Koopman Tori, Knots, Elliptic Curves and BSD Conjecture

This article will focus on the continuing work on Koopman operators and the incorporation into the Feynman-Kac quantum fields and AdS space, with Boltzmann Machine correlations for machine learning and AI. Previously we demonstrated a proof of the Koopman Feynman-Kac Mellin Space with a D2-brane in the DBI action and compacted onto a chaotic KAM Torus.

The work of Asakawa et. al focuses on the D2-brane fermionic string as a basis for the qubit, the fundamental unit for quantum computing. We will introduce the Birch Swinnerton Dyer (BSD) Conjecture and its relationship to this research.

Our approach is inclusive with possible applications to quantum gravity, gravitational waves-cosmic strings, fluid dynamics, plasma fusion energy, ion propulsion, fluid neural networks, metamaterial cloaking, electromagnetic singularities, robotic systems, econometrics, stock pricing, and many others.

Normally string theory or loop quantum gravity (LQG) is integrable. In special cases such as the Kolmogorov-Arnold-Moser (KAM) Theorem, it is non-integrable. This is where space is compacted on a torus with an irrational winding number, or how many times the string is wrapped around the torus.

In 4-dimensional phase space, the 2-torus (Poincare Circle) with irrational winding number will survive, while the rational or resonant tori will decompose into elliptical and hyperbolic points.

The elliptical points are defined on an elliptic curve. An elliptic curve is a smooth curve with a cubic third order polynomial described on a plane including a singular infinite point (singularity pole, knot).

Normally these are on a complex space and embedded into a simple circular. In our case, we are using a complex space and the KAM torus is the choice. The Weierstrass version has a meromorphic singularity pole in the 2d space.

Elliptic curves figure into many important aspects of mathematics, including being instrumental in solving Fermat’s Last Theorem, one of the Millenium Prize problems. There is also recent work on using elliptic curves and modular surfaces in helping to solve the Langlands program, the "Unified Theory" of all mathematics.

We will expand on the relationships between the work in these articles, Fermat’s Last Theorem, the Langlands program, and Seiberg-Witten Theory, a critical 4D field theory, in future articles.

 Elliptic curves are crucial to the development of modern cryptography and cybersecurity, especially by using prime number factorization in the roots of the curve.

The Birch-Swinnerton-Dyer (BSD) conjecture is another Millenium Prize problem that has yet to be solved for a general case. The crux of the BSD Conjecture is the rational solution of elliptic curves, when the rank is based on whether there are finite or infinite number of basis points on the curve. A rank of 0 means there is a finite number of rational points.

A rank of 1 means there is an infinite number of rational points. Mathematicians Bryan John Birch and Peter Swinnerton-Dyer developed the conjecture using a computer.

The solution of a general case will be a game changer in cryptographical systems and number theory including prime numbers.

In our work to correlate elliptic curves to KOT and KKOGFAB, we will use KAM Tori on complex space to define elliptic curves, there is numerous example of this in the literature. The Hausdorff space, the space used for KKOGFAB D2 Brane, is special case with a one-dimensional torus S1.

The correlation between complex tori in BSD Elliptic Curves and KKOGFAB D2 Brane is the focus of this work, with the Riemann surface genus 1 and the Hausdorff as special. The knot-strings nodes on the tori are the meromorphic Weierstrass complex tori poles (singularities) on the tori.

In the case of a BSD rank with infinite rational spaces, the KOT Dynamic Mode Decomposition (DMD) will be used to reduce the dimensions to a linear finite space.

The KOT applied to data science/machine learning-AI converts nonlinear data to linear but with infinite dimensions. Dynamic Mode Decomposition (DMD) reduces dimensions by Principal Component Analysis (PCA) and Fourier series.

I have deliberately left out mathematical equations, theorems, and proofs in order to make the material shorter and more accessible. These are available in various places including papers published in Academia.edu, Google Scholar, and in my book KKOGFAB Theory available on Amazon.com KKOGFAB.

Here are some links:

Papers on Academia.edu and Google Scholar Papers | Outline | Primer

1.  Birch and Swinnerton-Dyer Conjecture at Clay Mathematics Institute

1.  Stewart, Ian (2013), Visions of Infinity: The Great Mathematical Problems, Basic Books, p. 253, ISBN 9780465022403, 

1.  Cremona, John (2011). "Numerical evidence for the Birch and Swinnerton-Dyer Conjecture" (PDF). Talk at the BSD 50th Anniversary Conference, May 2011., page 50

1.  Tate, John T. (1974). "The arithmetic of elliptic curves". Invent Math. 23 (3–4): 179–206. Bibcode:1974InMat..23..179T. doi:10.1007/BF01389745., page 198

1.  Cremona, John (2011). "Numerical evidence for the Birch and Swinnerton-Dyer Conjecture" (PDF). Talk at the BSD 50th Anniversary Conference, May 2011.

1.  Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag. ISBN 0-387-97966-2.

7.     Arthaud, Nicole (1978). "On Birch and Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication". Compositio Mathematica. 37 (2): 209–232. MR 0504632.

Additional References are listed at the end of each article.

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3. Brown, Ezra (2000), "Three Fermat Trails to Elliptic Curves", The College Mathematics Journal, 31 (3): 162–172, doi:10.1080/07468342.2000.11974137, S2CID 5591395, winner of the MAA writing prize the George Pólya Award

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1. Wiles, Andrew (2006). "The Birch and Swinnerton-Dyer conjecture" (PDF). In Carlson, James; Jaffe, Arthur; Wiles, Andrew (eds.). The Millennium prize problems. American Mathematical Society. pp. 31–44. ISBN 978-0-8218-3679-8. MR 2238272. Archived from the original (PDF) on 29 March 2018. Retrieved 16 December 2013.

43.   Williams, M.O., Kevrekidis, I.G. & Rowley, C.W. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. J Nonlinear Sci 25, 1307–1346 (2015). https://doi.org/10.1007/s00332-015-9258-5

44.   Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.

Other references

Milan Korda, Yoshihiko Susuki, Igor Mezić,. Power grid transient stabilization using Koopman model predictive control. IFAC-PapersOnLine, Volume 51, Issue 28,2018, Pages 297-302, ISSN 2405-8963.

https://doi.org/10.1016/j.ifacol.2018.11.718.

(https://www.sciencedirect.com/science/article/pii/S2405896318334372)

INTERNATIONAL COLLABORATIONS:

E.Deotto (MIT, USA); M.Reuter (Mainz University, Germany); A.A.Abrikosov (jr) (ITEP, Moscow); I.Fiziev (Sofia University, Bulgaria).