Talk:Hodge conjecture

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Contents 1 Unsolved problems in mathematics 2 Announcement of counterexample 3 References 4 phrase

[edit] Unsolved problems in mathematics

Please, can someone write a brief description of the conjecture? These two or three phrases (written as one or more questions) are needed for adding to the article the following template:

{{unsolved|mathematics|Brief description}}

That will put a tag with the brief description and a link to the list of Unsolved problems in mathematics, as well as putting this article in the Category:Unsolved problems in mathematics. There are more articles that must be tagged. Thanks! --surueña

The conjecture states that for a smooth projective complex algebraic variety, every Hodge cycle in a space H^p,p is a rational linear combination of classes of algebraic cycles of codimension p. Charles Matthews 15:06, 29 July 2005 (UTC)

Thank you, but I don't understand a word... :-) Sorry for not get me explained correctly, but in my opinion the series of lists of Unsolved problems is intended for popular science, so the brief description of the unsolved problem should be understood more or less by everyone. Sometimes the text of the tag is not needed to explain the whole problem but only say why it is important and its consecuences. For example, see the tags in the following articles:

• Theory of everything
• Turbulence

Maybe the first step is to write the list of "popular science" questions regarding every unsolved problem in the list of Unsolved problems in mathematics, as done in Unsolved problems in physics. Anyway, maybe some of these mathematical problems are very, very difficult to explain to everyone, so thanks again for your brief description, and sorry for the inconvenience. --surueña 16:33:35, 2005-07-29 (UTC)

Well in mathematics some problems can be stated simply, and some can't. That hasn't much to do with their importance. The Hodge conjecture relates topology with algebraic geometry, i.e. polynomial equations in n variables. It won't mean much to anyone not conversant with both those fields. But your idea about 'popular science' may not be everybody's. Surely a list of unsolved problems is basically a list of problems that are important to solve, popular or not? Charles Matthews 22:00, 29 July 2005 (UTC)

Yes, I'm agree with you completely. But in my opinion the lists about unsolved problems can be useful for everyone: for experts because they will discover the list of other unsolved problems --experts that probably will contribute with high quality contents to those articles--, and to the unskilled, because they will understand a bit of the problem and why it is important, attracting more readers --and maybe they will help with maintenance works in that articles (fixing typos, adding wikis and fixing wrong links, improving the images, ...) like me--. In summary, they are good for Wikipedia, and, if done right, the brief explanations will not make the articles "less serious". For example, in the article about the theory of everything the tag doesn't explain anything about that complex theory, but gives useful info. --surueña 09:01:08, 2005-07-30 (UTC)

[edit] Announcement of counterexample

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). Roush is or was at Alabama State University. So, not an algebraic geometer, on the face of it. It would certainly be interesting to know what the technique applied is. Charles Matthews 15:08, 12 August 2006 (UTC)

200.199.213.124 18:11, 3 September 2006 (UTC) I have removed the countexample claim now they have recanted it; some time has passed since and it is not news anymore. I follow the precedent of the Riemann Hypothesis. 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.

[edit] References

The only source for this article is to a non-online reference. Can anyone provide something clickable and checkable? wikipediatrix 19:46, 22 August 2006 (UTC)

[edit] phrase

Should the phrase:

" 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 "

be actualy written as:

" 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"

~~