Image: PXhere.com

The Intricate, Beautiful World of Math and Mathematicians

{NOTE: As Maryna Viazovska of Ukraine becomes only the second woman ever to win a Fields Medal — the Nobel of Mathematics — I recall this old post on why this award is special, and on the arcane and ‘irrationally interesting’ field of mathematics. Resharing here with much delight. Let the fun multiply by dividing it with other people :) }

Most of the recent headlines about winners of the Fields Medal in Mathematics have centered on how great this “Nobel of Math” is (which is given every four years), the episode of one of the medals being stolen right there at the awards venue or the Indian origin of another winner (Akshay Venkatesh) — but the mention of Fields Medal once again transported me into the arcane, intriguing world of Math and mathematicians.

While I know how frustrating and exhilarating Math can be at the same time (being a one-time Math aspirant myself), I first heard of Fields Medal when I happened to watch a YouTube video in which Prannoy Roy of NDTV interviewed an amusingly shy Manjul Bhargava — one of the four winners in 2014 (the International Mathematical Union or IMU, which awards these medals, has a thing for four, and you can read about it )

In that interview, Roy called Bhargava a “bewildering genius”. And when he read out the citation, that Bhargava won the award for “developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves”, he added a twist, “Simple as that, right? You got it? Simple stuff, yaar!”

The audience of students gathered to hear the interview burst out laughing.

The citations of the winners of this year’s Fields Medals are no less bewildering. To continue with our ‘Indian connection’, let’s look at the citation for Venkatesh. “For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects,” it says.

Now, that’s what is given as the ‘short citation’ on the Mathunion.org website. There is a long citation as well — and even that notes that this is “a small sample” of his major achievements. In that, I think the comparison with a Nobel prize (for literature at least) is understandable: the award is to recognize the body of work and, in the case of Fields Medal as stated on the site, “the promise of future achievement”.

Reading the above, I swing my head from left to right on what these young Mathematical geniuses — Fields Medal winners have to be below 40 — might unleash by way of achievement in the head-scratching years that lie ahead of them.

As you are still trying to wrap your head around homogeneous dynamics and elliptic curves, let me throw the citations of the other three winners of this year’s awards at you.

Caucher Birkar: “For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.”

Alessio Figalli: “For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.”

Peter Scholze: “For transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories.”

This last one is my favorite, with no disrespect or partiality to the others. If you are someone like me (a non-Math-genius), you might know why — just read the citation a couple of times and repeat after me:

p-adic fields.

Perfectoid spaces.

Galois representations.

And, ahem, cohomology theories.

All enmeshed together within the linguistic space of a single, English-sounding sentence.

Wow.

Each one of these cryptic phrases deserves a paragraph of its own, a separate chapter perhaps — maybe even a book. I don’t know. My own contribution, as you can see, has been the three-letter word: wow.

I mean, in other global awards, even if they are a bit technical in nature, one can somehow translate the achievement in words that a lot of people (if not most people) can understand. But how do you do that for Mathematical complexities that involve p-adic fields, Galois representations and cohomology theories?

Digging deeper, I did some Googling, and came up with some interesting episodes about this “most exact” of sciences and the creatures so absorbed in it — the mathematicians.

I found, for instance, that the Russian mathematician Grigory Perelman refused to accept the Fields Medal which was offered to him for solving one of the most intractable problems in Mathematics — the Poincare Conjecture. Not only that, Perelman also refused the sum of $1 million dollars offered by the Clay Mathematics Institute (CMI, claymath.org) as part of its offer of the money to anyone who could solve one of the seven most difficult problems that mathematicians have been struggling with at the turn of the millennium. Dubbed the Millennium Prize Problems, one of the seven is Poincare Conjecture; the other six remain unsolved and are known as Yang-Mills and Mass Gap, Riemann Hypothesis, P vs NP Problem, Navier-Stokes Equation, Hodge Conjecture and Birch and Swinnerton-Dyer Conjecture.

While I’m no liker of press releases, the one from CMI announcing the award in 2010 to Perelman does a remarkable job of simplifying what Poincare Conjecture is and how its proof in three dimensions eluded mathematicians for over a century. Curiously, two Fields Medals were awarded, one each to Stephen Smale (1966) and Michael Freedman (1986), for proving the Poincare Conjecture in five or more dimensions and in four dimensions respectively. (The branch of Mathematics concerned with the conjecture is called topology; those tenacious enough to go through the press release can view it )

But why did Perelman refuse those prizes? The question led me to, how should I put it, “the beautiful enigma” that Perelman turned out to be and, unfortunately, to the saga of an obnoxious battle for power in Mathematical circles (pun intended). For an engrossing account of this, read the longish New Yorker piece (for an appetizer, know that it is co-authored by Sylvia Nasar, who had earlier written mathematician John Nash’s biography, A Beautiful Mind, that was made into a film starring Russell Crowe).

Ordinary folks might need months or even years to understand some of the most pressing problems of Math or their significance, let alone solve them.

But then, mathematicians are no run-of-the-mill people.

Take Fermat’s last theorem, as another example, and the guy who cracked it — this one after more than three centuries!

Well, at least the theorem is relatively simpler to state. In number theory, according to Wikipedia, Fermat’s last theorem states that no three positive integers a, b and c satisfy the equation a(to the power n) + b(to the power n) = c(to the power n) for any integer value of n greater than 2.

Most of us (who paid some heed to trigonometry in middle school) can pick out the Pythagoras theorem from the above, for n=2, that is, a squared + b squared = c squared (true for a right angled triangle, where c is the hypotenuse).

I can almost hear some of you chuckle: Now you are talking Math!

But the guy who really, really talked Math here — tons of it, in fact — is Andrew Wiles, who proved Fermat’s last theorem in the mid-1990s. Too old at that time to be given a Fields Medal (the IMU did give him a silver plaque in recognition), he was awarded an equally distinguishing Math honor, the Abel Prize, in 2016 (what took the Norwegian Academy of Sciences and Letters so long to award Wiles is something I chose not to dwell on at the moment of writing this.)

When the seventeenth-century French mathematician Pierre de Fermat proposed this theorem, he is said to have scribbled a note in the margin of his book that claimed that the proof was “too long” to fit there.

One look at the size of Wiles’s proof and you would think that probably Fermat was right: a New Scientist article notes that Wiles’s version spanned “several hundred pages of cutting-edge 20th century Mathematics.”

The tale of more than three centuries of mathematical quest has been crafted wonderfully well by British author Simon Singh in the book Fermat’s Last Theorem. And oh, by the way, Singh himself happens to be a theoretical and particle physicist (I think I’m going to read that book one day.)

Guys, where are the girls? Don’t tell me there are no ladies of Mathematics, the same old stereotypical thing!

As far as the Fields Medal is concerned, only one woman has thus far been awarded — Maryam Mirzakhani, an Iran-born professor of Mathematics at Stanford University. Sadly, she died of breast cancer in 2017, at the age of 40.

But there is no dearth of “world famous women Mathematicians” as Google helpfully told me through this nicely illustrated scroll of honor. And I was happy to recognize a few names I was familiar with — Ada Lovelace and Grace Hopper among them (the latter and more names would appear if you clicked the right arrow in the Google results page).

Mathematics is indeed an enchanting science, mesmerizing enough for its most expert practitioners to be termed as Mathemagicians.

Fantastic and complex as its domains, theorems and conjectures may be, the other scientific and technological fields would not have made all those advances — giving us the modern marvels of living such as automobiles, genetic medicine, smartphones and airplanes — without progress in the Big M: Mathematics. And of course, without the labor of love of hundreds of mathematicians over millennia of human civilization.

I’m reminded of this famous quote by German mathematician Leopold Kronecker: “God made the integers; all else is the work of man.”

Originally published at https://www.linkedin.com.

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