Deligne conjecture
Encyclopedia
In mathematics
, there are a number of so-called Deligne conjectures, provided by Pierre Deligne
. These are independent conjecture
s in various fields of mathematics.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, there are a number of so-called Deligne conjectures, provided by Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
. These are independent conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s in various fields of mathematics.
- The Deligne conjecture in deformation theoryDeformation theoryIn mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
is about the operadic structure on Hochschild cohomology. It was proved by Kontsevich–SoibelmanYan SoibelmanYakov Soibelman born 15 April 1956 is a Russian mathematician, professor of Kansas State University , member of the , founder of Manhattan Mathematical Olympiad.-Scientific work:...
, McClure–Smith and others. It is of importance in relation with string theoryString theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
- The Deligne conjecture on special values of L-functions is a formulation of the hope for algebraicityAlgebraic numberIn mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
of L(n) where L is an L-functionL-functionThe theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
and n is an integer in some set depending on L.
- There is a Deligne conjecture on 1-motives arising in the theory of motivesMotive (algebraic geometry)In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
- There is a Gross–Deligne conjecture in the theory of complex multiplicationComplex multiplicationIn mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
.
- There is a Deligne conjecture on monodromyMonodromyIn mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
, also known as the weight monodromy conjecture, or purity conjecture for the monodromy filtration.
- There is Deligne conjectureE7½ (Lie algebra)In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in orderto fill the "hole" in a dimension formula for the exceptional series En of simple Lie algebras. This hole was observed by Cvitanovic, Deligne, Cohen and de Man...
in the representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the exceptional Lie groups.
- There is a Deligne–Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.