Sunday, April 12, 2020

In Memory of John Conway

John Conway, the legendary mathematician, has passed.  Conway will be remembered for his deep work in pure mathematics, especially in cellular automata and Monstrous moonshine.  When asked about one of his greatest life’s mysteries, he expressed his desire for the 196883 representation of the Monster to be properly understood.  The quest goes on, and hopefully, through physics this mystery can be solved.

Friday, April 03, 2020

Eric Weinstein’s Geometric Unity

At long last, Eric Weinstein has released footage of his 2013 talk at Oxford on his candidate unified theory—geometric unity.  The talk was hosted by Marcus du Sautoy.  Enjoy!

Tuesday, May 14, 2019

Goro Shimura and M-theory

The great Goro Shimura passed on May 3 in Princeton, at age 89.  More than a number theorist, Shimura's mathematics also tied into the geometry of hermitian symmetric domains, via the Shimura variety.  The full implications of his research takes us into the cutting edge of the study of M-theory.

In pure mathematics, Shimura's work was pivotal in setting up Andrew Wiles' proof of Fermat's last theorem.  What was actually proven by Wiles is the Taniyama-Shimura conjecture (the modularity theorem), which as a corollary gives the treasured proof of Fermat's famous assertion.

The power of Shimura's work is amplified when combined with the work of Grothendieck.  Stay tuned.

Monday, September 24, 2018

Atiyah's proposed proof of the Riemann Hypothesis


After much anticipation, M. Atiyah unveils his proposed proof of the Riemann hypothesis.  It depends on an interesting function called the Todd function.  His approach of course still must be properly evaluated and time will reveal if all assumptions are sensible.

Update: Sir M. Atiyah has passed on January 11, 2019.  He leaves a brilliant mathematical legacy behind, and a spirit of inquiry even E. Witten admired.

Wednesday, March 14, 2018

Hawking's Legacy

On this 14th day of March, 2018, the world honors Stephen Hawking as an extraordinary scientist, teacher and futurist.  From A Brief History of Time to The Universe in a Nutshell, it is clear Hawking's deepest fascination has been the quest for a unified theory of physics.  In particular, Hawking was anticipating the completion of M-theory.

When can we expect the completion of M-theory?  Is it really an 11-dimensional completion of 11D supergravity and 10D superstring theory?  Or is it much more?  Ed Witten once mused, "String theory is 21st century physics that fell accidentally into the 20th century."  So the joke continues, in that 22nd century mathematics is needed to solve M-theory.  Might M stand for motive?  Time will tell.


Wednesday, November 22, 2017

Exceptional Periodicity

As was hinted at in a previous post, it is possible to view the exceptional Lie algebras as the tip of an infinite algebraic spectrum.  This novel concept, we coined Exceptional Periodicity (EP), is now available for download on the arXiv: arXiv:1711.07881 [hep-th].

This EP structure was inspired by certain Yang-Mills-like gradings of the exceptional Lie algebras, as well as higher dimensional spin groups, used in approaches to unification.  It differs from the conventional infinite dimensional generalizations of e8 in that the Jacobi identity is not in general obeyed by these higher algebras, yet do retain structure similar to lattice vertex algebras.  Moreover, building on the "Magic Star" projection of e8, each of these higher algebras can be projected to higher Magic Stars, that generalize that of e8.  At the six inner vertices of the star, the cubic Jordan algebras are generalized to a cubic ternary algebra, first envisioned by Vinberg, dubbed T-algebras.  Such T-algebras are reminiscent of spin factors and Peirce decompositions of cubic Jordan algebras.

The e8 Magic Star thus encodes the exceptional Jordan algebra on its six star vertices, which exhibits triality, from its off-diagonal 8D components.  These higher stars do not retain this triality, as the bosonic off-diagonal parts do not grow as fast as the spinor part, which grows exponentially.

So what can be done with these higher EP Magic Stars?  The T-algebras appear to encode a rich matter sector, that generalize the 16, 32, 64 and 128 spinors found in the exceptional Lie algebras.  Such higher stars can be used to design higher mathematical universes, in a periodic, algebraic fashion.  More details will be given in a series of upcoming papers.  Stay tuned.