While the problem is easily understood by the layman, the techniques used to solve it are incredibly deep and you would probably describe them as esoteric. There must be a layman's definition! who knows little maths by saying something like "The hodge conjecture postulates that joined together continuously." Thank you for the link, unfortunately my mathematical understanding is insufficient to understand his actual answer, but the summary gives an idea why it is considered important. While this may be important to mathematicans working in this area, it seems frankly a rather esoteric subject. polynomial equations in n variables.
You don't have to understand all these words, just look at how it changes the previous point-counting formula: instead of simply having a power [; q^i ;] corresponding to an i-cell [; \mathbb{A}^i ;], you have complex numbers of absolute value [; q^i ;], as much as the dimension of the cohomology in dimension 2i.
I'll undo them in a few days.--Enyokoyama (talk) 06:56, 16 February 2013 (UTC), I think that the written paragraphs were irrelevant, but the assumption in the Hodge conjecture that X is algebraic cannot be weakened. Maybe that is why it’s so disturbing to notice that there are still mathematical problems that have no solution, ... Here’s a number of the unsolved millennium problems and different conjectures that have not been confirmed.
You have the description of n-dimensional projective space over a field as a union of a point, a line, a plane, etc: [; \mathbb{P}^n_{/K} = \mathrm{pt} + \mathbb{A}^1_{/K} + \mathbb{A}^2_{/K} + \cdots + \mathbb{A}^n_{/K}.
Ask a science question, get a science answer.
wikipediatrix 19:46, 22 August 2006 (UTC), " the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the COHOMOLOGY classes of subvarieties ", " the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the HOMOLOGY classes of subvarieties", The section on the integral Hodge conjecture has been oversimplified to the point of incorrectness - we need to distinguish between classes in integral cohomology (Hodge's original formulation) and classes in rational cohomology which are the images of classes in integral cohomology. I'm thinking particularly of the reference to Atiyah and Hirzebruch's 'Vector bundles and homogeneous spaces'.
More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties.
Charles Matthews 22:00, 29 July 2005 (UTC), As far as I can see, the announcement by Fred W. Roush (with Kim) is from people who provided a counterexample to another conjecture, the Williams conjecture, nearly a decade ago (in symbolic dynamics). Carovingian (talk) 12:28, 15 July 2019 (UTC), integral Hodge conjecture section needs improvement, abelian varieties paragraph needs some improvements, More specific references to counterexamples, No explanation in the introduction for the layman, Category:Unsolved problems in mathematics, https://en.wikipedia.org/w/index.php?title=Talk:Hodge_conjecture&oldid=906373302, Creative Commons Attribution-ShareAlike License, This page was last edited on 15 July 2019, at 12:28.
In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety.
In that case, only the current claim that goes unverified is mentioned; all others are reduced to a cursory note as they do not add anything to the current knowledge on the subject.
If your field K is finite of size q, then you get point counting estimates:[; |\mathbb{P}^n(K) |= 1 + q + q^2 + \cdots + q^n. It won't mean much to anyone not conversant with both those fields. The signs are not technical oddities, but rather a fundamental notion about how point counting works, reminiscent of Euler characteristic (I can explain this in a separate post).
The fantastic idea that emerged from Grothendieck's school is that this generalises to arbitrary smooth proper varieties, and you have to replace the above formula by cohomological invariants (the Lefschetz trace formula for l-adic étale cohomology). Press J to jump to the feed. Would this have anything to do with the study of an 11 dimensional universe? I don't consider Fermat's Last Theorem or its solution to be esoteric. Roush is or was at Alabama State University. In fact, though not introduced in this article, Zucker showed in 1977 that Complex Tori with non-analytic rational cohomology of type (p,p) turn to be an counterexample to Hodge conjecture.
Can anyone provide something clickable and checkable? It's not really a conjecture about a mathematical theorem; it's a conjecture about new kinds of tools. You cannot define mirror symmetry without Hodge theory.
Thus, if the Hodge and Tate conjectures are true, we know that there are profound connections between those two worlds: we can pass information from one to the other through the medium of algebraic cycles. That only a small subset of the population is able to understand a solution does not mean that the rest of the population does not care about it, provided they can understand the question itself. For example, if the word of god came down and told us that yes, indeed the Hodge conjecture was true/false, it would be nice but not nearly as groundbreaking as a proof for it. The essence to understand from this is that the point counting goes from easy "have a q to the power of the dimension" to having to add complicated complex numbers of a certain absolute value together.
Otherwise, what cohomology class is being talked about?
any surface of any imaginable shape (most shapes existing in more dimensions than the usual three spatial dimensions) can be described by sets of curved lines and straight lines
The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: This page was last edited on 14 August 2018, at 02:02. I haven't checked the other references, but this is particularly poor 218.215.42.121 (talk) 23:56, 26 December 2016 (UTC), Further digging around suggests that it is in fact a different 1961 paper of Atiyah and Hirzbruch that provides the counterexample.
Charles Matthews 15:08, 12 August 2006 (UTC).
The only source for this article is to a non-online reference.
Why is the Hodge conjecture so important? I'd draw the comparison with proving that, say, the Euler-Mascheroni constant is transcendental. 131.111.24.26 (talk) 16:55, 14 December 2008 (UTC), In the section on Hodge loci, should "the locus of all points on the base where the cohomology of a fiber is a Hodge class" actually read something like "the locus of all points on the base where a particular cohomology class in the fiber is a Hodge class", or something like that?
This makes for a bad stand-alone reference.
I'll add this fact into the relevant position in this article.--Enyokoyama (talk) 16:56, 16 February 2013 (UTC), I've already add the above mention into the paragraph "Reformulation in terms of algebraic cycles." Gluons12 (talk) 21:07, 26 May 2016 (UTC).
However, if the question itself is only intelligble to a small group I would probably classify it as esoteric, without any negative connotations intended. So, not an algebraic geometer, on the face of it. For instance, you'd expect that every time you do have a [; q^i ;] in the previous formulas, it's because you actually do have some underlying geometry explaining it, similar to [; \mathbb{A}^i ;] inside your variety, which explains it.
Not the whole cohomology (for instance because these algebraic cycles always land in even degree, never in odd degree), but essentially everything it could attain.
-- Spireguy (talk) 02:52, 10 June 2009 (UTC), add an expository article by Claire Voisin as an external link.--Enyokoyama (talk) 01:37, 16 March 2013 (UTC), I think the paragraph abelian varieties needs some improvements. hgutahw linked to an application in an entirely different direction as well. Otherwise eg Kollar's example doesn't make sense.
Visions of Infinity: The Great Mathematical Problems. - MathOverflow.
In a nutshell, following Grothendieck's school, cohomology has become a fundamental tool in understanding the geometry of algebraic varieties, which are simply sets of solutions to polynomial equations. The Hodge conjecture . Either way, creating new math and connecting existing math are the real reasons why solving open problems is important.
But your idea about 'popular science' may not be everybody's.
Thanks!