International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 1||Issue|| 6 ||Pages|| 64-67 || June 2018|| ISSN: 2581-4540
Proofs and Disproof of the Hodge Conjecture from 2016-20172018 James T. Struck BA, BS, AA, MLIS 1. INTRODUCTION The Godel incompleteness Theorem argues “your set of axioms will always be incomplete-they won’t be sufficient to prove all the facts. In mathematics as in life, parts of the truth are destined to remain forever elusive.” [1] Kurt Godel was a famous Austrian Mathematician and some appreciation for his incompleteness theorem should be given. Hodge Conjecture could be disproved in a few words, as axioms often are incomplete. If we take a set of irrational algebraic cycles, the result will be an irrational combination of classes. We cannot add irrational algebraic cycles and make up a rational linear combination. Yes, Hodge as we show can make up fantasies, inventions, or ideas like the Hodge Class, but that is only a definitional proof. When tested on real irrational algebraic cycles, the definition is just that a definition and not really a description of all algebraic geometric cycles like irrational shapes. The irrational algebraic cycle easily disproves the conjecture, as combinations will not be rational! I have worked on and off on the Hodge Conjecture from 2016-2018. Here I present the Historical Disproof that Hodge Classes Arose in a Non-Linear Relationship to Algebraic Cycles in the Past and Nomenclature Disproof that a Hodge Class is disproved to be a Set of Points as the Names and Words are Distinct. The problem we are faced with is the “Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” We also present the definitional proof that Hodge can define a conjecture statement “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles” that is provable as that is how he is defining the problem and he can conjecture something as a statement of proof.
II. METHODOLOGY We consider historical discussion, definitional statement and linguistic concepts to consider the Hodge Conjecture. If someone defines a problem such as “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles”, then we can see their definition as a type of proof as that is what they defined the class to be. Somewhat similar I can define a Saturn asteroid as Hidalgo and Chiron because that is how I defined the group even though Chiron has also become a comet with gas emissions recently. We consider irrational algebraic cycles to show that combinations do not need to be rational.
III. DISCUSSION Proof #1 Algebraic cycles as points. If we combine algebraic cycles as points on top of each other, we can see a Hodge class as a pile of points. The pile of points on top of each other does appear to be a linear combination of algebraic cycles or points. Proof #2 If Hodge defined “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles” then we can accept his definition. Scientists are permitted to discover new ideas or concepts. Similarly, I can define a “Saturn asteroid” as Hidalgo and Chiron because that is how I defined the group even though as around the orbit of Saturn, even though Chiron has also become a comet with gas emissions recently. Scientists are permitted to name their discoveries like “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” This presents the concept of a definitional proof. Disproof #1 We take algebraic cycles as points distant from each other. We define this collection of points as not Hodge Cycles as the points can be seen as sets of points but not Hodge cycles.
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Proofs and Disproof of the Hodge Conjecture… Disproof #2 The linear combination of the distant points is not a Hodge Cycle as the Kaehler manifold argues that the Hodge Cycle must always be changing as manifolds are always changing arguably or by definition what a manifold is. Disproof #3 We define a Hodge Cycle that is a Kaehler manifold, but not an algebraic variety. As we have defined the cycle as not an algebraic variety but as a Kaehler manifold, we have disproved the conjecture. Disproof #4 We see algebraic variety of points on top of each other. As the combination of points appears to be a point, the combination is a point and not a Hodge Cycle. Disproof #5 We define the algebraic variety as a group of points that are always changing then not one to one. The combination of these points then is not linear hence the conjecture disproved as the points are always changing and not one to one. Disproof #6 I argue that William Vallance Douglas Hodge working in the 1930-1940 period was a particular mathematician whose nomenclature of Hodge Class should not apply or be connected to language like a combination of points or algebraic variety. The algebraic variety of points or any algebraic variety does not have to be connected to a Hodge Cycle or Hodge Class. This presents the concept of historical disproof. Disproof #6 William Vallance Douglas Hodge was a Scottish mathematician in the last century working around 1930-1940. We do not need to see a connection or a linear relationship between his Hodge classes or cycles and the many varieties of algebraic varieties. We can see algebraic varieties as unrelated to Hodge Cycles as Hodge Cycles were invented in the 1930’s and not during the invention of all the algebraic varieties throughout history. Different concepts like Hodge cycles and Algebraic variety were not invented at the same time and can be seen as different. “Hodge class is NOT a rational linear combination of classes cl(Z) of algebraic cycles” as the concepts were invented at different eras. Disproof #7 An algebraic cycle can be using certain symbols and letters and an algebraic cycle can be using certain other symbols and letters. As the other letters do not have to be the same an algebraic cycle does not need to relate to an algebraic variety as the nomenclature or letters used to replace the parts of the equation do not need to be similar or related. Concluding Thoughts: Proof and disproof is possible regarding conjectures and here we show that one can easily both disprove and prove a conjecture. William Vallance Douglas Hodge’s invented a particular type of cycle or class in the 1930’s and 1940 period which can be proved and disproven to be a linear combination of algebraic cycles. One of the most interesting disproof here is the historical disproof. William Hodge worked in the 1930’s and a linear combination of algebraic varieties earlier in history does not make a Hodge Cycle as the combination occurred in the past before Hodge’s work in the 1930’s and 1940’s. Nomenclature distinctions between what is a pile of points and a Hodge Class can be used as both proof and disproof. Hodge Conjecture-If the shape is always altering or oscillating, there does not have to be an algebraic variety which makes up the shape. For example, for a virtual shape, there is no need for an algebraic variety. For an anti-shape, there would be no need for an algebraic variety. For the absence of a shape which can be seen as a shape, there does not have to be an algebraic variety. More Proofs of the Hodge Conjecture 2016 Hodge Conjecture. “On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” We can prove the statement by the same group or categories of proofs we provided elsewhere in proofs of Poincare and P=NP. Linguistically a statement can be seen as able to be said or verified. Grammatically a statement can be involving nouns and verbs and therefore grammatically provable. By Faith, a statement can be verified. I can ask myself, “Do you believe in the Hodge Conjecture?” and say “Yes I believe in the Hodge Conjecture.” The faith proof is just as real as the linguistic proof. The Hodge Conjecture can be said and the Hodge Conjecture can be believed on faith. Proof by math theory.: Combine a large number of parts or cycles and you get a class. If we think or draw small cycles we can combine them into a group or class. Disproof by Math theory: “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles” What if the algebraic cycles involve irrational numbers or irrational combination of classes? Irrational numbers exist
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Proofs and Disproof of the Hodge Conjecture… “Irrational number From Wikipedia, the free encyclopedia
The mathematical constant pi (π) is an irrational number that is much represented in popular culture.
“The number” Square root of 2 “is irrational” In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two;[1][2][3] in fact all square roots of natural numbers, other than of perfect squares, are irrational.” (1) So since irrational numbers can be part of algebraic varieties and algebraic cycles, we can say that we do not need to have a “Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” We do not need to have rational combinations as algebraic cycles can involve irrational numbers therefore partly disproving the conjecture. 2017 Disproof Disintegrating, disappearing numbers provide a type of disproof as the classes disintegrate and disappear not showing equality. It is not clear if Hodge only defined one class. Hodge's name does not appear in a number of Encyclopedias of mathematics. As one cannot easily do background reading on Hodge, the lack of background material can be seen as a type of disproof? Color disproof. We can have algebraic varieties of one color that do not relate directly to Hodge classes of a different color.
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Proofs and Disproof of the Hodge Conjecture… Algebraic varieties do not need to be made of algebraic cycles. I just looked in a math encyclopedia where algebraic varieties were mentioned with no mention of algebraic cycles.
ACKNOWLEDGEMENTS I would like to thank my mother Jane Frances back Struck who died on 7/15/2017 as she left a chest at the nursing home where she died that I picked up just now giving me the time to work on this problem and helping me be confident in my abilities. I would also like to thank Jane Frances Back Struck for defending the right to speak against Illinois and Cook County abuse.
REFERENCES 1. Devlin, Keith, The Millennium Problems The Seven Greatest Unsolved Mathematical Puzzles of our Time, Basic Books, New York, 2002. 1. https://en.wikipedia.org/wiki/Irrational_number accessed on 4/13/2018 James T. Struck BA, BS, AA, MLIS 773 680-7024 PO BOX 61 Evanston IL 60204
James T. Struck BA, BS, AA, MLIS. “Proofs and Disproof of the Hodge Conjecture from 20162017-2018.” International Journal of Modern Research in Engineering & Management (IJMREM), vol. 1, no. 6, 25 June 2018, pp. 64–67., www.ijmrem.com.
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