Ido Kaminer's talk Exp, Math, talk. Sept. 28, 2023

Title:
The Ramanujan Machine 2.0: algorithm-assisted discovery of an intrinsic order among
mathematical constants

Abstract:
Recent years have shown a rise in the number of discoveries in fields of mathematics that are being assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. Now that computers and algorithms become more powerful, their ability to augment human intuition is expected to lead to new kinds of discoveries.

To contribute to efforts toward this vision, we started the Ramanujan Machine project [1], relying on a network of volunteers from different fields to develop algorithms that facilitate mathematical discoveries. We specifically focused on the search for new formulas of mathematical constants, and had particular success with formulas in the form of continued fractions [1-2], in analysis of integer sequences [3], and in discovery of efficient Diophantine approximations [1-4].

For our most recent work [4], we developed a massively parallel computer algorithm [5] that utilize thousands of personal computers worldwide, discovering an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas unveiled a novel mathematical structure that we call the conservative matrix field. Such matrix fields unify thousands of existing formulas and generate infinitely many new formulas. Most importantly, the matrix fields lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we used a matrix field to reconstruct the celebrated proof by Apéry for the irrationality of ζ(3) and generalize it to more cases.

The seminar will present the most recent progress in this area and discuss future directions, especially highlighting important open questions.

 

[1] Raayoni G., Gottlieb S., Manor Y., et al. Generating conjectures on fundamental constants with the Ramanujan Machine, Nature 590, 67–73 (2021)

[2] Ben David N., Nimri G., Mendlovic U., Manor Y., Kaminer I., On the Connection Between Irrationality Measures and Polynomial Continued Fractions, arXiv 2111.04468 (2021)

[3] Razon O., Harris Y., Gottlieb S., Carmon D., David O., & Kaminer I., Automated Search for Conjectures on Mathematical Constants using Analysis of Integer Sequences, International Conference on Machine Learning, PMLR, 28809 (2023)

[4] Elimelech R., et al., Algorithm-assisted discovery of an intrinsic order among mathematical constants, arXiv 2308.11829 (2023)

[5] https://rnma.xyz/boinc/