlands conjecture

In a significant advancement in mathematics, a team of nine mathematicians has successfully proved the geometric Langlands conjecture, a central aspect of the Langlands program, which has been under investigation for 30 years. This proof, comprising over 800 pages across five papers, was led by Dennis Gaitsgory and Sam Raskin. The Langlands program, initiated by Robert Langlands in the 1960s, aims to create a comprehensive framework connecting number theory, geometry, and function fields, likened to a "Rosetta stone" for mathematics. The recent proof is considered the most comprehensive result in the geometric aspect of this program, with experts expressing confidence in its correctness due to its internal consistencies.

The geometric Langlands conjecture relates to compact Riemann surfaces and their fundamental groups, proposing a correspondence between these mathematical structures and certain complex functions known as eigensheaves. The proof builds on decades of foundational work and innovative ideas, particularly those of Alexander Beilinson and Vladimir Drinfeld, who linked the conjecture to conformal field theory. Gaitsgory's long-term dedication to this problem culminated in this breakthrough, which is expected to have profound implications for various fields within mathematics. The proof not only resolves a longstanding question but also enriches the understanding of the relationships between different mathematical domains, reinforcing the Langlands program's status as a unifying theory in mathematics.