Originally posted by XanthosNZholy cats Xanthos -
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.
You not only started the thread, but there are 32 posts when I woke this morning. Amazing.
For anyone really interested here is a site to walk you through some of this
http://www.ma.utexas.edu/~dafr/HodgeConjecture/Hodge%20Conjecture_files/frame.htm
Xanthos you say are attempting to disprove ... is that because you don't believe the conjecture is true - or that you think that is just a more interesting way to invest your time and work?
Facinated by the algebraic cycle, but I don't have enough math to make the conceptual jump. Maybe someday.
Originally posted by Rosa KorganeI might have already had all this figured out, if I had known that this web page only works with Internet Explorer Browser. I tried it last night. It wouldn't load with firefox. So I gave up. Now that I know I am so good at solving problems let me work on this conjunction gizmo a while. But I am starting to agree with ZanthosNZ and starting to believe already that this thing couldn't possibly work because who could understand it enough to ever believe that it could be true.
holy cats Xanthos -
You not only started the thread, but there are 32 posts when I woke this morning. Amazing.
For anyone really interested here is a site to walk you through some of this
http://www.ma.utexas.edu/~dafr/HodgeConjecture/Hodge%20Conjecture_files/frame.htm
Xanthos you say are attempting to disprove ... is that because you don' ...[text shortened]... the algebraic cycle, but I don't have enough math to make the conceptual jump. Maybe someday.
Here's what my frantic google search uncovered, for all neophytes:
"The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles."
Originally posted by Rosa KorganeI've tried both proving and disproving, but currently as a new tool has emerged I thought I'd try to use it to disprove.
Xanthos you say are attempting to disprove ... is that because you don't believe the conjecture is true - or that you think that is just a more interesting way to invest your time and work?
Facinated by the algebraic cycle, but I don't have enough math to make the conceptual jump. Maybe someday.
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Originally posted by XanthosNZcan you give a synopsis of what would happen mathematically
I've tried both proving and disproving, but currently as a new tool has emerged I thought I'd try to use it to disprove.
speaking if you prove or disprove it? What areas of math would be
affected? Packing theory? Mapping?
BTW did you read the math article in the latest Scientific American,
about the limits of proofs in maths? It seems some conjectures
have to be taken on faith till otherwise disproven which in some
cases is impossible.
Originally posted by sonhouseMost of the affected areas are extremely esoteric but there are quite a number of papers which start with "Assume the Hodge Conjecture" or "Assume a disproof of the Hodge Conjecture" and go on to show what that would mean. There aren't as many as in the Riemann Hypothesis (As almost everyone believes that to be true) but there would still be a great deal of new ground to cover if it were proven or disproven.
can you give a synopsis of what would happen mathematically
speaking if you prove or disprove it? What areas of math would be
affected? Packing theory? Mapping?
BTW did you read the math article in the latest Scientific American,
about the limits of proofs in maths? It seems some conjectures
have to be taken on faith till otherwise disproven which in some
cases is impossible.
As for the limits of proofs if you are going to assume a conjecture or hypothesis (mathematicians don't take things on faith they assume, there is a difference) then you better hope that it is impossible to disprove it (otherwise your work becomes useless). As I said, many papers have been written working on the basis that something until now unproven is true (or false). There is nothing wrong with that and in fact in some cases it can lead to a proof of the assumed thing (If I assume A and then prove a result B and can show that B could only be true if A was true then a second proof of B not involving A would be sufficient to prove A).
Originally posted by XanthosNZOK got it - took additional readings - thanks
(If I assume A and then prove a result B and can show that B could only be true if A was true then a second proof of B not involving A would be sufficient to prove A).
B could only be true if A was true ...second proof of B
