Around the same time as Sasha’s arrest in 1940, Alexander and his mother were moved to the Rieucros Camp near the city of Mende in the south-east of France as “undesirable dangerous foreigners” (Pragacz, 2004). At 13 yeras old Alexander managed to escape the camp, but was soon taken back. The two were separated when Hanka was later transferred to the Gurs internment camp in the Pyrenees Mountains. Alexander, then 13 years old, found shelter in a hostel for refugee children called Secours Suisse, routinely having to flee into the forrest during Gestapo raids:

“Quand on était averti par la police locale qu’il y aurait des rafles de la Gestapo, on allait se cacher dans les bois pour une nuit ou deux, par petits groupes de deux ou trois, sans trop nous rendre compte qu’il y allait bel et bien de notre peau“ — From Récoltes et Semailles (Grothendieck, 1983–86)

Alexander was able to continue his education at the school now known as Le Collège-Lycée Cévenol International, then run by local Protestants pacifists and anti-war activists in the mountain town of Le Chambon-sur-Lignon.

“Chaim Honig once asked Grothendieck why he had gone into mathematics. Gothendieck replied that he had two special passions, mathematics and piano, but he chose mathematics because he thought it would be easier to earn a living that way” (Jackson, 2004a)

Education (1942–53)

Le Collège-Lycée Cévenol International (1942–44)

Alexander first discovered his passion for mathematics in his early teens, while attending the Le Collège-Lycée Cévenol International in south-eastern France, stating in his later autobiography Récoltes et Semailles that he “learned from an inmate, Maria, the definition of the circle [the set of points located at the same distance from a point]. It had impressed me with its simplicity and its obviousness, whereas the perfect roundness of the circle appeared to me as a mysterious reality”.

Even while still only in his early teens, Grothendieck was notoriously curious about the nature of geometry. As retold by Piotr Pragacz (2004), he was barely thirteen years old when he questioned to himself “How do you accurately measure the length of a curve, area of a surface and volume of a solid?“

Université de Montpellier (1945–1948)

“In Grothendieck’s days, […] not a proper place for studying great mathematical problems” — Jean Dieudonné (1990)

By the time the war in Europe ended in 1945, Grothendieck was 17 years old. He was reunited with his mother Hanka and the two went to live in a village outside of Montellier, where he enrolled at Université de Montpellier. There, he was awarded a scholarship which him and his mother survived on, in addition to her doing housework and him doing seasonal work in the grape harvest of the local wine fields (Jackson, 2004a).

Grothendieck did not blossom in Montpellier, as he found the lectures at the university to be uninspired repetitions of textbooks. Rather than attend, he would instead devote most of the three years of his undergraduate studies catching up on what he had missed in high school. This included “providing a satisfactory definition of length, area and volume” (Jackson, 2004a), which in essence involved re-discovering for himself the result known as the Lebesgue measure, first proved in 1902. Regarding his time there, Grothendieck himself later wrote

“I learned then, in solitude, the thing that is essential in the art of mathematics — that which no master can really teach”

École Normale Supérieure (1948–1949)

After Montpellier, Grothendieck continued on to Paris, where he spent a year attending courses at the prestigious École Normale Supérieure, the foremost research institution for mathematics in France at the time. His teacher of calculus in Montpellier, Monsieur Soula, recommended Grothendieck travel there and make contact with his former professor Élie Cartan (1869–1951), the father of the Professor Henri Cartan (1904–2008) at École Normale Supérieure (ENS).

Before he left, Grothendieck applied for a scholarship which required an interview with a French education official. The official later recalled being “astounded” at his meeting with the young Grothendieck:

“Instead of a meeting of twenty minutes, he went on for two hours explaining to me how he had reconstructed, ‘with the tools available’, theories that had taken decades to construct. He showed an extraordinary sagacity” — André Magnier

The official noted that he immediately recommended Grothendieck for the scholarship. He also noted, however, that he had been left with an impression of “an extraordinary young man” who already then appeared “imbalanced by suffering and deprivation”.

Grothendieck arrived in Paris in the fall of 1948. He began attending lectures in a legendary seminar held by Henri Cartan on algebraic topology and sheaf theory. There, he encountered for the first time many of the most prominent French mathematicians of the day, including Chevalley, Delsarte, Dieudonné, Godement, Schwartz and Cartan’s student, later Fields Medal winner Jean-Pierre Serre (Jackson, 2004a). Early leader of the Bourbaki group André Weil (1906–1998) also attended the seminar. Grothendieck, who had never before mingled with first-rate mathematicians was an outsider in the group. Not only was he a German in post-war France, his background as an undergraduate student from the Université de Montpellier was also sharply in contrast to that of the other attendees, most of which had their educational background from the prestigious ENS.

Despite this, as Jackson (2004a) writes, Grothendieck later recalled not feeling intimidated, but rather receiving what he called a “benevolent welcome” (Grothendieck, 1986 pp. 19–20) where he felt free to ask questions, but also found himself “struggling to learn things that those around him seemed to grasp instantly […] like they had known them from the cradle”. The contrast lead Grothendieck to eventually leave Paris, in October of 1949 on the advice of Cartan and Weil who recommended he instead travel to Nancy to work with Schwartz and Dieudonné on functional analysis.

University of Nancy (1950–53)

Although not as prestigious as the ENS in Paris, Nancy was indeed still one of the most prominent centers of mathematics in post-war France, having been central in birthing the Bourbaki movement. Studying under Laurant Schwartz (1915–2002), in Nancy Grothendieck immersed himself in functional analysis. Starting in 1950, his early papers (written starting when he was 22 years old) posed important questions regarding the structure of locally convex linear topological spaces such as the complete linear metric spaces. Grothendieck worked so hard during this period that his supervisor Schwartz reportedly told another graduate student, Paulo Ribenboim, “You seem to be a nice, well-balanced young man; you should make friends with Grothendieck and do something so that he is not only working” (Jackson, 2004).

His efforts paid off. During his time in Nancy, Grothendieck published an astounding six papers before completing his Ph.D.:

All six papers regarded the theory of function spaces spearheaded twenty years before by Polish mathematician Stefan Banach (1892–1945).

Dieudonné and Schwartz were running a seminar in Nancy on topological vector spaces. As Dieudonné explained, by this time Banach spaces and their duality were well understood, but locally convex spaces had only recently been introduced, and a general theory for their duality had not yet been worked out. In working in this area, he and Schwartz had run into a series of problems, which they decided to turn over to Grothendieck. They were astonished when, some months later, he had solved every one of them and gone on to work on other questions in functional analysis. - Excerpt, "Comme Appelé du Néant - As If Summoned from the Void: The Life of Alexandre Grothendieck" by Allyn Jackson (2004a)

“When, in 1953, it was time to grant him a doctor’s degree, it was necessary to choose from six papers he had written, any one of which was at the level of a good dissertation” Dieudonné (1989) later wrote. For his Ph.D. thesis, dedicated to his mother Hanka, Grothendieck chose a 1952 paper which would later be published as:

In the paper, starting from the Schwarts kernel theorem, Grothendieck proposes the notion of a nuclear space, which was later shown to have wide applications. Nuclear spaces are topological vector spaces whose properties have much in common with finite-dimensional vector spaces. That is, the topology on nuclear spaces can be defined by a family of seminorms whose unit balls decrease rapidly in size.