My $0.02 on Field Medalist Dr.Atiyah solving the (RH) Riemann Hypothesis
Disclaimer: Electrical Engineering graduate with math training including multivariable calculus, linear algebra and differential equations and geometry. Above are needed to understand this article in depth. I do not deal with mathematics for work so not everything in this article is expected to be 100% correct, but every sentence is original. My goal is to explain this scientific break-through in the most thorough way possible and in laymen’s terms.
Background: Field Medalist Dr.Atiyah, the most famous mathematician of the 21st century claimed solving the RH by accident, while attacking the math behind the fine structure constant in physics, which is related to the electromagnetic interaction force between particles. The RH is one of the most important math problems deemed by the renowned math society, due to its contribution to the modern number theory. Cryptography is one of its applications. For instance RSA key generation is related to factoring a large number into 2 prime numbers. Since the sequence of prime numbers is random, it makes decoding nearly impossible. On the other hand, RH can be broken down to product of infinite series (called zeta function), hence product of primes. (more evidence to show here) This leads us to finding the zero’s of the zeta function, whose significant solutions contains a real part and an imaginary part. The hypothesis says all the non-insignificant solutions must have the real part as 1/2 — dubbed the critical line. Mathematically proving RH can give us a direction to look for the behavior of prime numbers, not necessarily cracking the code. BTW, RH is worth $1,000,000 dollars.
Status Quo: Michael Atiyah’s talk at Heidelberg Laureate Forum today (9/24) showed the proof in just one slide here (https://www.heidelberg-laureate-forum.org/blog/video/lecture-monday-september-24-2018-sir-michael-francis-atiyah/). It was so called “short” and “easy”, due to deriving from the Todd function. However this brand new assumption of the Todd function/Todd mapping is the real elephant in the room. Luckily I also found his paper on “The Fine Structure Constant”, where the Todd Mapping was defined in detail. Atiyah used contradiction in this proof, unlike many failed attempts previously. My opinion is first, contradiction is a valid proving method; second even though no progress was made in the number theory, a special dimension which can map from infinite to finite, from discreet to continuous is already much to explore; third even if RH is proven this way, more progress can be made based on it being proven; fourth, is Todd Map really valid?
Mystery lies in the Todd Function — a weakly analytic function. (a converging infinite series) The Todd function is based on the work of Von Neumann (best known as father of computer scientists), where the brand new notion of “weakly” is based on the work of Hirzeburch. Again, most of today’s breakthroughs stand on shoulders of giants.
“Weakly” essentially means the multiplicative process which could extend from rational numbers to real numbers. Hence, real numbers is the weak closure of rational numbers. (need more evidence to show here) And complex numbers are just adding the imaginary part to it. Hirzeburch already showed Todd Function had the exponential generating function as the Bernoulli function, however he didn’t need the convergence property of the Bernoulli function for his previous work until Atiyah derived the “weakly” property from Bernoulli function being analytic and converging. It is worth to note that Atiyah mentioned Hirzeburch’s work as magic-like, whereas the following as truly deep.
Todd function assumes a dimension A defined in real numbers, which contains only unique isomorphism (differential functions can map themselves) and takes the weak closure of the Hilbert space (no missing points). The function itself is basically the sum of infinite powers. Note only traces (diagonal inner product of matrix) and isomorphisms are linear, but also the Todd function itself. Since the Todd function is defined in real numbers and works on compact set or closed space as shown above, it is continuous and finite to start with. Rational numbers are infinite and discrete. Then we can map from an infinite and discrete dimension to a finite and continuous dimension using the trace and isomorphism’s linear property and the “weakly” property. (This is mostly words rather than just numbers or letters some of you might expect to see. But I can hardly find any in Atiyah’s paper either.)
Below is how Atiyah explained the Todd Mapping in his original paper:
Conclusion: We know the RH proof is by contradiction and based on the properties of the Todd Function. The correctness of the Todd Mapping still needs to be peer reviewed rigorously. Worth to note Todd was the math teacher of Atiyah. Personally, I hope the proof is good for both fine structure constant and RH. And again, the beauty of mathematics is really about understanding other people’s genius work, and being able to link among them. Sherlock Holmes once said 99% of the things in the world are known to someone, we just have to look.
