Deligne conjecture

In mathematics, there are a number of so-called Deligne conjectures, provided by Pierre Deligne. These are independent conjectures in various fields of mathematics.

• The Deligne conjecture in deformation theory is about the operadic structure on Hochschild cohomology. It was proved by Kontsevich–Soibelman, McClure–Smith and others. It is of importance in relation with string theory.

• The Deligne conjecture on special values of L-functions is a formulation of the hope for algebraicity of L(n) where L is an L-function and n is an integer in some set depending on L.

• There is a Deligne conjecture on 1-motives arising in the theory of motives in algebraic geometry.

• There is a Gross–Deligne conjecture in the theory of complex multiplication.

• There is a Deligne conjecture on monodromy, also known as the weight monodromy conjecture, or purity conjecture for the monodromy filtration.

• There is Deligne conjecture in the representation theory of the exceptional Lie groups.

• There is a Deligne–Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.