## 2014 Abel Prize

The Norwegian Academy of Science and Letters this morning announced the recipient of its 2014 Abel Prize, the twelfth one awarded. With mathematicians so rarely in the news, I have made it a point here at Ron’s View each year to write a post about the award. (Click on the following links for 2009, 2010, 2011, 2012, and 2013.) This year’s recipient is Yakov G. Sinai, a professor of mathematics at Princeton University and researcher at the Landau Institute of Theoretical Physics outside Moscow.

As I explain each year, the Abel Prize was established in 2001 by the Norwegian government to be the counterpart in mathematics to the Nobel Prizes in other disciplines. It has been awarded by the Norwegian Academy of Science and Letters each year since 2003 to one or two outstanding mathematicians and honors the great, early-nineteenth-century Norwegian mathematician Niels Abel.

Regarding Sinai, here is a passage from the press release:

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2014 to Yakov G. Sinai (78) of Princeton University, USA, and the Landau Institute for Theoretical Physics, Russian Academy of Sciences, “

for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics”.[snip]

Yakov Sinai is one of the most influential mathematicians of the twentieth century. He has achieved numerous groundbreaking results in the theory of dynamical systems, in mathematical physics and in probability theory. Many mathematical results are

named after him … .Sinai is highly respected in both physics and mathematics communities as the major architect of the most bridges connecting the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. During the past half-century Yakov Sinai has written more than 250 research papers and a number of books. He has supervised more than 50 Ph.D.-students.

Yakov Sinai has trained and influenced a generation of leading specialists in his research fields. Much of his research has become a standard toolbox for mathematical physicists. The Abel Committee says, “His works had and continue to have a broad and profound impact on mathematics and physics, as well as on the ever-fruitful interaction between these two fields.”

The Abel Prize website has a brochure on this year’s prize that is a bit more informative about Sinai’s work and career. Here are the first two paragraphs of an expanded description of his work.

Ever since the time of Newton, differential equations have been used by mathematicians, scientists and engineers to explain natural phenomena and to predict how they evolve. Many equations incorporate stochastic terms to model unknown, seemingly random, factors acting upon that evolution. The range of modern applications of deterministic and stochastic evolution equations encompasses such diverse issues as planetary motion, ocean currents, physiological cycles, population dynamics, and electrical networks, to name just a few. Some of these phenomena can be foreseen with great accuracy, while others seem to evolve in a chaotic, unpredictable way. Now it has become clear that order and chaos are intimately connected: we may find chaotic behavior in deterministic systems, and conversely, the statistical analysis of chaotic systems may lead to definite predictions.

Yakov Sinai made fundamental contributions in this broad domain, discovering surprising connections between order and chaos and developing the use of probability and measure theory in the study of dynamical systems. His achievements include seminal works in ergodic theory, which studies the tendency of a system to explore all of its available states according to certain time statistics; and statistical mechanics, which explores the behavior of systems composed of a very large number of particles, such as molecules in a gas.

See also the short article by Arne Sletsjøe (Norwegian mathematician and former Olympic sprint canoer) at the end of the brochure in which he explains what dynamical systems and entropy are, leading to a glimpse into Sinai’s contributions. I’ll close with an excerpt.

A dynamical system is a description of a physical system and its evolution over time. The system has many phases and all phases are represented in the phase space of the system. A path in the phase space describes the dynamics of the dynamical system.

A dynamical system may be deterministic. In a deterministic system no randomness is involved in the development of future states of the system. A swinging pendulum describes a deterministic system. Fixing the position and the speed, the laws of physics will determine the motion of the pendulum. When throwing a dice, we have the other extreme; a stochastic system. The future is completely uncertain, the last toss of the dice has no influence on the next.

In general, we can get a good overview of what happens in a dynamical system in the short term. However, when analyzed in the long term, dynamical systems are difficult to understand and predict. The problem of weather forecasting illustrates this phenomenon; the weather condition, described by air pressure, temperature, wind, humidity, etc. is a phase of a dynamical system. A weather forecast for the next ten minutes is much more reliable than a weather forecast for the next ten days.

Yakov Sinai was the first to come up with a mathematical foundation for quantifying

the complexity of a given dynamical system. Inspired by Shannon’s entropy in information theory, and in the framework of Kolmogorov’s Moscow seminar, Sinai introduced the concept of entropy for so-called measure-preserving dynamical systems, today known as Kolmogorov–Sinai-entropy. This entropy turned out to be a strong and far-reaching invariant of dynamical systems.

## Lee Lorch

Lee Lorch, mathematician and civil rights leader, has died. I don’t have much to add to what has been written elsewhere. The NYT has a lengthy obituary that I recommend. Some excerpts:

Lee Lorch, a soft-spoken mathematician whose leadership in the campaign to desegregate Stuyvesant Town, the gargantuan housing development on the east side of Manhattan, helped make housing discrimination illegal nationwide, died on Friday at a hospital in Toronto. He was 98.

His daughter, Alice Lorch Bartels, confirmed the death. Mr. Lorch had taught at York University in Toronto, and had lived in Toronto since 1968.

By helping to organize tenants in a newly-built housing complex — and then inviting a black family to live in his own apartment — Mr. Lorch played a crucial role in forcing the Metropolitan Life Insurance Company, which owned the development, to abandon its whites-only admissions policy. His campaign anticipated the sit-ins and other civil rights protests to come.

But Mr. Lorch’s lifelong agitation for racial equality, not just in New York but later in Tennessee and Arkansas, led him into a life of professional turmoil and, ultimately, exile.

[snip]

Mr. Lorch became vice chairman of a group of 12 tenants calling themselves the Town and Village Tenants Committee to End Discrimination in Stuyvesant Town.

“When you got into Stuyvesant Town, there was a serious moral dilemma,” he recalled in a 2010 interview with William Kelly of the Stuyvesant Town-Peter Cooper Village Video Project. “In the concentration camps of Nazi Germany, people had seen the end results of racism.”

Some 1,800 tenants eventually joined the group. “Stuyvesant Town is a grand old town; but you can’t get in if your skin is brown,” went one of its chants, wrote Charles V. Bagli of The New York Times in a book about Stuyvesant Town’s history. A group of 3,500 residents petitioned Mayor William O’Dwyer to help eliminate the “no Negroes allowed” policy, and supported anti-discrimination legislation before the City Council.

But Metropolitan Life held firm. And in early 1949, Mr. Lorch paid the price. Despite the backing of a majority of colleagues in his department, the appointments committee at City College blocked his promotion, effectively forcing him to leave.

[snip]

in September 1950, he accepted a new academic post, becoming one of two white professors at Fisk University, the historically black institution in Nashville, Tenn. His wife, a longtime activist herself — she had led the Boston School Committee in its effort to stop women from being fired as teachers the moment they married, as she had been — returned to Stuyvesant Town, where the Teamsters union supplied protection for protesting tenants.

In January 1952, as tenants barricaded themselves in their apartments and picketed outside City Hall and Metropolitan Life’s headquarters, the company compromised: Mr. Lorch and two other organizers would move out, but the Hendrixes got to stay.

Seven years later, only 47 blacks lived in Stuyvesant Town. But the frustration the campaign helped unleash culminated in the Fair Housing Act of 1968, which prohibited discrimination in the sale, rental, or financing of housing.

At Fisk, Mr. Lorch taught three of the first blacks ever to receive doctorates in mathematics. But there, too, his activism, like his attempt to enroll his daughter in an all-black school and refusal to answer questions before the House Un-American Activities Committee about his Communist ties, got him in trouble. In 1955, he was again let go.

One of those three students at Fisk who received doctorates in math is Gloria Hewitt, who came to my own department here at the University of Washington to study algebra and went on to a distinguished career at the University of Montana.

I heard Lorch speak many years ago. I can’t remember where. Perhaps at an American Math Society meeting. That’s the extent of my contact with him. The video embedded above is a film by Rachel Deutsch, produced by Science for Peace,, which “explores his experiences with: social justice, civil rights, de-segregation, communism, housing, boxing, music, activism, love, memories, change.” I’ve watched part of it. I look forward to seeing it all.

## Paul Sally

[Sharat Ganapati, The Chicago Maroon]

University of Chicago mathematician Paul Sally died last Monday at the age of 80. I got to spend time with him during my year in the department a few decades ago and have since been immensely fond of him. Aside from being a superb research mathematician, he was an inspiring teacher, a leader in mathematics education at all levels, one of the funniest people I’ve ever known, and a generous spirit.

From the university website:

Known for his contributions to the field of harmonic analysis and his passionate commitment to teaching, Prof. Paul J. Sally, Jr. built a legacy of love for mathematics at the University of Chicago for nearly 50 years. …

Sally taught at the University since 1965 and served as chairman of the mathematics department from 1977 to 1980. …

“Paul had a fierce belief in mathematics and in people,” wrote Professor Shmuel Weinberger, chair of mathematics, in a note to faculty. “I will miss him deeply.”

Sally’s impact in the classroom was legendary. He produced 19 PhD students and was director of Undergraduate Studies in the Mathematics Department for decades. He pioneered outreach programs in mathematics for elementary and secondary teachers and students. From 1983 to 1987, Sally served as the first director of the University of Chicago School Mathematics Project, home of the nation’s most widely used university-developed mathematics curriculum. In 1992, he founded Seminars for Elementary Specialists and Mathematics Educators (SESAME), a first-of-its-kind program for elementary school teachers from Chicago Public Schools.

Diane Herrmann, the co-director of Undergraduate Studies in Mathematics and a Senior Lecturer, described Sally as “a force of nature.” Herrmann worked with Sally as a teacher, mentor and then as a colleague for the past 30 years.

“He was passionately interested in mathematical education at all levels,” said Herrmann, who with Sally co-founded the Young Scholars Program, a groundbreaking enrichment program for mathematically talented seventh through 12th graders.

One student who benefitted from the Young Scholars Program starting in seventh grade was Maryanthe Malliaris, who is now an assistant professor in UChicago’s department of mathematics. She recalled the experience as “exhilarating” and “decisive for my future in mathematics.”

“He had an incredible psychological astuteness, and a forceful clarity,” Malliaris wrote in an email. “He devoted a great deal of his time to creating possibilities for others. He concerned himself with the field as a whole. He would be there on Saturdays, on evenings, in the summer. His door was always open. He would show by example what it is to be a great human being.”

David Vogan, AB, SM, ’74, another former student of Sally’s, went on to be a Professor of Mathematics at the Massachusetts Institute of Technology and president of the American Mathematical Society.

“What distinguished Paul Sally was not only his passion for mathematics, but also his love and care for everyone studying mathematics,” Vogan said. “He had an appreciation for all the different levels of mathematics. He was a remarkable individual who seemed to have an unlimited supply of energy.” …

“He overcame tremendous obstacles to provide education and outreach at the University, in the city of Chicago and nationally” said Robert Fefferman, the Max Mason Distinguished Service Professor in Mathematics. “He lost both legs and lost his eyesight to childhood diabetes and it did not stop him at all.”

A Boston native, Paul was a high school basketball star. From a Boston Globe 2007 article:

Sally has, by all accounts, had an unconventional journey to the upper pinnacles of the mathematical world. His early numerical life was built around 2’s and 3’s on the basketball court – he starred at Boston College High School – and it wasn’t until he was nearly 30 that he found his step in the mathematics.

“I was very late by the standards of this field,” he says. “Mathematicians are supposed to do their best work at 21. When I was that age, I was still dribbling a basketball down Dorchester Avenue.”

After graduating from BC, Sally ambled around town, driving a red cab in Brookline, loading furniture in Downtown Crossing, teaching at BC High, playing hoops in “every gym in the city,” until, in 1957, he claims he sneaked in the door with the first class of graduate students at Brandeis University. That’s where he met his wife, Judy Sally, who recently retired as a math professor at Northwestern University – “When you’re at Brandeis, and you meet an Irish lass named Judith Flanagan Donovan, it’s all over,” he says. After finishing his doctorate, he made his way to the University of Chicago. He was awarded tenure in 1969, and has been there ever since.

Though he’s technically a research mathematician – he’s done important work in reductive groups, an algebraic concept – Sally’s passion has always been standing at the blackboard. He loves his students and, by all accounts, they adore him.

“He’s unique because he’s this big powerful man, but his hallmark is that he nurtures people,” said Phil Kutzko, a math professor at the University of Iowa … . “The jokes are funny, but the reason his students and colleagues love him is that he’s been there for people.”

…

While his health has repeatedly betrayed him – the macular degeneration in his right eye is now so bad that he’s legally blind – Sally, whose accent retains a strong trace of Roslindale, says he has no plans of slowing down.

“I’ll keep teaching as long as I can find the blackboard,” he says.

One more quote, from a 2009 interview of Paul (PS) by Supriya Sinhababu (SS) in the university student newspaper:

PS: Six-three, 200, is the best you could be, let me tell you. Well I can’t say that—when I was a senior in college, I was about six-three, 185, and since then I’ve sort of put it on. A lot of it’s muscle. No, I don’t want to be anything but six-three. Now the other side of that is, I don’t want to be five-eight. This really angers five-eight people when I say that, but I don’t want to be five-eight! I’ve been six-three my whole life. As a matter of fact, when I had my second leg cut off, my surgeon and prosthetist got together and said, “Look, Paul, if we lower your center of gravity, you’ll have much more balance.” I said, “Are you kidding me?” They really thought they were going to shorten my height by about five or six inches. When you learn to exist and address the world at a certain height—and six-three is a very nice height to address the world from—you want to stay there.

About an hour after our conversation, I went back to Paul Sally’s office to ask him a final question.

SS: You told me that you’re 95-percent blind. I imagine this requires you to do an enormous amount of math in your head.

PS: I do.

SS: How do you do that?

PS: I’m one powerful son of a bitch.

That he was.

## The Good Old Days

[From AMS, courtesy of Alan Tucker]

I received my copy of the latest American Mathematical Monthly today. The Monthly is a publication of the Mathematical Association of America, which describes itself as “the largest professional society that focuses on mathematics accessible at the undergraduate level.” (They complement the American Mathematical Society, whose mission is to “further the interests of mathematical research and scholarship.”)

An article in the new Monthly caught my eye, Alan Tucker’s “The History of the Undergraduate Program in Mathematics in the United States.” This is not likely to interest you as much as me, so you may not be too disappointed to learn that for non-members it sits behind a pay wall, available at JSTOR for $12. Thus you’re likely to miss out on the following paragraph:

In the early 1950s faculty at many leading research departments still saw teaching as their primary mission. Even senior administrators often taught two courses per semester. When my father, A. W. Tucker, was chair of the Princeton mathematics department in the 1950s, not only did he have the same teaching load as other senior faculty, but every other semester he was also in charge of the freshman calculus course taken by almost all students. When I questioned him years later why he took on this huge extra obligation, he responded, “The most important thing that the Princeton Mathematics Department did was teach freshman calculus and so it was obvious that as chair, I should lead that effort.”

Just as well. I wouldn’t want you to get any crazy ideas.

(It may be useful to explain that at many large research universities, including mine, the math department chair has no teaching obligations.)

## Abraham Nemeth

[National Federation of the Blind]

Mathematician and Braille pioneer Abraham Nemeth died last Wednesday. From tomorrow’s NYT obit:

Abraham Nemeth, whose frustrations in pursuing an academic career in math prompted him to develop the Nemeth Code, a form of Braille that drastically improved the ability of visually impaired people to study complex mathematics, died on Wednesday at his home in Southfield, Mich. He was 94.

The cause was congestive heart failure, said his niece Dianne Bekritsky.

Blind since he was an infant, Dr. Nemeth grew up on the Lower East Side of Manhattan, the grandson of a kosher butcher. He was a bright child who taught himself to play the piano using Braille music books and was increasingly drawn to what he later called “the beauty of mathematics.”

Yet as his math skills increased, he found that Braille could take him only so far. It was too easy to confuse letters and numbers in certain situations and too cumbersome to constantly clarify. The more complicated math became, the more limited Braille became.

“There was no way of doing square roots, partial differentials, et cetera,” said Joyce Hull, who worked with Dr. Nemeth for many years, refining and writing manuals for his code. “That’s one of the reasons they said, ‘No, blind people can’t do math.’ ”

Dr. Nemeth knew that they could. Even as college advisers steered him in other directions — he earned his master’s in psychology from Columbia in 1942 — he began tinkering with the six-dot cell that is the foundation of Braille. By the late 1940s, while working in the shipping department of the American Foundation for the Blind (and playing piano in Brooklyn bars to make extra money), he had come up with a customized Braille code for math; he made symbols for the basics of addition and subtraction but also for the complexities of differential calculus. He even made a Braille slide rule.

He began informally sharing his new symbols with others, and the code quickly caught on. In 1950, he presented it to the American Joint Uniform Braille Committee. By the mid-1950s, the Nemeth Code had been adopted by national groups and incorporated into textbooks, providing him with a new career. In 1955, he was hired by the University of Detroit to teach math — to sighted students, using a chalkboard.

[snip]

Dr. Nemeth received his doctorate in mathematics from Wayne State University in Detroit. He began studying computer science in the 1960s and later started the university’s computer science program. He retired in 1985. For two years he served as the chairman of the Michigan Commission for the Blind.

Throughout his life, he dedicated much of his spare time to creating Braille versions of Jewish texts, including helping to proofread a Braille Hebrew Bible in the 1950s. He also helped develop MathSpeak, a method for communicating math orally.

Dr. Cary Supalo, a professor at Illinois State University who is blind and works to make science and science laboratories accessible to the blind, said Dr. Nemeth was revered among educators focused on the blind.

And from a National Federation of the Blind news release:

Dr. Marc Maurer, president of the National Federation of the Blind, said: “Dr. Nemeth had a great mind and a wonderful sense of humor. His invention of the Braille code that bears his name has enabled many blind people to learn, work, and excel in scientific, technology, engineering, and mathematics, and his tireless Braille advocacy work undoubtedly changed countless lives. He will be sorely missed and his contributions will be valued by generations to come.”

The news release also links to a 1991 interview with Nemeth, from which I’ve taken the photo above.

## Math Love

Perhaps you have already read Manil Suri’s op-ed piece How to Fall in Love With Math in today’s NYT. It’s currently ranked #1 in their list of most e-mailed articles, so it certainly has gotten a fair bit of attention. But if you missed it, follow the link and have a look.

Suri is both a successful mathematician and distinguished novelist. Not a common combination, but not unheard of either.* He maintains separate math and fiction websites.

**There’s Eric Temple Bell, a prominent American mathematician of the first half of the twentieth century who wrote influential works of math history and—under the pseudonym John Taine—was a pioneer in science fiction. Bell received his Master’s degree from the University of Washington and returned as a faculty member after receiving his PhD at Columbia, moving on to Caltech a few years later.*

Suri opens his op-ed with a tale familiar to mathematicians.

Each time I hear someone say, “Do the math,” I grit my teeth. Invariably a reference to something mundane like addition or multiplication, the phrase reinforces how little awareness there is about the breadth and scope of the subject, how so many people identify mathematics with just one element: arithmetic. Imagine, if you will, using, “Do the lit” as an exhortation to spell correctly.

As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.

We all have stories like this. Many, having asked what we do and learning that it’s math, are momentarily at a loss, but then may point out that they were never good at it. A few decades ago, University of Chicago mathematician Paul Sally told me his favorite reply, one that echoes Suri’s comment about doing the lit: “I was never good at reading.”

Occasionally I adopt Suri’s tack and endeavor to explain that there’s much more to math, but it generally doesn’t end well. Undeterred, Suri carries on:

Gaze at a sequence of regular polygons: a hexagon, an octagon, a decagon and so on. I can almost imagine a yoga instructor asking a class to meditate on what would happen if the number of sides kept increasing indefinitely. Eventually, the sides shrink so much that the kinks start flattening out and the perimeter begins to appear curved. And then you see it: what will emerge is a circle, while at the same time the polygon can never actually become one. The realization is exhilarating — it lights up pleasure centers in your brain. This underlying concept of a limit is one upon which all of calculus is built.

Suri concludes on an optimistic note:

Fortunately, today’s online world, with its advances in video and animation, offers several underused opportunities for the informal dissemination of mathematical ideas. Perhaps the most essential message to get across is that with math you can reach not just for the sky or the stars or the edges of the universe, but for timeless constellations of ideas that lie beyond.

Speaking of today’s online world, Vi Hart has achieved renown over the last couple of years with her youtube series of enticing mathematical videos. I have embedded one at the top.

## Book of the Week

I can’t pass up mention of a new book, Beyond the Quadratic Formula, which appeared for sale just three days ago at the eBooks Store of the Mathematical Association of America. I understand that a print version will be available in July. Here’s the blurb:

The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial’s coefficients can be used to obtain detailed information on its roots. A closing chapter offers glimpses into the theory of higher-degree polynomials, concluding with a proof of the fundamental theorem of algebra. The book also includes historical sections designed to reveal key discoveries in the study of polynomial equations as milestones in intellectual history across cultures.

Also, there are links to the Contents and the Preface.

Could be interesting. I might buy a copy or two.

## The Flamethrowers 2

I went to a great lecture earlier this month. Richard Tapia, a renowned mathematician at Rice University, spoke on “Math at top speed: exploring and breaking myths in the drag racing folklore.” The abstract:

In this talk the speaker will identify elementary mathematical frameworks for the study of old and new drag racing beliefs. In this manner some myths are validated, while others are destroyed. The first part of the talk will be a historical account of the development of drag racing and will include several lively videos and pictures depicting the speaker’s involvement in the early days of the sport.

It turns out that Tapia and his brother Bobby were drag racing pioneers half a century ago. Bobby would beat the great Art Arfons in a match race in 1959, set records in the 1960s, and be inducted into the National Hot Rod Association Hall of Fame in 2002. Richard would focus on math and receive honors of his own, including the 2010 National Medal of Science and election to the National Academy of Engineering (the first Hispanic so elected). He is a national leader in preparing women and underrepresented minorities for PhDs in science, math, and engineering. And, at heart, still a drag racer.

I didn’t grow up following drag racing, but I did follow the quest for the land speed record, which received lots of coverage in the 1960s. Arfons and Craig Breedlove were regularly in the news, with their latest efforts at the Bonneville Salt Flats in Utah.

Which brings me to the point of this post. I wrote a few days ago about having started Rachel Kushner’s new novel The Flamethrowers. I’m now just past the halfway point, and was pleasantly surprised to find that in Kushner’s tale, the narrator arrives at the salt flats to participate in some speed racing herself.

It’s the 1970s. The narrator has graduated from the University of Nevada, Reno, and moved to New York, where she has met an older artist who is a member of the Italian Valera family, maker of motorcycles and tires. I don’t want to describe too much of the plot. Suffice to say that there is a marvelous scene in which she has arrived in Utah in time to watch the Valera team prepare for its latest assault on the land speed record, with famed driver Didi Bombonato at the wheel.

With that as background, I can give an example of Kushner’s fabulous prose, a single paragraph in which our narrator describes Didi:

Each morning, I watched Didi out the window of the trailer as he put on his driving gloves and stretched his fingers, open and fisted, open and fisted, as if he were communicating some kind of cryptic message in units of ten. After his hand stretches, a crew member brought him a little thimble of espresso, which he took between deerskin-gloved finger and thumb, tilted his head back, and drank. He had pocked, sunken cheeks, thin bluish lips, and eyes like raisins, which made him seem angry and also a little dimwitted. Not everyone can be a great beauty, and I’m not exactly a conventional beauty myself. But there was a special tragedy to Didi’s looks: his hair, which was lustrous and full, feathered into elaborate croissant layers. Somehow the glamorous hair brought his homeliness into relief, like those dogs with hair like a woman’s. There was that advertisement on television where you saw a man and a woman from behind, racing along in an open car. The driver and his companion, her blond hair flying on the wind, the American freedom of a big convertible on the open highway, and so forth. The camera moves up alongside. The passenger, it turns out, is not a woman. It’s one of those dogs with long feathery hair, whatever breed that is. Didi’s breed. After drinking his espresso, Didi would flip his hair forward and then resettle it with his fingers, never mind that he was about to mash it under a helmet. It would have been better to skip the vanity and primping and instead use his face as a kind of dare, or weapon:

I’m ugly and famous and I drive a rocket-fueled cycle. I’m Didi Bombonato.

She can write. And the salt flats scene ends with a wonderful surprise, which I leave for you to discover when you read the novel.

## Mathematics in Ancient Iraq

I started Eleanor Robson’s 2008 Mathematics in Ancient Iraq: A Social History two days ago. Not my usual reading. I needed to find out what she had to say on certain issues, but the book was checked out of the library, so I decided, what the heck, I’ll just buy it and give it a try. Through the wonders of Amazon, I had a copy 48 hours later, and I’m glad.

From the book jacket:

This monumental book traces the origins and development of mathematics in the ancient Middle East, from its earliest beginnings in the fourth millennium BCE to the end of indigenous intellectual culture in the second century BCE when cuneiform writing was gradually abandoned. Eleanor Robson offers a history like no other, examining ancient mathematics within its broader social, political, economic, and religious contexts, and showing that mathematics was not just an abstract discipline for elites but a key component in ordering society and understanding the world.

The region of modern-day Iraq is uniquely rich in evidence for ancient mathematics because its prehistoric inhabitants wrote on clay tablets, many hundreds of thousands of which have been archaeologically excavated, deciphered, and translated. Drawing from these and a wealth of other textual and archaeological evidence, Robson gives an extraordinarily detailed picture of how mathematical ideas and practices were conceived, used, and taught during this period. She challenges the prevailing view that they were merely the simplistic precursors of classical Greek mathematics, and explains how the prevailing view came to be. Robson reveals the true sophistication and beauty of ancient Middle Eastern mathematics as it evolved over three thousand years, from the earliest beginnings of recorded accounting to complex mathematical astronomy.

The study of Babylonian mathematics is generally associated with Otto Neugebauer, an Austrian who began studying mathematics in the early 1920s in Göttingen, then one of the great centers of mathematics in the world. Soon his interests changed to history, and he wrote his doctoral thesis on Egyptian mathematics. The work on Babylonian mathematics that occupied him next led to the truly monumental three-volume work *Mathematische Keilschrift-Texte* (or *Mathematical Cuneiform Texts*), in which he analyzed cuneiform tablets from museums throughout the world, translating and interpreting their texts. Neugebauer’s work (partly in collaboration with Abraham Sachs; see their book *Mathematical Cuneiform Texts*) would have enormous influence on subsequent studies in the history of ancient mathematics and science. He eventually came to the US, where he worked at Brown University and then at the Institute for Advanced Study in Princeton, dying in 1990. (I must add to my ever growing list of missed opportunities that I succeeded in spending a year at the IAS, near the end of his life, without doing anything to take advantage of his presence.)

In recent years, some of Neugebauer’s analyses and conclusions have been re-examined, with new interpretations given for individual tablets and, more broadly, the purpose of the entire enterprise. This is what I have been learning about in the last week, most notably through the writings of Jens Høyrup and Eleanor Robson.

Unfortunately—missed opportunities again—I’m way too late to go back to New York and see an exhibition that was put on a little over two years ago at NYU’s Institute for the Study of the Ancient World, called Before Pythagoras: The Culture of Old Babylonian Mathematics. Here’s the website description of the show:

Since the nineteenth century, thousands of cuneiform tablets dating to the Old Babylonian Period (c. 1900-1700 BCE) have come to light at various sites in ancient Mesopotamia (modern Iraq). A significant number record mathematical tables, problems, and calculations. In the 1920s these tablets began to be systematically studied by Otto Neugebauer, who spent two decades transcribing and interpreting tablets housed in European and American museums. His labors, and those of his associates, rivals, and successors, have revealed a rich culture of mathematical practice and education that flourished more than a thousand years before the Greek sages Thales and Pythagoras with whom histories of mathematics used to begin.

This exhibition is the first to explore the world of Old Babylonian mathematics through cuneiform tablets covering the full spectrum of mathematical activity, from arithmetical tables copied out by young scribes-in-training to sophisticated work on topics that would now be classified as number theory and algebra. The pioneering research of Neugebauer and his contemporaries concentrated on the mathematical content of the advanced texts; a selection of archival manuscripts and correspondence offers a glimpse of Neugebauer’s research methods and his central role in this “heroic age.”

Edward Rothstein reviewed it in the NYT, writing that the institute

has gathered together a remarkable selection of Old Babylonian tablets from the collections of three universities — Columbia, Yale and the University of Pennsylvania — that cover a wide mathematical range. Made between 1900 and 1700 B.C., they include student exercises, word problems and calculation tables, as well as more abstract demonstrations. Under the curatorship of Alexander Jones, a professor at N.Y.U., and Christine Proust, a historian of mathematics, the tablets are used to give a quick survey of Babylonian mathematical enterprise, while also paying tribute to Neugebauer, the Austrian-born scholar who spent the last half of his career teaching at Brown University and almost single-handedly created a new discipline of study through his analysis of these neglected sources.

Only about 950 mathematically oriented tablets survived two millenniums of Babylonian history, and since their discovery, debate has raged over what they show us about that lost world. Every major history of Western mathematics written during the last 70 years has at least started to take Babylonians into account.

Rothstein even mentions Robson’s book:

In a fascinating 2008 book, “Mathematics in Ancient Iraq: A Social History” (Princeton), Eleanor Robson even suggests that many tablets like these of the second millennium B.C. “were essentially ephemera, created to aid and demonstrate recall, destined almost immediately for the recycling bin.”

But as Ms. Robson also points out, these tablets’ word problems about digging and construction, their use in teaching record keeping and calculation, and their implicit affirmation of the importance of scribes and teachers, also reveal a highly organized, bureaucratic society, an “ordered urban state, with god, king and scribe at its center.”

Among the tablets on view was Plimpton 322, the most famous of all the mathematical tablets. Here it is, as photographed by co-curator Christine Proust:

Why is it of such interest? From the website again:

Plimpton 322 reveals that the Babylonians discovered a method of finding Pythagorean triples, that is, sets of three whole numbers such that the square of one of them is the sum of the squares of the other two. By Pythagoras’ Theorem, a triangle whose three sides are proportional to a Pythagorean triple is a right-angled triangle. Right-angled triangles with sides proportional to the simplest Pythagorean triples turn up frequently in Babylonian problem texts; but if this tablet had not come to light, we would have had no reason to suspect that a general method capable of generating an unlimited number of distinct Pythagorean triples was known a millennium and a half before Euclid.

Plimpton 322 has excited much debate centering on two questions. First, what was the method by which the numbers in the table were calculated? And secondly, what were the purpose and the intellectual context of the tablet?

I sure wish I took an interest in this subject three years ago. I could have read Robson’s book, then arranged for us to be in New York during the exhibition.

Anyway, I’m reading the book now, and Robson concludes (I don’t usually jump to the end, but I did this time) with a compelling argument for taking an interest in the subject:

Compared to the difficulties of grappling with fragmentary and meagre

nth-generation sources from other ancient cultures the cuneiform evidence is concrete, immediate, and richly contextualised. We can often name and date individuals precisely; we have their autograph manuscripts, their libraries and household objects. This opens a unique window onto the material, social, and intellectual world of the mathematics of ancient Iraq that historians of other ancient cultures can only dream of.

## 2013 Abel Prize

[Cliff Moore]

The Norwegian Academy of Science and Letters this morning announced the recipient of its 2013 Abel Prize, the eleventh one awarded. With mathematicians so rarely in the news, I have made it a point here at Ron’s View each year to write a post about the award. (Click on the following links for 2009, 2010, 2011, and 2012.) This year’s recipient is Pierre Deligne, professor emeritus at the Institute for Advanced Study in Princeton.

As I explain each year, the Abel Prize was established in 2001 by the Norwegian government to be the counterpart in mathematics to the Nobel Prizes in other disciplines. It has been awarded by the Norwegian Academy of Science and Letters each year since 2003 to one or two outstanding mathematicians and honors the great, early-nineteenth-century Norwegian mathematician Niels Abel.

Regarding Deligne, here is a passage from the announcement of the award:

Pierre Deligne is a research mathematician who has excelled in finding connections between various fields of mathematics. His research has led to several important discoveries. Deligne’s best known achievement is his spectacular solution of the last and deepest of the Weil conjectures. This earned him both the Fields Medal (1978) and the Crafoord Prize (1988), the latter jointly with Alexandre Grothendieck.

Deligne’s brilliant proof of the Weil conjecture made him famous in the mathematical world at an early age. This first achievement was followed by several others that demonstrate the extreme variety as well as the difficulty of the techniques involved and the inventiveness of the methods. He is best known for his work in algebraic geometry and number theory, but he has also made major contributions to several other domains of mathematics.

The Abel Committee says: “Deligne’s powerful concepts, ideas, results and methods continue to influence the development of algebraic geometry, as well as mathematics as a whole”.

[snip]

Deligne was only 12 when he started to read his brother’s university math books. His interest prompted a high-school math teacher, J. Nijs, to lend him several volumes of “Éléments de mathématique” by Nicolas Bourbaki, the pseudonymous grey eminence of French mathematics. For the 14-year old Deligne this became a life changing experience. His father wanted him to become an engineer and to pursue a career that would afford him a good living. But Deligne knew early on that he should do what he loved, and what he loved was mathematics. He went to the University of Brussels with the ambition of becoming a high school teacher, and of pursuing mathematics as a hobby for his own personal enjoyment. There, as a student of Jacques Tits, Deligne was pleased to discover that, as he says, “one could earn one’s living by playing, i.e. by doing research in mathematics”.

Deligne announced the proof of the last and most difficult part of the Weil Conjectures when I was a graduate student. His work was well over my head, but everyone was talking about it. I had friends who were algebraic geometers, my advisor was himself a leading algebraic geometer, I was taking courses in the field. And one thing was clear, that Deligne was one of the great mathematical geniuses of our time. Just over a dozen years later, I would spend the year at the Institute, with Pierre a constant and inspiring presence. His selection enhances the prize as much as it honors him.