The lives of Alexander Groathendieck mathematical visionary

By Edward Frenkel

Nov 25 2014.

The New York Times

Alexander Grothendieck, who died on Nov. 13 at the age of 86, was a visionary who captivated the collective psyche of his peers like no one else. To say he was the No. 1 mathematician of the second half of the 20th century cannot begin to do justice to him or his body of work. Let’s resist the temptation to assign a number to a man of numbers. There are deeper lessons to be learned from this extraordinary human being and his extraordinary life.

In mathematics, he revolutionized the field known as algebraic geometry. Ever since Descartes, we have known that geometric shapes can be described by equations. When we write x^{2}+y^{2}=1, we wish into existence a perfect circle. Indeed, each solution of this equation is nothing but a pair of coordinates, x and y, of a point of the unit circle on a plane.

This is an example of an algebraic equation, one that involves only products of powers of coordinates, such as

x^{2} or x^{3}y^{5}. Since the number of coordinates can be arbitrarily large, such equations may be quite daunting. But they are fundamental, and many can be found in nature. Algebraic geometry is about them and the geometric shapes, or spaces, they describe.

Right away, one encounters a problem. The above equation gives rise to a circle only if we consider the solutions in the domain of real numbers. But there are many other domains, such as the complex numbers (which involve an imaginary number, the square root of minus 1).

One can show that the solutions of the same equation in complex numbers are points of an entirely different space; namely, a plane with one point removed. For another domain, the space of solutions could be a family of circles of different sizes: Visualize a living and breathing circle evolving in time.

Thus, for a given equation we get a whole zoo of spaces. How are they related to one another and to the equation itself? Which came first, the equation or the space? These questions had perplexed mathematicians for centuries.

Grothendieck’s genius was to recognize that there is a “being” hiding behind a given algebraic equation (or a system of equations) called a scheme. The spaces of solutions are mere projections, or shadows of this scheme. Moreover, he realized that these schemes inhabit a rich world. They “interact” with one another, can be “glued” together and so on.

The concept of a scheme was one of the cornerstones in the gargantuan effort led by Grothendieck to rebuild this vast subject. The thousands of pages meticulously composed over a decade starting in the late 1950s became commonly known as EGA and SGA, the abbreviations of their French titles. Monumental like Euclid’s “Elements,” they have not been surpassed to this day in clarity, generality, technical mastery and conceptual perfection.

They are the fruits of endless discussions, 12-hour seminars, solitary thinking — of work, in a word, for that’s what it takes: the obsessive, sustained search for truth in its most universal and abstract form. With no compromises, ever.

As Pierre Deligne, a former student of Grothendieck’s and himself a mathematical maestro, put it in Le Monde, Grothendieck “had to understand things from the most general possible point of view,” and once he achieved that, everything “became so clear that proofs seemed almost trivial.” Perhaps that’s why Grothendieck’s ideas “penetrated the unconscious of mathematicians.”

Though one might ask if there are any real-world applications of his work, the more important question is whether having found applications, we also find the wisdom to protect the world from the monsters we create using these applications. Alas, the recent misuse of mathematics does not give us much comfort.

For example, according to published reports, the National Security Agency inserted a back door in a widely used encryption algorithm based on “elliptic curves” — mathematical objects illuminated by Grothendieck’s research. Though that specific algorithm was developed much later, Grothendieck recognized the potential dangers of such misuse of math and sounded the alarm. He was incensed when he learned that IHES, the mathematics institute near Paris where he worked, received funding from the French Ministry of Defense. In protest, he resigned from the institute in 1970 at the height of his power. He had hoped that his colleagues would follow him, but none did.

So began his estrangement from the academic community, which he lambasted as lacking ethics and integrity. He refused to go to Moscow in 1966 to collect his Fields Medal, the highest award in mathematics, to protest prosecution of dissidents in the Soviet Union. He declined the prestigious Crawford Prize in 1988, calling the scientific world “fundamentally unhealthy.” He devoted himself to ecological issues long before it became fashionable, helping to found the group Survivre et Vivre, “an international movement for the survival of humanity,” in 1970.

Reading the group’s newsletter, one can see Grothendieck confronting the world’s ills with his signature rigor and passion. He fought against the injustice he saw, accepting no compromises.

A party of one, he was unafraid to be himself and to speak his truth. The man who had advanced mathematics in the most profound ways did not believe that math was the answer to everything. He taught us that life is more valuable than any equation.

Edward Frenkel, a professor of mathematics at University of California, Berkeley, is the author of “Love and Math: The Heart of Hidden Reality.”

I will write “what is analysis and why do analysis” after this…

More later,

Nalin Pithwa

Alexander Grothendieck passes away at 86 — one of the greatest mathematicians of the 20th century

Alexander Grothendieck died November 13 2014 at the age of 86. His extraordinary mathematical vision, which brought an entirely new viewpoint to algebraic geometry in the 20th century, remains a fertile source of inspiration today. He was a highly original thinker whose later non-mathematical writings exhibit radical and penetrating insights about life and spirituality. Born in Berlin in 1928, he endured dislocation and trauma during World War II, including the death of his father in Auschwitz. After the end of the war he made his way to Paris and there discovered others who shared his passion for mathematics. He earned his PhD at the Université de Nancy in 1953 and in 1958 was appointed professor at the newly created Institut des Hautes Études Scientifiques just outside Paris. There he spent his most fruitful mathematical years, becoming the epicenter of a revolution that attracted leading mathematicians of the day and many of the most outstanding students. He was awarded the Fields Medal in 1966 and was also for a time a member of the Bourbaki group. In 1970 he abruptly left the IHÉS and began to work on issues of ecology and anti-militarization. Grothendieck joined the faculty at the University of Montpellier in 1972 and retired in 1990. Not long thereafter he essentially became a hermit, living in the French Pyrenees, where he devoted himself to thinking and writing.

Winfried Scharlau has written a 2-part biography of Grothendieck, in German. The first volume has been translated into English, and a translation of the second is under way. See this web page for more information. Alexandre Grothendieck: A Mathematical Portrait, containing pieces by various contributors and edited by Leila Schneps, appeared this year.

Scharlau’s article “Who is Alexander Grothendieck?” appeared in the AMS Notices (September 2008). Also in the Notices: “Comme Appelé du Néant: As If Summoned From the Void,” by Allyn Jackson, part 1 and part 2, which appeared in 2004; and “Reminiscences of Grothendieck and His School,” by Luc Illusie, with Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, et al.

From Grothendieck’s monumental work, Récoltes et Semailles (English translation follows):

Et toute science, quand nous l’entendons non comme un instrument de pouvoir et de domination, mais comme aventure de connaissance de notre espèce à travers les âges, n’est autre chose que cette harmonie, plus ou moins vaste et plus ou moins riche d’une époque à l’autre, qui se déploie au cours des générations et des siècles, par le délicat contrepoint de tous les thèmes apparus tour à tour, comme appelés du néant.

And every science, when we understand it not as an instrument of power and domination but as an adventure in knowledge pursued by our species across the ages, is nothing but this harmony, more or less vast, more or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the delicate counterpoint of all the themes appearing in turn, as if summoned from the void.

– See more at: http://www.ams.org/news?news_id=2461#sthash.5qvjwKoT.dpuf

(the above article is from the american mathematical society website)

Regards,

Nalin Pithwa