Geometric Langlands Conjecture: An Introduction Coleman Dobson CMC3 South October 13, 2018 Grothendieck taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra by Yuri Manin
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Coleman Dobson CMC3 South October 13, 2018Coleman Dobson CMC3 South October 13, 2018 Grothendieck taught that to do geometry one does not need a space, it is enough to have a category
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Geometric Langlands Conjecture: An Introduction
Coleman Dobson CMC3 South October 13, 2018
Grothendieck taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra by Yuri Manin
IdeaWe will explore the magical contributions of Peter Scholze (University of Bonn) and Edward Frenkle (UCB) to the Geometric Langlands Conjecture, by way of p-adic number theory, perfectoid spaces, curves over finite fields, Riemann surfaces (curves over the field of complex numbers) and Galois representations.
Frenkle:“ It is tempting to think of it as a “grand unified theory” of mathematics, since it ties together so many different disciplines”
Andrew Wiles:“There exists a deep analogy between number theory and the
geometry of complex algebraic curves, with the theory of
algebraic curves over finite fields appearing as the go-between.
The Original TitansGeometric Langlands theory originated from the ideas of four people:
A. Beilinson, P. Deligne, V. Drinfeld and G. Laumon
Langlands Correspondence
Classical Langlands:
Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles
Geometric Langlands:
Equate representation of Galois Groups and p-adic modular forms. Method: Replace number fields with function fields and apply derived algebraic geometry
Categorical Langlands:
The categorical geometric Langlands conjecture is a categorical version of the geometric Langlands conjecture. It is formulated as an equivalence of stable (infinity,1)-categories of D-modules on the derived stack of G-bundles on a curve X, and ind-coherent sheaves on the derived stack of LG-equivariant local systems on X, where X is a smooth complete curve over a field of characteristic zero, and G is a reductive group and LG is its Langlands dual.
Quantum Q-Langlands: quantum K-theory of Nakajima quiver varieties