SOLVING FERMAT'S ENIGMA: THE APPROACH

3/1/2019

Hitler's ascent to power in 1933 marked a profound change in the balance of power in mathematics. Not only did German-Jewish mathematicians escape to Britain and America, but a revival of French mathematics began under the influence of young scholars who had gone to Germany to study. Lamenting the lack of suitable modern textbooks, they created their own using the pseudonym Nicolas Bourbaki, the name of a French General in the Franco-Prussian War.

The Greek origin of the name suited the intended style of a multi-volume work under the heading 'Elements of Mathematics', inspired by Euclid's previous ancient work. Following their first meeting in 1934, the Bourbaki group's influence went international, with new volumes of 'Elements' published over 50 years. After WWII, they produced a series of advanced monographs by individual authors based on public lectures given from the Sorbonne in Paris, the new emerging center for mathematics.

During this time a brilliant young boy named Alexander Grothendieck (pronounced Groth-then-deck), born in Berlin in 1928, was hidden on a farm in northern Germany. His father, Alexander Schapiro, a Russian anarchist, had taken part in the failed 1905 Russian Revolution. He lost an arm escaping from prison ten years later; leaving Russia on forged papers, he reached Berlin in 1922. When the Nazi's took power in 1933 Schapiro escaped to France and was later joined by Alexander's mother, Hanka Grothendieck. Both parents joined the effort that became the Spanish Civil War. After returning to France in 1939, Schapiro was interned in Camp Vernet in the French Pyrenees, before the Vichy regime handed him over to the Nazi's upon disappearing into Auschwitz. That year, Alexander's mother decided to leave Germany, putting her son Alexander on a train to France, where he would later join his mother, living in various camps for displaced persons.

After the war, Alexander Grothendieck became a student at Montpellier. There his professor told him that a previous protege named Henri Lebesgue, a French mathematician who died in 1941, had already solved all the problems of mathematics but his ideas and methods were too difficult to teach. Undeterred, Alexander Grothendieck rediscovered much of Lebesgue's great work. He achieved his first mathematical success in isolation, honestly believing that he was the only mathematician in the world.

Alexander was a phenomenon, unique by any standard. After writing a dense and profound article that fueled the research of a whole school of scholarly thought, he abandoned the discipline altogether. He late said of himself that he was destined to be the builder of houses he would never inhabit. Aged 27, he started a new chapter in his life, turning the subject of algebraic geometry into something far grander.

Roughly speaking, algebraic geometry studies curves and surfaces representing solutions to algebraic equations. For example, solutions to the equation X+Y=Z determine a surface in three dimensional space framed by the coordinates x,y,z. This particular equation inspired Fermat's last theorem, which states that when (n, meaning any number) is greater than 2 there are no non-trivial solutions for which x,y,z are whole numbers. In other works, the surface defined by that equation contains no such points.

Fermat's famous conjecture, remained unresolved for more then 350 years until Alexander Grothendieck tackled it. Its resolution is extraordinarily difficult: direct number-theoretic approaches never succeeded, nor did direct geometric approaches. Its solution finally came from a viewpoint at a far higher level of abstraction.

His experiences, hidden in Germany for years, then escaping to France, losing a father who battled Tsarist Russia and the Communists and who was finally murdered by the Nazi's, gave Alexander a yearning for extreme abstraction. He would have nothing to do with physics, nor with any kind of military support for mathematics. When he resided at the infamous Institue des Hautes Etudes Scientifiques in Paris he discovered that some of his work was defense related he abandoned that brilliant research center.

In 1970, he went to Montpellier, when he had once been a student. It was the beginning of the end of his mathematical work, and after retirement from academia he went to live in the French Pyrenees, not far from the internment camp where his father had lived before deportation to Auschwitz. Grothendieck's address and telephone number were known only to a select few, sworn to secrecy.

In happier days, the Bourbaki group had been ready and willing to help Alexander. In particular, Jean Dieudonne and another mathematician continued to encourage him to write and tackle contemporary mathematical problems. Alexander began his magnum opus containing all his life's work. With the help of his 12 personal disciples, Alexander Grothendieck's magnum opus on algebraic geometry spanned ten thousand pages.

Like his mathematical predecessors, Gauss and Riemann, and the physicist Einstein, Grothendieck was fascinated by the concept of space. For him, the key ingredient to his work remained the concept of a 'point', something resembling Euclid's notion of something having no dimension. He drew algebraic geometry into a broader context embracing not just curves and surfaces but much of number theory, creating highly sophisticated mathematical tools to handle his specific abstract terrain.

It departed from the usual more concrete concerns about equations and their geometric representations. Eventually Grothendieck himself departed from mundane concerns altogether. He abandoned his best disciples, his five children from three different mothers, and reached for ever deeper abstraction. This man, who eschewed war, gave mathematical seminars to North Vietnamese during U.S. bombing raids! He fought long custody battles for his eldest son and a later court case involving an enormous home he built on his mothers property.

He would finally return to his Montpellier home, retreat under the Pyrenees and write theology.

Alexander Grothendieck, died in November 2014 at the age of 86.

His commentaries along with his magnum opus reveal a contemporary Pythagoras, known to those who can understand him.

NOTE: Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety.