* The Hodge conjecture

The Hodge conjecture is a problem in mathematics that remains unsolved.

Loosely speaking, the Hodge conjecture asks:

The modern statement of the Hodge conjecture is:

:: Hodge conjecture.

Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle.

With this notation, the Hodge conjecture becomes:

The first result on the Hodge conjecture is due to.

In particular, the Hodge conjecture is true for H < sup > 2 </ sup >.

However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.

If the Hodge conjecture holds for Hodge classes of degree p, p < n, then the Hodge conjecture holds for Hodge classes of degree 2n − p.

Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2n − 2.

This proves the Hodge conjecture when X has dimension at most three.

The Lefschetz theorem on ( 1, 1 )- classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:

: is generated by Hdg < sup > 1 </ sup >( X ), then the Hodge conjecture holds for X.

Hodge may refer to:

At a keynote speech to the Institute for Public Policy Research on 26 November 2004, Hodge strongly defended the idea of greater state regulation of individuals ' choices, stating that " some may call it the nanny state but I call it a force for good ".

Hodge defined a Mickey Mouse course as " one where the content is perhaps not as rigorous as one would expect and where the degree itself may not have huge relevance in the labour market "; and that, furthermore, " simply stacking up numbers on Mickey Mouse courses is not acceptable ".

203 ) In Hodge v. Canada ( Minister of Human Resources Development ) ( 2004 ), it was noted that a court may reject a claimant's choice for a comparator group, and that choosing the wrong comparator group may cause the rights claim to fail.

For varieties over fields of positive characteristic, the dimension of the Albanese variety may be less than the Hodge numbers

Charlie Hodge may refer to:

In the Protestant traditions some of the earliest writings opposing unorthodox groups like Swedenborg's teachings, can be traced back to John Wesley, Alexander Campbell and Princeton theologians like Charles Hodge and Benjamin Breckinridge Warfield.

By contrast, the Maxwell equations describing electromagnetism can be understood to be the Hodge equations of a principal U ( 1 )- bundle or circle bundle π: P → M with fiber U ( 1 ).

Based on the works of the 19th century thinker Charles Hodge, the group was devoted to strengthening the faith of students faced with what members believed to be the " theological disarray " of the times, and to providing an example of how to oppose " false and destructive ideas.

* Some definitions require that the holonomy be exactly equal to SU ( n ) rather than a subgroup of it, which implies that the Hodge numbers h < sup > i, 0 </ sup > vanish for 0 < i < dim ( M ).

observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure.

On 20 July 2006, Broadcasting Minister Shaun Woodward and Industry Minister Margaret Hodge announced that Whitehaven would be the pilot site for the switchover to digital terrestrial television in the United Kingdom.

Since the space of alternating linear forms in k arguments on a vector space is naturally isomorphic to the dual of the space of k-vectors over that vector space, the Hodge dual can be defined for these spaces as well.

As with most constructions from linear algebra, the Hodge dual can then be extended to a vector bundle.

Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle ( i. e. the space of differential forms on a manifold ) where it can be used to construct the codifferential from the exterior derivative, and thus the Laplace-de Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.

This holds for all vector spaces, and can be used to explain the Hodge dual.

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

* John R. Hodge ( 1893 – 1963 ), American general not to be confused with U. S. General Courtney Hodges

Manly asked Hodge to estimate the dimensions of the wings, tail and propellers, the mechanism of the propeller drive and the nature of the construction, which Manly thought might be too weak.

" Hodge reported the machine did not appear to be airworthy.

Panton then passed the letter to the press which planned to publish it, only to be judicially restrained from doing so at the instruction of Hodge.

Hodge was the first person to be Children's Minister when the post was created in 2003 but suffered difficulties after the Islington controversy ; her resignation was called for on several occasions by the press and parliamentary opposition.

In January 2010, Margaret Hodge announced that Royal Parks, who manage Richmond Park and Bushy Park in the borough of Richmond on Thames, would be allowed to charge drivers £ 2 per visit.

I recollect him one day scrambling up Dr. Johnson's breast, apparently with much satisfaction, while my friend smiling and half-whistling, rubbed down his back, and pulled him by the tail ; and when I observed he was a fine cat, saying, ' Why yes, Sir, but I have had cats whom I liked better than this ;' and then as if perceiving Hodge to be out of countenance, adding, ' but he is a very fine cat, a very fine cat indeed.

He was well known as a philanthropist, using much of his fortune for charity work including setting up the Jane Hodge Foundation in memory of his mother and as a benefactor for Cardiff University from 1970 onwards ( one of the halls of residence now bears his name-Hodge Hall ).

Richard manipulates Keith's mind by warning him that Bree might be using him as a diversion from her actual feelings after her divorce from Orson Hodge.

As vectorial product can be expressed in a tensorial way using the Hodge dual of:

Unfortunately, the date of eruption that Hodge calculated using the potassium-argon method was 25 million to 27 million years ago.