The Mathematician’s Shiva Read online

  CHAPTER 21

  From A Lifetime in Mathematics

  by Rachela Karnokovitch: Turbulence

  I am a mathematician and have been one since I was nine years old. I suppose such a statement implies that I am rational and cold, but anyone who makes that inference would be incorrect. Love certainly isn’t anything rational or cold and I am definitely capable of love. I also believe in miracles fervently, or at least in one miracle, the creation of life itself. There is random chance of course as well, and there is a strong tendency to interpret fortuity as being something miraculous. The war made me a mathematician. That’s not a miracle. It’s simply a response to a dramatic, life-changing event.

  Perhaps my hunger-induced somnambulence over that winter in the Arctic Circle had been beneficial in a way. For whatever reason, the summer’s daily bear meat awoke my mind almost magically. I felt so fresh and free mentally. My ability to visualize mathematics was heightened to such a degree that I felt I could understand anything. Grozslev, my tutor, sadly was gone, but he had shown me the language of mathematics over the year we studied together. He was a gifted man. Without him, I would have never been able to do anything in those early years. I wouldn’t have had the necessary formal entryway into my discipline and wouldn’t have known what had and hadn’t been solved before me. I would have had to use my own invented mathematical language, raw and ill-suited to making any real breakthroughs. I use the word “gifted” when describing my first mathematics tutor. I don’t use that word lightly. There are individuals, rare indeed, who possess a mammoth ability for invention.

  I don’t see the need for modesty at this time in my life, and perhaps I’ve never felt that need, as others who know me have said not infrequently. At one time, I possessed the gift of creation, just as at one time my first mentor possessed it. I had a unique vision, I knew that it was unique, and I could see the world I had invented with a clarity that allowed me to identify every detail. Like an old pianist whose fingers will no longer let him create magic in a recital, I can no longer achieve what I was once able to do, at least not with any consistency. No mathematician can be expected to do so past a certain age. But because I have experienced this magic, I can certainly mentor someone who possesses the ability to create something new and vital, just like Kolmogorov and Grozslev did for me.

  We, unlike all life that came before us, do have the ability to create and be Godlike, albeit in a minor way in comparison to God himself. Why not call these creations of ours miracles? They certainly feel miraculous to the creators. From out of nothing, a group of initially incoherent thoughts and images, something magically appears.

  Like Kolmogorov, Grozslev was interested in the great unknown in the world of physics and mathematics, turbulence. But by the 1930s, his time had come and gone for original thinking. He could describe the problem, he could cast about for a suitable language to attack the problem, and he could, like Kolmogorov, make some definitions. But to solve any major problem in mathematics or physics requires a fresh mind, one that sees only possibilities. A mathematician, even the best of the best, will fail far more often than he succeeds. Over time, those failures crowd out the possibility for future success.

  Feynman called turbulence the last great unknown in physics, and in terms of the mathematics of physical processes, it is also the last great unknown. When it is solved, one major canvas of mathematics will be complete. I’m sure minor work will continue in this area for centuries after, as it always does for mature fields.

  Grozslev introduced me to the field of turbulence in 1940. Somehow, in the godforsaken land near the Barents Sea, he was managing to work on defining the essence of fluid mechanics in a mathematical sense. He was even corresponding with Kolmogorov about this work. But these two were both mature mathematicians, and who with a fresh mind was Grozslev going to find to make headway? In desperation he showed me the progress he had made, and I worked under his mentorship. It was exciting to work on something completely new and uncharted. Some of this work was published by Kolmogorov in 1941. Neither Grozslev nor I were given authorship, but this was not Kolmogorov’s fault. The Soviets were not going to allow an Enemy of the People or the daughter of an Enemy of the People to receive recognition for their intellectual discoveries. Still, it made me wary, this omission, about letting anyone know about any future work of mine until it was complete.

  We like to think that our lives are ordered. It satisfies us to believe that there is cause and effect, that we can make corrections to our lives as easily as we change batteries in a radio. In fact, much of our life is chaotic. There are patterns, yes, but they are unpredictable. Very little can be improved. There are no simple batteries that can fix illnesses, wayward children, poverty, hurt feelings, war, and government calamities. In response, our minds become irrational and do their best to distort our actual world. We make our present and future seem more positive and less calamitous than they are in reality. Psychologists of today, I’m told, understand our tendency to look through rose-colored glasses. The great Proust understood this well before them when he noted, “To make reality endurable, we are all obliged to encourage in ourselves a few small foibles.”

  Similarly, to make the physical world around us endurable, we like to emphasize that the gases and liquids that move our oceans, allow our planes to fly, and make our boats sail follow predictable, orderly paths. Often they don’t, and this difference between our wish for an orderly universe and the reality of the calamity of the natural world makes us deny reality. While denial of our personal difficulties, and I know this better than Proust, can actually help us live our daily lives and give us the strength to survive under the worst conditions, our denial of the lack of order in the physical world creates havoc. Planes crash because of our false optimism. Boats turn over. As my son well knows, hurricanes destroy homes that we refuse to leave. The water and air around us can be destructive and we, more often than not, underestimate the magnitude of that destruction.

  But imagine a world where we could understand these destructive forces fully, where we weren’t either blindly scared of turbulence or, more commonly, blindly optimistic that we will avoid its effects. Understanding a physical phenomenon like turbulence ultimately means predicting its behavior, or at the very least understanding just what can and what cannot be predicted over time. Prediction implies quantification of the basic physical processes that drive turbulence. This is something we cannot currently do with any reasonable degree of sophistication.

  From the standpoint of a mathematician, not an engineer or a physicist, the problem of turbulence is fundamentally one of being certain, to prove, that there exists in the universe a set of descriptors, in this case partial differential equations, that can encompass the behavior of fluids as they move at velocities that cause chaos. We have, in fact, such an equation that we think should work. But we don’t know for sure that it does. It is a conjecture that the equation we use to describe the motion of fluids when they flow as lazily as rivers also describes the motion of fluids when they dance around an airplane wing.

  It is a reasonable conjecture. But we may be fooling ourselves when we think that what works for one state of fluid motion works for all states. Like Newton’s equations of motion, it may be that the equations we use, the Navier-Stokes equations, are simply approximations. If that is so, they have utility only in a practical sense. In a mathematical sense, they are dross.

  Even as a girl living in squalor and deprivation above the Arctic Circle, I was fascinated by the great unknown of turbulence. Fluids behave so predictably when they move slowly. But when they speed up, they change character completely. Predicting even the average behavior of millions of fluid particles, much less the path of one elemental particle, is essentially impossible at this point in time. Who, with an intellectual focus, doesn’t want to try to make what seem to be impossible problems tractable?

  Early on in my work with turbulence we defined a basic lexi
con. These are known today as Kolmogorov’s numbers, and given that I view their creation as something rather trivial, I have no interest in trying to obtain credit. I would, however, dearly love to have Grozslev’s name attached to them. The man was kind, gentle, and brilliant. The Soviets murdered him and managed to bury and ignore much of his intellectual achievement. I hope that Grozslev does, eventually, get his due with regard to this and much other work in mathematics.

  Unlike many mathematicians I know, my inspiration comes not simply from the equations that define the language of mathematics. No, I need to tie these equations to visual images. Sometimes that visualization is an abstract thing. A variable in an equation will enter my mind as a colorful object on a landscape and travel along a path in the hills and valleys or across the rivers that I have created for it. I will draw these images on paper and somehow the physical act of drawing opens up even more possibilities.

  But other times, and this is what happened to allow me to understand turbulence, the image is something real. One might think that these efforts at drawing—whether fanciful or realistic—are simply an entryway into understanding, but that wouldn’t be right. One does not have to use the language of mathematics to make mathematical proofs. One can use simple pictures sometimes. Perhaps an example is in order for you to understand just what mathematicians mean when they say they are solving a proof.

  For example, everyone has learned and many have actually used the Pythagorean Theorem, which states that for a right triangle with sides a and b, the length of the hypotenuse, c, can be described by the following relation:

  c2 = a2 + b2

  Suppose we wanted to prove that this relationship exists for all right triangles. A triangle consists, of course, of three sides. Let’s make each of those sides one side of a square:

  In order to compare the areas of the three squares, a2, b2, and c2 and whatever relationship might exist between them, let us look at the larger square defined by the sum of lengths a and b. Its interior and exterior can be drawn in many ways. Here are two:

  Visual inspection of the two sets of figures indicates that the interior square on the right is of length c and area c2. The area of the square of length a plus b is (a+b)2, which equals the area of the interior square, c2, plus the sum of four interior triangles of area 0.5×(ab), or 2ab. Therefore:

  a2 + b2 + 2ab = c2 + 2ab

  We have proven visually that indeed Pythagoras was right, if we had any doubt.

  Symbolically, mathematics was easier in the time of Pythagoras, although those who created the foundations of mathematics, like God creating the first cell, were likely far more brilliant than any mathematician I have known. I cannot make simple drawings to prove anything new. But they can still be an aid. Without a visual image, I simply cannot do my work. It was so when I was nine and learning my trade on my own. It is still so today.

  Two visual images that came to me back to back were the foundation of all my significant work on turbulence. At face value, they are trivial images. But that’s how it always works. Just like life has its origins in ordinary elements, the foundation for human creativity is ultimately quotidian.

  It all happened on one summer day. We were studying geography at the time and I suppose that our teacher wanted to make the maps of our lessons come to life. She asked all of us, “Who has seen the ocean?” No one raised a hand. Later I found out that my teacher had never seen the ocean herself. I had seen the Black Sea, of course, when we briefly lived in Odessa. But the ocean seemed to be of an unfathomable scale to my mind. It seemed improbable that I would ever be able to touch its waters anywhere, much less in the remote location to which we had been banished.

  Somehow Mrs. Sharekhova, the woman whose observant eyes were responsible for the strong foundation in mathematics given to me, managed to gather three military vehicles to take us to the coast. It was a distance of approximately twenty kilometers along a crude road.

  Many of us were excited, even giddy, on this trip. It was perhaps the first time any of us felt like we were real children, not adults of small size, since the war began. When the ocean came into view, I could not believe what my eyes took in. I did at first think I was seeing a mirage. You can talk about the vastness of the ocean, but until you actually see it, any discussion is inherently meaningless.

  We climbed out of the vehicles onto the sand. I reached down and felt its coarseness with my fingers. I had never touched anything like it, damp and full of organic debris from the sea. The air also was new, like nothing I had smelled before, a mix of sulfur, moisture, and salt. Mrs. Sharekhova led us on a walk to the ocean while the drivers stayed near the vehicles, smoking and chatting. For us, it was magical, truly, to spend this little time on a beach, something that seemed like a vacation. It was a wonderful respite from our squalor. The wind blew against my face as I looked out at the waves crashing in the distance. The sand was mixed with seaweed. Wind-weathered branches were scattered along the shore.

  We got closer and closer to the waves and I could feel the adrenaline surge in me as I looked at the water. It was so noisy and dramatic. I watched the waves swell to what seemed to be an impossible height and then crash perhaps ten meters offshore, their progress apparently impeded by an invisible sandbar.

  Our teacher walked with us to the edge of the water. I could sense that she felt brave doing this. I could also sense, or perhaps I’m projecting, the pride that she felt for finding the wherewithal and tenacity to take us all there.

  Mrs. Sharekhova reached down and felt the water against her fingertips. Most of us followed her lead. She took the salt water from her fingers to her lips and tasted it. I’ll never forget my own taste, so remarkable it was to sense the ocean this way. I watched the waves crash and crash again. This was turbulence, the movement of water ascending and curling, the cascades coming one after another. How can one possibly solve the problem of turbulence without actually seeing it first, not in a picture, but in real life in all its dimensions and scope? I would have made no progress at all that summer and fall had Mrs. Sharekhova not made the extraordinary effort to let us witness this wonder of nature.

  I don’t know how long we actually stayed there. Perhaps we walked, ever so carefully, along the shore for an hour. The weather was not pleasant, but we didn’t care. This visit was a true gift for us, a cherished freedom. We might have stayed longer, but the soldiers were getting so impatient that they shot their rifles into the air, and we knew it was time to go back. I cannot describe how wonderful I felt as we made that final walk.

  As we approached the vehicles, insects, which apparently didn’t like the salt water at all, returned to hover around our faces. They were as voracious as ever. I watched their motion intently, like I had never watched them before, because it was something fresh again and strangely unexpected. No, their paths were not, in any pure sense, turbulent. But these little creatures did not move independently. They were constrained, somehow, by the motion of their neighbors, who were in turn constrained by others more distant. Their dance was somewhat akin to the dance of fluid particles. Like the particles that made up the waves in the ocean, their ascent and descent required some information to transfer from one insect to the next. I, of course, did not like the presence of these biting bugs one bit, but I was learning something from their patterns of flight. The equations that could describe insect flight behavior were not the same as those that described fluid flow, but they were related. The horrible bites had a silver lining. If there had been no clouds of insects as we left the beach (and in our camp always), I likely wouldn’t have been able to start to truly understand turbulence. Insects and an ocean. I needed both.

  I drew pictures of what I saw at the ocean that day. I can still recall those images on paper from when I was young. They became the foundation of decades of work. At the age of eleven years, far from any semblance of civilization, while my father did all he could to keep us alive another day, I
began an intellectual quest to understand turbulence. More formally, I wished to prove the following: in three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes equations. It would be both the most difficult and most private achievement of my life.

  CHAPTER 22

  The Truth, Sort Of, Comes Out

  DAY 2

  “How does she do it?” Yakov asked, slurping his bowl of ukha at the kitchen table. I sat across from him. He looked like the happiest man on earth.

  “Recipes, I guess,” I said.

  “There are recipes and then there is talent. This woman has a feel for Russian food. Beautiful, too. I talked to her when she came this morning. She has a head on her shoulders. You sure she really is an American? Maybe she came here when she was very young.” Yakov again sounded hopeful, despite the long odds.

  “Definitely, she was born here. I think she was going for a Ph.D. in chemistry before she had kids. You in love?”

  “Maybe. There is not one woman in all of Nebraska like her. I know. I’ve already looked.” Yakov looked down at the soup with a baleful expression, as if the reflection from the bowl was playing a movie of his failed attempts at romance.

  “I don’t think you’ve looked that hard, Yakov. Not to diminish the allure of Jenny Rivkin.”