What is Langlands correspondence?

In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory.

What is geometric representation theory?

Geometric representation theory is a relatively new field which has attracted much attention. The general idea is to use geometric methods to construct classically algebraic objects, such as representations of Lie groups and Lie algebras.

Is the ABC conjecture proved?

The abc conjecture was shown to be equivalent to the modified Szpiro’s conjecture. Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still regarded as unproven.

Is algebraic geometry useful for physics?

“Whereas with Einstein it was differential geometry that was the most relevant, for modern theoretical physics it is algebraic geometry. Algebraic geometry is the central aspect of geometry for the physicists now.”

How do you prove the Hodge conjecture?

The strongest evidence in favor of the Hodge conjecture is the algebraicity result of Cattani, Deligne & Kaplan (1995). Suppose that we vary the complex structure of X over a simply connected base. Then the topological cohomology of X does not change, but the Hodge decomposition does change.

What is the global Langlands correspondence?

In 2018, Vincent Lafforgue established the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields. Philip Kutzko ( 1980) proved the local Langlands conjectures for the general linear group GL (2, K) over local fields.

Is there a Langlands correspondence for reductive groups over non-Archimedian fields?

We begin by giving a brief overview of the local Langlands correspondence for reductive groups over local non-archimedian fields, such asFq..t//, the field of Laurent power series over a finite fieldq. We wish to understand an analogue of this correspondence whenFqis replaced by the fieldCof complex numbers.

What is the Langlands program in math?

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands ( 1967, 1970 ), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory…

What is Langlands’s group theory?

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.