Every harmonic differential form (of a cetain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles.
In this thread discuss concepts you believe useful in proving or disproving this, your inclinations as to it's truth, progress you have made and potential proofs.
I will try to answer any questions non-mathmaticians have.
Currently I am in the process of trying a new attempt at disproving the conjecture and have made not a small amount of progress. The recent proof of Salmon's Theorem has helped me a great deal. Has anyone else found it useful?
EDIT: Sorry, I forgot the 'popular version' of the Hodge Conjecture which may be of use to the less good mathematicians among us who wish to follow along:
Let X be a projective algebraic manifold and p a positive integer. Also, let H[2p](X,Q)alg union H[2p](X,Q) be the subsapce of algebraic cocycles, i.e., the Q-vector space generated by the fundamental classes of algebraic subvarieties of codimension p in X. The Hodge conjecture assert that one can "compute" the subspace H[2p](X,Q)alg using Hodge theory, specifically H[2p](X,Q)alg = H[p,p](X) intersection H[2p](X,Q).
Originally posted by XanthosNZWow, is that the recipe for "irrational" cookies?
Every harmonic differential form (of a cetain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles.
In this thread discuss concepts you believe useful in proving or disproving this, your inclinations as to it's truth, progress you have made and potential proofs.
I will try to a ...[text shortened]... lg using Hodge theory, specifically H[2p](X,Q)alg = H[p,p](X) intersection H[2p](X,Q).[/i]
Originally posted by skeeterGo read some poetry and leave the important stuff to the rest of us. Perhaps you could make some tea?
.....which, given your present progress may be "a bridge too far" for you, in my opinion.
skeeter
EDIT: Flex, I haven't the time nor the energy to devote to explaining complex and extremely abstract mathmatical concepts to anyone. The whole thing took me months to understand and even now I learn more everyday. May I suggest Five Golden Rules: Great Theories of the 20th Century - And Why They Matter as a good starting place.