Why is Hodge Theory important?

Why is Hodge Theory important?

Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.

Is the Hodge star linear?

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. This map was introduced by W. V. D. …

What is differential form calculus?

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

Is Hodge conjecture true?

It turns out that the Hodge Conjecture is true in low dimensions due to a result of Lefschetz in 1924 from before Hodge even made the conjecture in 1950. Lefschetz proved it for codimension 1. In other words, every Hodge class in H²(X, ℚ) is algebraic.

What is algebraic geometry used for?

In algebraic statistics, techniques from algebraic geometry are used to advance research on topics such as the design of experiments and hypothesis testing [1]. Another surprising application of algebraic geometry is to computational phylogenetics [2,3].

What is Hodges?

Hodge. (hɒdʒ) n. (Agriculture) a typical name for a farm labourer; rustic.

What is exterior calculus?

Exterior calculus is a branch of mathematics which involves differential geometry. In Exterior calculus the concept of differentiations is generalized to antisymmetric exterior derivatives and the notions of ordinary integration to differentiable manifolds of arbitrary dimensions.

Is Hodge conjecture solved?

Some Progress. It turns out that the Hodge Conjecture is true in low dimensions due to a result of Lefschetz in 1924 from before Hodge even made the conjecture in 1950. Lefschetz proved it for codimension 1. In other words, every Hodge class in H²(X, ℚ) is algebraic.

What is the problem of Hodge conjecture?

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.