Calabi-Yau Research Papers - Academia.edu (original) (raw)

In this paper, we consider a relationship between projection and dimension in geometry. We discuss the relation from projective geometrical viewpoint by paying attention to higher and lower dimensional spaces. Then, we categorize those projections among different dimensional ones by their characteristics.
We also express the classification by group theory and verify that there exists the automorphism.
p.s. Any point projected into other dimensional space always keeps its original freedom degrees. In other words, the freedom degree of point is invariant after projections or in a series of the group actions, so that it has "symmetry" in this sense.

Dear colleagues,
In 1872, Felix Klein announced an influential geometric program also known as Erlangen Program. He insisted that any type of geometry be considered as projective geometry operated by group theory. He was the first to inform us of the relationship between geometry and group theory. This thought has succeeded in physics rather than mathematics or geometry. For example, Afterward, Emmy Noether established the so called Noether's theorem. She so to say replaced projective geometry by another term, "symmetry" and considered symmetry as operated by group. So she insisted as follows; If there is a consecutive symmetry (but discrete symmetry is also possible in quantum area, for example, as selection rule) in a system, there exists a physical conservation. Hermann Weyl pushed the limits and introduced gauge symmetry based on representation of group in quantum theory. Yang Mills theory and other main theories in the quantum field theory are based on the gauge theory.

However, today's particle physics like super string theory is unnecessary described by symmetry. it applies so many mathematical techniques to the string theory, but it seems even to be more and more confused. I have so far, from mathematical viewpoint based on Felix Klein and his followers' spirits, researched a relationship among projective geometry, symmetry, and group theory or representation of group. Certainly, at this time, it is not necessarily in the mainstream of today's higher dimensional physics like super string theory. On the other hand, it is also the fact that there are physicists and mathematicians skeptical to the stringers' research direction. For example, I recently read a bestseller "Not Even Wrong: The Failure of String Theory". He is an excellent mathematician who once majored in particle physics. In his book, I found an interesting part of the preface to be (the last paragraph of page xix, you could read it at "Look Inside" on the website of amazon.) "Taking it out".

"The positive argument of this book will be that historically, one of the main sources of progress of particle theory has been the discovery of new symmetry groups of nature, together with new representations of these groups. The failure of superstring theory program can be traced to its lack of any fundamental new symmetry principle. Without unexpected experimental data, new theoretical advances are likely to come about only if theorists turn their attention away from this failed program and toward the difficult task of better understanding the symmetries of the natural world."

Now, let us remember my words above. His claim is coincidentally the same as I have insisted on so far through my research. His words motive me further to seek the possibility and potential in mathematics and physics. I think that the direction of my research can be significant. Please see the research paper via the link below.
https://www.academia.edu/16715934/Projections_and_Dimensions_trilogy_of_dimensionality_1_

Although projection from higher to lower dimensional space is not explained enough in this paper above, another example will make a good case for it:
https://www.academia.edu/29017770/A_Group_Representation_Notations_for_Projections_and_Dimensions_trilogy_of_dimensionality_3_ (: see Claim 2 on the page 26 to 27.)

What I'd like to say in this research is that any point projected into other dimensional space always keeps its original freedom degrees. In other words, the freedom degree of point is invariant after projections or in a series of the group actions, so that it has symmetry in this sense. Additionally, it will be more reasonable idea than Kaluza-Klein theory or Braneworld theory in higher dimensional physics, I believe.

Lately, athrough further research to scrutinize the property of transformation group mentioned above by the modeling on the number line, I got the results written in another paper, please see it also via the links below. Frankly, it'll be controversial especially for group theoreticians because it suggests a new finite simple group.
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_

https://www.academia.edu/16717180/A_Transformation_Group_and_Its_Subgroups_trilogy_of_dimensionality_2_full_paper_

It might be equivalent with Abelian group in Z, but actually non-commutative. By the way, if any subgroups in a group are normal like Abelian group, it's called Dedekind group. Additionally, if the group is non-commutative, Hamilton group. However, the transformation group In my paper is neither of them. Strictly speaking, it’s a “groupoid”. What I’m surprised at is that many of math professors seem not know about groupoid(: algebraic, not in category theory.) As you know, groupoid is partially functional, so unnecessary claimed binary operation. As mentioned above, I take it also as a group for a reason and that’s why my papers’ titles are written ‘group’. See also my paper in this A Transformation Group being Finitely Simple:
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_
https://www.researchgate.net/publication/313024517_A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper

By the way, a math professor saw my paper replied me that the contents of my papers impressed him a geometric matter well discussed by geometers for more than a century, Yes, that'd be true like I mentioned above. Additionally, at least we should have got back to 400 years ago when Rene Descartes established the coordinate geometry. Because, what I'm discussing in my paper titled "Projections and Dimensions" is also to rethink of it with a concept of space and dimension. Although we generally consider that lower dimensional space is subspace of higher dimensional space, we have to do them disjoint each other. Means, for example, one dimensional space is not subspace of two dimensional space. They are disjoint each other.

Based on this viewpoint, let us think differently from Rene Descartes's idea. Assume that a point in the one dimensional space were transported into two dimensional space, it should have been "point" also in the two dimensional space? Rene Descartes would say "yes", but I definitely does "no". Since the point originally existed in the one dimensional space has only one freedom degree, it can't decide the direction it moves in the two dimensional space. In other words, if the point had two freedom degrees, it could "randomly" or "arbitrary" decide the direction in the two dimensional space. Therefore, the transported point would radiate if it moved within the two dimensional space because of no choice for arbitrary direction. For the detailed explanation, I made a short document. Please see it also via the link below.
https://www.academia.edu/35228325/Dear_Prof.pdf

When I noticed it in my early 20s, I began to rethink of what space dimension is. Thank you for reading all my bullshits anyway, but I'm wishing you could feel what I memtioned above with my research papers worthy of attention.
With best regards,
Euich Miztani

p.s. About the paper entitled "A Transformation Group and Its Subgroups" which is mentioned above, I'm picky though, in the proof of proposition 2.1., strictly speaking, the right cosets Hg={2k-1T2n} "where 2n is fixed" and H'g'={2kT2n-1} "where 2n-1 is fixed." See also “a Transformation Group Being finitely Simple”.

p.p.s. Again, they are the trilogy above:
1. Projections and Dimensions( published in 2011)
https://www.academia.edu/16715934/Projections_and_Dimensions_trilogy_of_dimensionality_1_

2. A Transformation Group and Its Subgroup( published in 2015)
https://www.academia.edu/16717180/A_Transformation_Group_and_Its_Subgroups_trilogy_of_dimensionality_2_full_paper_

A Transformation Group being Finitely Simple(: the digest for the #2, published in a book by selected math papers in 2017)
https://www.academia.edu/33780859/A_Transformation_Group_Being_Finitely_Simple_trilogy_of_dimensionality_2_digest_paper_

3. A Group Repreentation: Notations for “Projections and Dimensions”( published in 2016)
https://www.academia.edu/29017770/A_Group_Representation_Notations_for_Projections_and_Dimensions_trilogy_of_dimensionality_3_