Originally posted by XanthosNZSo its in a field of conjectures that may lead in a thousand
Most 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 o ...[text shortened]... ue if A was true then a second proof of B not involving A would be sufficient to prove A).
differant directions and nobody knows the real world use yet or
is too early for a real world situation that Hodge would be used.
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).
If we take the canonical bundle, for example, p to the power of 4 produces a complete intersection threefold, i.e. Calabi-Yau threefold. Let X be the gen. complete intersection threefold extrapolates to infinite rational curves using Parson's Gate method of triangular coligration. These infinite curves roll to an infinite rank hexgroup if 2CH(Q)squared over the Roland constant at RTC, therefore we conject the trigonal curves and/or plane quintics could result in a hyperelliptic curve whereupon the vector space anomaly of H[2p] = C.div, i.e quadrantry relation to the Cheener index. But, if X based on a singularity then we preclude ourselves back to the ridiculous debate of infinity/0. Why you have chosen to ask the question, XanthosNZ, I am uncertain, for the analogous semi-simple polotard algebraic sphere is certainly so far above cataclysmic critical morphoses that even our greatest computers struggle in solving anything involving these sorts of infinitesimally ubiquitious fields. What would be much more interesting than proving the hodge conjecture would be in studying leading mathematical physicists' supposed secrecy regarding this conjecture and the considerable amount of Wan Karder's network research time that has been invested in investigating its slight and subtle variation to More's titte principle.
Originally posted by hopscotchYou've gone wrong when using functions involving powers with complex components (x^z where z can be real, imaginary or a combination). This is a common failing of many attempts at solving both this and the Riemann Hypothesis.
If we take the canonical bundle, for example, p to the power of 4 produces a complete intersection threefold, i.e. Calabi-Yau threefold. Let X be the gen. complete intersection threefold extrapolates to infinite rational curves using Parson's Gate method of triangular coligration. These infinite curves roll to an infinite rank hexgroup if 2CH(Q)squared ove ...[text shortened]... as been invested in investigating its slight and subtle variation to More's titte principle.
You do bring up an interesting point with the canonical bundle approach, that may come in useful during my work. I'll be sure to credit you.
Originally posted by XanthosNZWhy on earth would anyone intelligent want to spend their life trying to prove or disprove something that has no known importance. I mean maybe there is some financial gains to the one who solves it, or egotistical recognition motivations, but what else, how can it possibly apply to the real world? Can someone please explain that to me? Or does it have some real importance that I am missing? Could I have an example? Is the answer to this harder than the problem that is trying to be solved? Be careful something of this magnitude may cause you to become delusional.
You've gone wrong when using functions involving powers with complex components (x^z where z can be real, imaginary or a combination). This is a common failing of many attempts at solving both this and the Riemann Hypothesis.
You do bring up an interesting point with the canonical bundle approach, that may come in useful during my work. I'll be sure to credit you.
Originally posted by cashthetrashWell, I wouldn't imagine this thread is doing much to prove the Hodge conjecture.
Why on earth would anyone intelligent want to spend their life trying to prove or disprove something that has no known importance. I mean maybe there is some financial gains to the one who solves it, or egotistical recognition motivations, but what else, how can it possibly apply to the real world? Can someone please explain that to me? Or does it have ...[text shortened]... trying to be solved? Be careful something of this magnitude may cause you to become delusional.
However, why do people paint or make music or write (or read) novels or play chess or cook food with (unnecessary) flavour?
Originally posted by royalchickenDo you think those are good comparisons. I don't. They all have a need and real value. Bahhh! Pure waste of intelligence. That causes me to question the true intelligence in the first place. Prove me right or prove me wrong. What does it matter?
Well, I wouldn't imagine this thread is doing much to prove the Hodge conjecture.
However, why do people paint or make music or write (or read) novels or play chess or cook food with (unnecessary) flavour?
Originally posted by cashthetrashHave you ever read anything serious about pure mathematics, or done any yourself?
Do you think those are good comparisons. I don't. They all have a need and real value. Bahhh! Pure waste of intelligence. That causes me to question the true intelligence in the first place. Prove me right or prove me wrong. What does it matter?
Originally posted by royalchickenLook, I am by no means criticizing mathematics. I can see a real need for algebra, calculus, geometry, trigonometry. Mathematics has a real true need and usage. I understand it is about solving problems. And math is both a science and an art. We all use it every day. What I am trying to get a grip on is what does this conjecture matter? I confess I have no understanding of it. You can call me conjecture illiterate. Even when reading about it. Even though I have a basic concept and I do mean basic. What I am asking from those who do somewhat understand it, is to explain its necessity in solving the problem. What would be their theory of what the importance or usage could be if it were solved? Now, if you can't explain it then stop trying to compare it to food and art, because it sounds silly. Have you ever stuck pencils in your nose and barked like a walrus?
Have you ever read anything serious about pure mathematics, or done any yourself?
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Originally posted by cashthetrashWho is to say what pure maths physics or engineering will at some point cannibalize to describe real life situations?
Look, I am by no means criticizing mathematics. I can see a real need for algebra, calculus, geometry, trigonometry. Mathematics has a real true need and usage. I understand it is about solving problems. And math is both a science and an art. We all use it every day. What I am trying to get a grip on is what does this conjecture matter? I confess I have rt, because it sounds silly. Have you ever stuck pencils in your nose and barked like a walrus?
An example you may understand, complex numbers were once purely mathematical. Nowadays you use them in plenty of real world problems. What if back when someone said, "Let's define sqrt(-1) as i." you were around to say, "Why? What's the point? Get a real job you hippy."?
EDIT: If you really don't want to discuss the Hodge Conjecture then get the hell out of my thread.
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Originally posted by XanthosNZI agree with the spirit of what you're saying, but you didn't choose the best example. In pure-mathematical hindsight, it is possible to construct the complex numbers rigourously (as the field R/(x^2+1)R, which is tantamount to doing what you said happened, for example), but historically it's hard to pinpoint when the complex numbers became recognised as a class of their own, since this happened differently in different contexts. For instance, Cauchy himself would probably have to look up the definitions of many parts of a modern statement of Cauchy's theorem in complex analysis. I don't know if it's the case, but it's mathematically plausible that large parts of complex analysis developed out of considerations of real vector fields, distinct from the recognition of complex numbers as formal solutions to rational polynomials.
Who is to say what pure maths physics or engineering will at some point cannibalize to describe real life situations?
An example you may understand, complex numbers were once purely mathematical. Nowadays you use them in plenty of real world problems. What if back when someone said, "Let's define sqrt(-1) as i." you were around to say, "Why? What's the you really don't want to discuss the Hodge Conjecture then get the hell out of my thread.
To see this, think of the different contexts, within mathematics, in which you think of the number 3. When viewing it as a prime factor of 12, its relation to the idea of 'magnitude' is shaky, because formally speaking, magnitude and divisibility are not related (not all ubique factorisation domains are Euclidean). Also, when 3 arises in analysis, it's quite rare to care that it's a prime factor of 12. It seems plausible, then, that if people are thinking about something which is a totally new concept, then they might not realise that two wildly different objects are manifestations of the same concept, or, if they realise this, they might seriously underrate its importance. I think this is what happened with i, since rational polynomials were being solved and curves in the plane being integrated along before anyone gave a coherent and complete account of complex numbers.
EDIT My apologies for having nothing to contribute to the Hodge Conjecture discussion that has not been said already 😛.
