There have been a couple news stories regarding proofs of major theorems. First, an update on Shinichi Mochizuki’s proof of the abc conjecture, then an announcement that Sir Michael Atiyah claims to have proven the Riemann hypothesis.
Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.
This break in the story of the ABC conjecture is sure to make that portion of Mike Miller’s upcoming math class on Gems And Astonishments of Mathematics: Past and Present at UCLA much more interesting.
The last couple months I’ve heard reports from several people claiming that arithmetic geometers Peter Scholze and Jakob Stix had identified a serious problem with Mochizuki’s claimed proof of the abc conjecture. These reports indicated that Scholze and Stix had traveled to Kyoto to discuss this with Mochizuki, and that they were writing a manuscript, to appear sometime this summer. It seemed best then to not publicize this here, better to give Mochizuki, Scholze and Stix the time to sort out the mathematics and wait for them to have something to say publicly. Today though I saw that Ivan Fesenko has put out a document entitled Remarks on Aspects of Modern Pioneering Mathematical Research.
Peter definitely predicted the Fields medal for Peter Scholze here.
The intrigue of this case is quite interesting. Take a look at some of the comments on these posts. Some border on religious zealotry, and even this when I know Peter heavily curates his comments section to make them useful.
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Suggested prerequisites: Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory.-Tuesday 7:00PM - 10:00PMLocation: UCLAInstructor: Michael MillerMATH X 451.44 | 362773Fee: $453.00
I’ve been waiting with bated breath to see what Dr. Miller would be offering in the evenings at UCLA Extension this Fall and Winter quarters. The wait is over, though it’ll be a few days before we can register.
If you’re interested in math at all, I hope you’ll come join the 20+ other students who follow everything that Mike teaches. Once you’ve taken one course from him, you’ll be addicted.
A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.
The biggest mystery in mathematics
This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)
The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.
Thanks to Rama for bringing this to my attention!