The average person keeping abreast of current events is aware that physicists have been striving mightily to discover the keys to a Grand Unified Theory — a coming together of all the forces and facets of the universe. This unification, it seems, was intact at the moment of creation as the Big Bang burst forth 14 billion years ago.
After a series of “symmetry breaking” events in the distant past, the altered reality we perceive at present having resulted in a curtain drawn over a truer understanding of the deeper and intrinsic correlations between electricity, magnetism, gravity, matter and everything else.
The greatest scientific talent on Earth spends many working days at the European for Nuclear Research (CERN) attempting to achieve this breakthrough, but it’s far less widely known that something akin to the same sort of campaign has been waged in the field of mathematics since the 1960’s.
Robert Langlands is the force behind this enterprise.
He is professor emeritus at the Institute of Advanced Studies at Princeton University and founder of the Langlands Program. Many preeminent mathematical authorities deem his efforts as worthy of note; he was awarded the immensely prestigious Abel Prize in 2018, along with the attendant three-quarter million dollar cash prize.
Dr. Langlands and others point to an astounding property of the various and myriad fields of mathematics, seemingly as different from one another as plus is to minus, but upon the closest inspection profound connections faintly take form. A recent great achievement in math took place in 1993 when Andrew Wiles solved the centuries-old enigma of Fermat’s Last Theorem.
What is as amazing as the triumph itself is the manner in which Wiles cracked this puzzle having to do with Pythagorean triples — a2 + b2 = c2 — but with exponents higher than squares, raised to the powers of three, four, five, etc.
The solution involved attacking the problem from an area of mathematics on the other side of the intellectual universe, a field known as modular forms. In fact, another similar pairing between these same modular forms and a group of mathematical statements known as elliptic equations was proved along the way as a first step — evidencing encapsulated connections within connections, not unlike Russian nesting dolls that fit unseen one inside the other.
It bears repeating that none of those ties would have appeared the least likely to have anything to do with each other. To ascertain that they plainly do should be as stunning as to discover that marshmallows and supernovas share an inherent bond.
Yet, these arcane pairings are popping up all over the mathematical landscape as researchers are attempting to match braid twists, clock arithmetic, prime numbers, sequences and series, topology, complex numbers and other supposed island galaxies in mathematics which seem nonetheless to be very much part of the same cosmos.
These links aren’t just interesting doodles for geniuses; their functional applications have proven quite practical in cryptography, the fields of DNA and enzymes, and quite plausibly could wind up being the linchpin in civilization’s next great leap forward.
The Langlands Program, however, has its work cut out, pushing directly against the ferociously powerful torrent of specialization that has been the watchword of the modern world for centuries.
As society becomes more and more sophisticated and advanced all the industrial, commercial, scientific and cultural underpinnings of an increasingly complex civilization demand experts in ever more shrinking fields of proficiency.
There is perhaps no discipline however more subject to fractured balkanization than mathematics; the diffusion into scattered fiefdoms is particularly intense, a fact that was highlighted as I recently met an esteemed mathematics professor in Cleveland while doing a booksigning there. When the subject of Wiles’ remarkable coup came up I couldn’t resist the opportunity to wheedle a free tutoring session with the expert at hand.
I explained that I’d made some sense of part of the solution but needed some assistance on the “modular forms” part of the answer. His reply was unexpected.
“Oh, I’m really not that versed in modular forms; you might know more about it than I do. My specialty is in bifurcation theory.”
My stunned look told him to continue so he did.
“You must realize, my colleagues, mathematics professors, all have about the same grounding in the subject up to first year graduate school level, but from there on it’s a very highly specialized discipline. All the sciences are that way, but mathematics takes that to extremes.”
Society might previously have intended as a compliment the term “jack of all trades” when it was first coined, but rather quickly the accolade morphed into a slight — with the “master of none” appended more often than not. Insofar as mathematics is concerned this is a proverb that may indeed merit another look.
David Nabhan is a science writer, the author of “Earthquake Prediction: Dawn of the New Seismology” (2017) and three previous books on earthquakes. Nabhan is also a science fiction writer (“Pilots of Borealis,” 2015) and the author of many scores of newspaper and magazine op-eds. Nabhan has been featured on television and talk radio all over the world. His website is www.earthquakepredictors.com. To read more of his reports — Click Here Now.
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