Sir Michael Francis Atiyah (22 April 1929 – 11 January 2019) wasn't just a mathematician; he was a magician of numbers, a wizard whose work could turn the most abstract mathematical ideas into something akin to pure magic. His groundbreaking contributions linked geometry and topology in ways that fundamentally transformed modern mathematics. Atiyah was not just any mathematician; he co-developed the Atiyah-Singer Index Theorem - a concept so profound that it's like discovering a hidden treasure map using nothing but a mathematical compass. This theorem creates a bridge between differential equations and the topological properties of shapes, making it one of the most intriguing ideas in the mathematical universe. Along the way, Atiyah collected nearly every top prize in mathematics, from the prestigious Fields Medal to the Abel Prize, firmly establishing himself as one of the grand wizards of the mathematical world.
The Enchanted Forest of Mathematics: Atiyah's 2018 Abel Prize Talk
In his 2018 Abel Prize talk at the International Congress of Mathematicians (ICM), Michael Atiyah did more than just give a lecture; he took his audience on a mesmerizing journey through the enchanted forest of mathematics. With the skill of a master storyteller, he wove together tales of numbers, geometry, and physics, revisiting some of his greatest mathematical adventures, including the renowned Atiyah-Singer Index Theorem. All that he did with humor.
Atiyah reflected on how these seemingly abstract concepts have surprising and deep connections to the real world, musings that felt like whispers of the universe's deepest secrets. With the charisma and flair of a seasoned wizard, he suggested that the most profound truths in mathematics are like hidden spells, waiting to be discovered by those brave and curious enough to explore them. It was a fitting farewell from one of the most brilliant minds to ever grace the field of mathematics.
The Fine Structure Constant: A Bold and Controversial Proposal
Atiyah's talk was not just a stroll down memory lane. He also presented a bold and speculative idea about one of the most mysterious constants in physics: the fine structure constant. This constant is crucial because it governs the strength of electromagnetic interactions, yet its precise value has long puzzled physicists.
Atiyah proposed an innovative approach to deriving the value of this constant, one that intertwined deep mathematical concepts with the mysteries of the physical universe. He traced his ideas back to the work of Euler, von Neumann's theory of factors, and Clifford algebras, seamlessly blending his lifelong passion for mathematics with his curiosity about the physical world. His discussion was both intriguing and controversial, sparking debate and reflecting his enduring spirit of exploration, even in his later years.
The Calculations: No Cigar, But Still a Magical Journey
Atiyah's proposal stirred quite a bit of discussion, particularly on forums like the Physics Forum, where physicists debated whether he had uncovered new links between arithmetic and physics. Viktor T. Toth, a notable contributor to the discussion, analyzed Atiyah's 2018 preprint using Maxima software, ultimately concluding that Atiyah's proposal did not match the expected value of the fine structure constant. Undeterred, I decided to dive into the calculations myself using Mathematica.
Using Atiyah's formula, the value of the inverse of the fine structure constant, 1/α (which he denoted by the Cyrillic letter Ж), came out as 0.160262—significantly different from the expected 137.0359992, making it about 854 times smaller than the expected value. In short, no cigar here. In fact, it was a true disaster.
But Atiyah's story doesn't end with a simple miscalculation. His paper delves into much more advanced mathematics, suggesting that Ж might be similar in nature to π. Atiyah speculated that just as Euler’s formula
links π with complex numbers, there might be a similar elegant formula for Ж involving noncommutative quaternions. From there, he ventured into von Neumann factors, traces, and infinite products, sketching ideas that are as challenging to comprehend as they are fascinating. While he didn’t provide all the answers, Atiyah’s speculations open up tantalizing possibilities that invite further exploration.
Conclusion: The Legacy of a Mathematical Wizard
Michael Atiyah's work remains a testament to the power of curiosity, creativity, and courage in the world of mathematics. While his attempt to link the fine structure constant to deep mathematical ideas may not have yielded the expected results, it exemplifies the adventurous spirit that drove him throughout his life. Atiyah's legacy is not just in the theorems he proved or the prizes he won, but in the magical way he connected different realms of mathematics and physics, inspiring future generations of mathematicians to explore the unknown. Whether or not his ideas about the fine structure constant will stand the test of time, they will undoubtedly continue to provoke thought and inspire wonder.
In the end, Sir Michael Atiyah leaves behind a legacy that, like all great magic, will continue to captivate and challenge those who dare to dream big in the world of mathematics.
Related links
- M. Atiyah , “The Fine Structure Constant” (2018)
- M. Atiyah, “Arithmetic Physics – Abel Lecture” (2018)
- “Atiyah and the Fine-Structure Constant“ on Sean Carroll's blog
- “Riemann hypothesis, the fine structure constant, and the Todd function” post on John D. Cook blog
- "A fantastic journey through Atiyah's mathematical mind", IMPA blog
P.S. 18-08-24 I thank to Viktor T. Toth for pointing it to me that Atiyah's Ж was supposed to denote 1/α, and not α, as I had it in the original version of this post.
No comments:
Post a Comment
Thank you for your comment..