Musings on science, history, philosophy and literature
Mathematics, And The Excellence Of The Life It Brings
Shing-Tung Yau and Eugenio Calabi
Mathematics and music have a pristine, otherworldly beauty that is very unlike that found in other human endeavors. Both of them seem to exhibit an internal structure, a unique concatenation of qualities that lives in a world of their own, independent of their creators. But mathematics might be so completely unique in this regard that its practitioners have seriously questioned whether mathematical facts, axioms and theorems may not simply exist on their own, simply waiting to be discovered rather than invented. Arthur Rubinstein and Andre Previn’s performance of Chopin’s second piano concerto sends unadulterated jolts of pleasure through my mind every time I listen to it, but I don’t for a moment doubt that those notes would not exist were it not for the existence of Chopin, Rubinstein and Previn. I am not sure I could say the same about Euler’s beautiful identity connecting three of the most fundamental constants in math and nature – e, pi and i. That succinct arrangement of symbols seems to simply be, waiting for Euler to chance upon it, the way a constellation of stars has waited for billions of years for an astronomer to find it.
The beauty of music and mathematics is that anyone can catch a glimpse of this timelessness of ideas, and even someone untrained in these fields can appreciate the basics. The most shattering intellectual moment of my life was when, in my last year of high school, I read in George Gamow’s “One, Two, Three, Infinity” about the fact that different infinities can actually be compared. Until then the whole concept of infinity had been a single concept to me, like the color red. The question of whether one infinity could be “larger” than another sounded as preposterous to me as whether one kind of red was better than another. But here was the story of an entire superstructure of infinities which could be compared, studied and taken apart, and whose very existence raised one of the most famous, and still unsolved, problems in math – the Continuum Hypothesis. The day I read about this fact in Gamow’s book, something changed in my mind; I got the feeling that some small combination of neuronal gears permanently shifted, altering forever a part of my perspective on the world.
Anyone who has seriously studied mathematics for any extended period of time also knows the complete immersion that can come with this study. In my second year of college I saw a copy of George F. Simmons’s book “Introduction to Topology and Modern Analysis” at the house of a mathematically gifted friend and asked to borrow it out of sheer curiosity. Until then mathematics had mainly been a matter of utilitarian value to me and most of my formal studies had been grounded in the kind of practical, calculus-based math that are required for solving problems in chemistry and physics. But Gamow’s exposition of countable and uncountable infinities had whetted my mind for more abstract stuff. The greatest strength of Simmons’s book is that it is entirely self-contained, starting with the bare basics of set theory and building up gradually. It’s also marvelously succinct, almost austere in the brevity of its proofs.
The book swept me off my feet, and the first time I started on it I worked through the theorems and problems right through the night; I can still see myself sitting at the table, the halo of a glaringly bright table lamp enclosing me in this special world of mathematical ideas, my grandmother sleeping outside this world in the small room that the two of us shared. The next night was not much different. After that I was seized by an intense desire to understand the fundamentals of topology – compactness, connectedness, metric and topological spaces, the Heine-Borel theorem, the whole works. Topology seemed to me like a cathedral – in fact the very word “spaces” as in “vector spaces” or “topological spaces” conjured up (and still do) an intricate, self-reinforcing cathedral of axioms, corollaries, lemmas and theorems resting on certain rules, each elegantly supporting the rest of it, being gradually built – or perhaps discovered – through the ages by its great practitioners, practitioners like Cantor, Riemann, Hilbert and Banach. It appeared like a great machine with perfectly enmeshed gears flawlessly fitting into each other and enabling great feats of mechanical efficiency and beauty. I was fortunate to find an enthusiastic professor who trained students for the mathematical olympiad, and he started spending several hours with me every week explaining proofs and helping me get over roadblocks. This was followed by many evenings of study and discussion, partly with a like-minded friend who had been inspired to get his own copy of Simmons’s book. I kept up the routine for several months and got as far as the Stone-Weierstrass theorem before other engagements intruded on my time – I wasn’t majoring in mathematics after all. But the intellectual experience had been memorable, unforgettable.
If even a lowly non-mathematician like myself could be so taken by the intricacies of higher mathematics, I can only dimly imagine the reveries experience by some of math’s greatest practitioners, one of whom is Shing-Tung Yau. Yau is a professor at Harvard and one of the world’s greatest mathematicians. His speciality is geometry and topology. Yau’s claim to fame is in bridging geometry and topology with differential equations, essentially founding the discipline of geometric analysis, although perhaps his greatest legacy would be forming novel, startling connections between physics and mathematics and opening up a dialogue that has had a long and often contentious history. For these efforts he won the Fields Medal in 1982, becoming the first mathematician of Chinese descent to do so.
The connection between algebra and geometry is an ancient one. In creating analytical or Cartesian geometry for instance, Rene Descartes had found a way to represent the elements of Euclidean geometry, entities like points and lines, as algebraic coordinates. This was a revolutionary discovery, allowing basic geometric entities like circles and ellipses to be described by algebraic equations. Analytical geometry lies at the very foundation of mathematics, enabling many other fields like multivariate calculus and linear algebra. The culmination of analytical geometry was in the field of differential geometry which uses techniques from algebra and calculus to describe geometric objects, especially curved ones.
The difference between geometry and topology is essentially that the former is about local entities while the latter is about global entities, about the big picture. Thus, in the context of the analogy often given to illustrate what topology is about, while a coffee cup and a donut are different geometric objects, they are identical topological objects because one can be converted into the other simply by stretching, expanding and contracting, without having to tear or cut any part. Perhaps Yau’s most interesting contributions to differential topology would be something called a Calabi-Yau manifold. Loosely speaking, a manifold is a topological space whose every point is essentially “flat” or “locally Euclidean”. A good analogy is with ancient views of the Earth as flat contrasted with modern views of a round earth; the discrepancy arises from the fact that even the round earth is locally flat or Euclidean. Manifolds are not just interesting mathematically but are of great importance in physics and especially in general relativity. For instance, Einstein used the theory of Riemannian manifolds to deal with the curvature of spacetime. Calabi-Yau manifolds are special manifolds that gained importance when they were found to represent “hidden” dimensions in string theory. But this is merely one of Yau’s many seminal contributions to math over a long and fascinating life as described in his memoirs, “The Shape of A Life“.
That life started in China in the 1950s, during the cultural revolution. Yau was one of nine children. His parents were both very intelligent and highly committed to the education of their children. His father in particular was a scholarly role model for Yau. He was a professor who taught many disciplines, including languages and history. He was well versed in poetry and philosophy and always had a ready store of Taoisms and Confucian parables for his children. Shing-Tung’s parents lost most of their property during the revolution, and like many others migrated to Hong Kong where a better life was found. This better life was still very hard. Yau’s parents moved several times, and most houses he lived in were either overcrowded or in the wilderness, without running electricity and water, sometimes infested by snakes and other animals. School was several miles away and had to be reached through a combination of walks and public transportation. And yet it seems to have been a generally happy childhood, sustained by stories and playmates in the form of several brothers and sisters, of whom Yau was especially close to a particular older sister. The poverty and hardscrabble life also engendered a tremendous capacity for persistence and hard work in Yau. This capacity was particularly enhanced after Yau’s father tragically passed away from cancer when he was fourteen. Yau was devastated by his amazing father’s passing, and he resolved to apply the lessons this role model had imparted as diligently as possible. His mother was a tremendous influence, and she worked at odd jobs to support her large family. Later she moved to the United States with her son and had the pleasure of watching him become successful beyond her dreams.
While not particularly prodigious in mathematics in his early years, Yau started shining in high school. By a quirk of fate in which he did less than ideally in a national examination by spending time with a street gang, Yau gained admission to a school named Pui Ching that remarkably enough produced no less than one future Nobel laureate, three future U.S. National Medal of Science winners and eight future members of the U.S. National Academy of Sciences. This is an astonishing record for a fairly provincial school in Hong Kong, similar to records of future eminent scientists from the Bronx High School of Science of New York City. One factor that played into the school’s success as well as that of the Chinese University of Hong Kong which Yau attended for college was the presence of visiting American professors or native-born professors who had studied at American universities. One such professor named Stephen Salaff recommended Yau for graduate school at the University of California in Berkeley, and Yau’s career was launched. A decisive factor in Yau’s admission was a strong recommendation by S. S. Chern, Berkeley’s eminent geometer and perhaps the leading Chinese mathematician outside the United States then. Chern’s relationship with Yau looms large in the book, perhaps too large, and throughout his life Chern was both father figure and mentor to Yau as well as nemesis and adversary. Here Yau also met his wife Yu-Yun, an accomplished physicist; curiously enough, in deference to Chinese traditional culture, while he saw her during his first week in the library, he waited several years to ask her out before someone made a formal introduction. The two also lived apart for several years while Yau, uncommonly for a mathematician, bounced between many universities like Stanford, the Institute for Advanced Study in Princeton and UCSD before finally settling down at Harvard. Their two sons are successful in their own regard, one being a biochemist and the other a doctor.
After graduating Yau made a variety of significant contributions to differential geometry and geometric analysis. This included proving the Calabi conjecture which entails proving the existence of Riemannian metrics with certain properties on complex manifolds. This was a years-lone struggle emblematic of great mathematical achievements, and like great mathematical achievements it involved some blind detours, including Yau’s mistaken early results that seemed to indicate counterexamples to the conjecture. A particularly key contribution by Yau of great relevance to physics was to come up with a purely mathematical proof of the so-called positive mass conjecture. This conjecture, taken at face value as obvious by physicists for a long time, said that the mass of any physical isolated system from both matter and gravitation is positive. This includes our universe. To prove this, Yau and his collaborator Richard Schoen constructed an ingenious argument: they first proved that if the average curvature of the spacetime corresponding to such a system is positive, then the mass is also positive. They then constructed a spacetime with positive curvature that had the same mass as our universe. Put together, the two results which showcased a classic argument by analogy showed that the mass of our universe must also be positive.
Yau and Schoen’s results inaugurated a new era of interaction between physics and mathematics. This relationship although long and profound had often been fraught; David Hilbert once famously said when asked if the relationship was bad that it wasn’t bad because for it to be so the two groups have to talk to each other. Most of the breakthroughs in twentieth century physics were made using what we might call 19th century mathematics – calculus, differential equations and matrix theory. Yau and others’ work showed that there were still novel approaches from pure math based on topology and geometry that could contribute to advances in physics. Roger Penrose who was trained in the classical tradition of British mathematics imbibed these fields, and he was able to use insights from them to make groundbreaking contributions to general relativity.
This line of discovery especially took off when physicists working on string theory in the 1980s discovered that the hidden dimensions postulated by string theory could essentially be modeled as Calabi-Yau manifolds. This was indeed one of those happy circumstances where a purely mathematical discovery made out of intellectual curiosity could have deep ramifications for physics. There are also examples from string theory that have spurred developments in pure mathematics. There was again precedent for such unexpected relationships – for instance the theory of Lie groups turned out to have completely unexpected connections with particle physics – but Yau and others’ work showed the great value of pure curiosity-driven research in mathematics that could spark a robust back and forth with physics. One aspect of string theory that is missing from Yau’s account is the increasing criticism of the field as being unmoored from experiment or even from experimental prediction. But notwithstanding this valid criticism, it is clear that string theory provides a great example of how, just like mathematics has traditionally contributed to physics, discoveries in physics can play back into pure mathematics.
Along with straddling the worlds of math and physics, Yau has also straddled two others worlds – those of China and the United States. Although he grew up in Hong Kong, his parents’ strong Chinese roots made him feel very strong connections to his ancestral homeland. He visited China several times a year, handpicked Chinese students to study in the US and collaborated extensively with Chinese researchers; in fact almost two-thirds of his students and collaborators are Chinese, and in what was a harbinger of current times, the CIA asked him about his students multiple times before realizing that their work was too obscure and pure to impact national security – one of the unexpected ancillary perks of working in pure mathematics. He also criticizes the Chinese system as being too enamored of prizes, money and fame rather than the pure intellectual satisfaction that comes from pursuing science for its own goals. After he won the Fields Medal, Yau’s Chinese sojourns became high profile and they got him into many controversies involving funding, favoritism and committee work, many involving his former mentor Chern. At least a dozen personal controversies dot the narrative in the book, and while they make for fascinating reading because they demonstrate that even the most abstract of mathematics is not free from the very human qualities of personal jealousies, feuds, nepotism and claims of credit, methinks that Yau sometimes doth protest too much, especially since we only hear one side of the story and seldom the other side.
Perhaps the most significant controversy came about when Sylvia Nasar who wrote the book “A Beautiful Mind” wrote in a widely read article in the New Yorker that Yau had tried, through his students, to steal credit away from the famously reclusive Russian mathematician Grigori Perelman and his stunning, completely unexpected proof of the century-old Poincare Conjecture. It turned out that Perelman had not worked out all the details of his proof and had built on the very important work done by the American mathematician Richard Hamilton. Yau recognized that Perelman’s results would have been impossible without Hamilton’s work, and went out of his way to praise Hamilton. He also recruited two Chinese students to work out a mammoth, 300-page exposition of the proof that filled in some gaps. There is no doubt that the proof was Perelman’s, but Yau’s extensive maneuverings made it sound like he was undermining Perelman’s efforts. In this case, because of Perelman’s self-imposed isolation from the community, it is easy to think that Yau deserves the criticisms, but he makes his side of the story clear and one gets the feeling that Nasar exaggerated the feud. And in spite of all these controversies, Yau has sustained warm friendships with many leading mathematicians.
Shing-Tung Yau’s life has been wholly dedicated to mathematics and its advancement. He sees mathematics much like Newton saw all of natural science:
“After much tumult in my early years, I was able to find my way to the field of mathematics, which still has the power to sweep me off my feet like a surging river. I’ve had the opportunity to travel upon this river – at times even clearing an obstruction or two from a small tributary so that water can flow to new places that have never been accessed before. I plan to continue my explorations a bit more and then, perhaps, do some observing – or cheerleading – from the riverbanks, a few steps removed.”
A little boy on the shore, playing with shiny pebbles, while the great ocean of truth lies undiscovered before him, ready to be explored.