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Caleb Ji

Part 1: Upbringing (1928–1949) Elements of Grothendieck’s life and work

Part 2: (1949–1970) Caleb Ji Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Four stages of life

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom (1970–1991) Part 4: (a) 1928-1949 (b) 1949-1970 Hermitage (1991–2014)

(c) 1970-1991 (d) 1991-2014 Parents

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Figure: Sascha and Hanka

“I will take your wife away from you!” Sascha to Alf Raddatz upon seeing a picture of Hanka Childhood

Caleb Ji Left in with the Heydorns, his foster family at the age of 5 Part 1: Upbringing Reunited with parents in at 11 (1928–1949) Part 2: Father soon murdered in Auschwitz, sent to concentration Mathematics (1949–1970) camps with his mother to live in poverty

Part 3: Freedom Once ran away intending to assassinate Hitler (1970–1991)

Part 4: Hermitage (1991–2014) Childhood (quotations from R´ecolteset Semailles)

Caleb Ji “I don’t recall anyone ever being bored at school. There was Part 1: Upbringing the magic of numbers and the magic of words, signs and (1928–1949) sounds. And the magic of rhyme, in songs or little poems.” Part 2: Mathematics (1949–1970)

Part 3: Freedom “I became thoroughly absorbed in whatever interested me, to (1970–1991) the detriment of all else, without concerning myself with Part 4: Hermitage winning the appreciation of the teacher.” (1991–2014) Student at (1945–1948)

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: What I found most unsatisfactory in my mathematics Mathematics (1949–1970) textbooks was the absence of any serious attempt to tackle the

Part 3: meaning of the idea of the arc-length of a curve, or the area of Freedom (1970–1991) a surface or the volume of a solid. I resolved therefore to make

Part 4: up for this defect once I found time to do so. In fact I devoted Hermitage (1991–2014) most of my energy to this when I became a student at the University of Montpellier, between 1945 and 1948. [...] In applying myself to this problem at the age of 17 and fresh out of the lyc´ee,I believed that I could succeed in my objective in a matter of weeks. As it was, it preoccupied me fully three years. Arrival in Paris (1948)

Caleb Ji Received a scholarship to study in Paris thanks to Andr´e Part 1: Magnier Upbringing (1928–1949) Attended the famous Cartan seminar Part 2: Mathematics (1949–1970) Part 3: I discovered the existence of a mathematical world when I got Freedom (1970–1991) to Paris in 1948, at the age of 20, with not much more in my Part 4: thin suitcase than a bachelor’s degree in science and a Hermitage (1991–2014) closely-written manuscript using both sides of the paper and without margins (paper was expensive!), representing three years of solitary reflections on what (I later learned) was known as “measure theory” or “the Lebesgue integral”. Since I had never met another, I believed until the day I arrived in the capital, that I was the only person in the world to “do mathematics”, the only . Cartan Seminar

Caleb Ji “[] welcomed me with that benevolent courtesy Part 1: Upbringing that was typical of him [..] He probably didn’t even realize the (1928–1949) extent of my ignorance [...] In any case, his benevolence was Part 2: Mathematics obviously directed towards my person, not towards my (1949–1970) knowledge or possible gifts, nor (later) towards reputation or Part 3: Freedom fame.” (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Henri Cartan and Jean-Pierre Serre Bourbaki

Caleb Ji

Part 1: Upbringing “[...] the rest of the Bourbaki gang would arrive together, like a (1928–1949) noisy of friends: Dieudonne, Schwartz, Godement, Part 2: Mathematics Delsarte.” (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Andre Weil Figure: PhD in Nancy: (1949-1953)

Caleb Ji Worked with Jean Dieudonn´eand Part 1: Upbringing Solved all 14 of their problems on locally convex (1928–1949) topological vector spaces Part 2: Mathematics Develops the concept of nuclear spaces in his thesis, (1949–1970) proceeding to revolutionize functional analysis Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Jean Dieudonn´e Figure: Laurent Schwartz PhD in Nancy: functional analysis (1949-1953)

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Figure: PRODUITS TENSORIELS TOPOLOGIQUES ET ESPACES NUCLEAIRES,´ 336 p., Memoirs of the AMS

“When, in 1953, it was time to grant him a doctor’s degree, it was necessary to choose from among six papers he had written, any one of which was at the level of a good dissertation.” – Jean Dieudonn´e A new life in Brazil: 1953-1955

Caleb Ji

Part 1: Upbringing (1928–1949) Spent two years in S˘aoPaulo Part 2: Mathematics Subsisted on milk, bread, and bananas (1949–1970)

Part 3: Frightful family fiasco on trip to Paris Freedom (1970–1991) Tying up loose ends, beginning to work on Part 4: Hermitage (1991–2014) “So now I can finally leave the field of topological vector spaces with no regrets, and start seriously working on .” New work in Kansas: 1955

Visiting research associate professor at the University of Caleb Ji Kansas in 1955 Part 1: Upbringing Wrote “A General Theory of Fibre Spaces with Structure (1928–1949) ” Part 2: Mathematics (1949–1970) Wrote “Sur quelques points d’alg`ebre homologique” Part 3: Began correspondence with Jean-Pierre Serre Freedom (1970–1991)

Part 4: Hermitage (1991–2014) The Tˆohokupaper

Sur quelques points d’alg`ebre homologique, published in Caleb Ji Tˆohoku,1957 Part 1: Upbringing Grew from Grothendieck’s attempt to reconstruct the (1928–1949) Cartan-Eilenberg book on Part 2: Mathematics (1949–1970) Introduced abelian categories Part 3: Gave a general framework for through derived Freedom (1970–1991) functors, which worked for sheaves Part 4: Hermitage (1991–2014) Return to

Caleb Ji

Part 1: Upbringing (1928–1949) Obtains a position at the CNRS, moves back to Paris in

Part 2: 1956 Mathematics (1949–1970) Continued to work deeper in topology and algebraic Part 3: Freedom (1970–1991)

Part 4: Hermitage ““I was sure something first-rate would come out of him. But (1991–2014) then what came out was even much higher than I had expected. It was his version of Riemann-Roch, and that’s a fantastic theorem. This is really a masterpiece of mathematics.” – Grothendieck-Riemann-Roch

Caleb Ji

Part 1: Theorem (Riemann-Roch) Upbringing (1928–1949) For a compact Riemann surface with canonical divisor K and Part 2: Mathematics genus g, for any divisor D we have (1949–1970)

Part 3: Freedom l(D) − l(K − D) = deg(D) − g + 1. (1970–1991)

Part 4: Hermitage (1991–2014) Theorem (Grothendieck-Riemann-Roch) Let X,Y be locally of finite type over S. Let F be a finite locally free sheaf on X of rank r. Let f : X → Y be a proper smooth . Then • • ch(f!F )td(Y ) = f∗(ch(F )td(X )). Grothendieck-Riemann-Roch

Caleb Ji ”Witches Kitchen 1971. Riemann-Roch Theorem: The final Part 1: Upbringing cry: The diagram is commutative! To give an approximate (1928–1949) sense to the statement about f : X → Y , I had to abuse the Part 2: Mathematics listeners’ patience for almost two hours. Black on white (in (1949–1970) Springer lecture notes) it probably takes about 400, 500 pages. Part 3: Freedom A gripping example of how our thirst for knowledge and (1970–1991) discovery indulges itself more and more in a logical delirium far Part 4: Hermitage removed from life, while life itself is going to Hell in a thousand (1991–2014) ways and is under the threat of final extermination. High time to change our course!” 1957-1958: the beginning of a new era

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: “1957 was the year in which I began to develop the theme Mathematics (1949–1970) “Riemann-Roch” (Grothendieck version) - which almost

Part 3: overnight made me into a big “movie star”. It was also the Freedom (1970–1991) year of my mother’s death and thereby the inception of a great

Part 4: break in my life story. They figure among the most intensely Hermitage (1991–2014) creative years of my entire life, not only in mathematics. I’d worked almost exclusively in mathematics for 12 years. In that year there was the sense that I’d perhaps done what there was to do in mathematics and that it was time to try something else. 1957-1958: the beginning of a new era

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics ”The following year (1958) is probably the most fertile of all (1949–1970) my years as a mathematician. This was the year which saw the Part 3: Freedom birth of the two central themes of the new geometry through (1970–1991) the launching of the theory of schemes (the subject of my Part 4: Hermitage paper at the International Congress of at (1991–2014) Edinborough in the summer of that year) and the appearamce of the concept of a “site”, a provisional technical form of the crucial notion of the .” The 1958 ICM address

Caleb Ji Inspired by Serre’s FAC, announces the introduction of cohomology into Part 1: Upbringing (1928–1949) Mentions the as a future goal Part 2: Introduced schemes and gives immediate applications Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Grothendieck’s address Schemes

Caleb Ji

Part 1: Definition (affine ) Upbringing (1928–1949) Given any A, define the affine scheme Spec A Part 2: Mathematics to be the locally ringed consisting of the prime ideals of (1949–1970) A equipped with the topology of closed sets being those Part 3: Freedom containing a fixed subset of A. The structure sheaf is defined (1970–1991) by O (D(f )) = A . A scheme is obtained by gluing affine Part 4: Spec A f Hermitage schemes along open subsets. (1991–2014)

”The very notion of a scheme has a childlike simplicity - so simple, so humble in fact that no one before me had the audacity to take it seriously.” Professor at the IHES (1958–1970)

Institut des hautes ´etudesscientifiques (IHES) created for Caleb Ji him as the French version of the IAS in Princeton Part 1: Upbringing Creates math 12 hours a day, 7 days a week (1928–1949)

Part 2: Leads his legendary SGA seminars Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Weil conjectures

Let X be a projective non-singular defined Caleb Ji over Fq. The zeta function ζ(X , s) is defined as Part 1: X Nm Upbringing ζ(X , s) := q−ms , (1928–1949) m Part 2: m≥1 Mathematics (1949–1970) where Nm is the number of points of X defined over Fqm . Part 3: Freedom Theorem (Weil conjectures) (1970–1991)

Part 4: −s Hermitage 1 ζ(X , s) is a rational function of T = q (1991–2014) nE −Es 2 ζ(X , n − s) = ±q 2 ζ(X , s) where E is the Euler characteristic of X . i/2 3 |αi,j | = q for 1 ≤ i ≤ 2n − 1 and all j. 4 If X is a (good) ”reduction (mod p)” of a non-singular Y defined over a number field embedded in the field of complex numbers, then the degree of Pi is the ith Betti number of the space of complex points of Y . Weil conjectures

Caleb Ji

Part 1: Upbringing (1928–1949) “ These utterly astounding conjectures allowed one to envisage, Part 2: for these new ” discrete varieties” ( or ”spaces”), the possibility Mathematics (1949–1970) for certain kinds of constructions and arguments which up to Part 3: that moment did not appear to be conceivable outside of the Freedom (1970–1991) framework of [topological spaces].” Part 4: Hermitage (1991–2014) ”The panorama that opened up before me, which I was obliged to make the effort to scrutinize and capture, greatly surpassed in scope and in depth the hypothetical needs for proving these conjectures, or indeed all the results that would follow from them.” Motives

Caleb Ji

Part 1: Upbringing ”In contradistinction to what one finds in ordinary topology, (1928–1949) one finds oneself in the presence of a disconcerting abundance Part 2: Mathematics of differing cohomological theories. One had the impression (1949–1970) that, in a sense that should be taken rather flexibly, all of these Part 3: Freedom theories ”boiled down” to the same one, that they ”gave the (1970–1991) same results”.” Part 4: Hermitage (1991–2014)

”This idea was developed, on the fringes of more fundamental and urgent tasks, under the name of the ”theory of motives”, or of ”philosophy (or ”yoga”) of the ”motives”, through the years 1963-69. It’s a theory of a fascinating structural richness, a large part of which remains purely conjectural.” EGA and SGA

EGA Caleb Ji 1 Le langage des sch´emas Part 1: Upbringing 2 Etude´ globale ´el´ementairede quelques classes de (1928–1949) morphismes Part 2: Mathematics 3 Etude´ cohomologique des faisceaux coh´erents (1949–1970)

Part 3: 4 Etude´ locale des sch´emaset des morphismes de sch´emas Freedom (1970–1991) SGA Part 4: Hermitage 1 Revˆetements´etaleset groupe fondamental (1991–2014) 2 Cohomologie locale des faisceaux coh´erentset th´eor`emes de Lefschetz locaux et globaux 3 Sch´emasen groupes 4 Th´eorie des topos et cohomologie ´etaledes sch´emas 5 Cohomologie l-adique et fonctions L 6 Th´eorie des intersections et th´eor`emede Riemann-Roch 7 Groupes de monodromie en g´eom´etriealg´ebrique Mathematical opus

Through 1970, includes: Caleb Ji 1 Topological tensor products and nuclear spaces Part 1: Upbringing 2 Derived categories, the 6 operations (1928–1949)

Part 2: 3 Grothendieck-Riemann-Roch, K-theory Mathematics (1949–1970) 4 Schemes Part 3: Freedom 5 Topos theory (1970–1991) 6 Etale´ cohomology and l-adic cohomology Part 4: Hermitage (1991–2014) 7 Motives, motivic Galois groups 8 , de Rham and Hodge coefficients 9 Topological algebra, ∞-stacks and afterwards: 10 Mediated topology 11 Anabelian geometry, Galois-Teichm¨ullertheory 12 Schematic or arithmetic Viewpoints for regular polyhedra The

Wins Fields medal in 1966 Caleb Ji Refuses to go to Moscow to accept it in protest over Part 1: human rights violations Upbringing (1928–1949) Donates prize money, auctions off medal to the Red Cross Part 2: of Vietnam Mathematics (1949–1970)

Part 3: Freedom ”Built on work of Weil and Zariski and effected fundamental (1970–1991) advances in algebraic geometry. He introduced the idea of Part 4: Hermitage K-theory (the Grothendieck groups and rings). Revolutionized (1991–2014) homological algebra in his celebrated ”Tohoku paper”.”

” is not 40 years old, and already the breadth of his work and the extent of its influence on contemporary mathematics are such that it is not possible to give anything else than a very distorted idea in such a brief presentation.” - J. Diuedonne Trip to Vietnam

Travels to Vietnam in November 1967 Caleb Ji Lectures on requirements for scientific research, schemes, Part 1: homological algebra, sheaf theory, and the Weil conjectures Upbringing (1928–1949)

Part 2: Mathematics “They have confidence in themselves, and that is the best (1949–1970) reason for us to have confidence in them and in their struggle Part 3: Freedom on all fronts, cultural as well as economic and military.” (1970–1991)

Part 4: Hermitage (1991–2014) Ho`angXuˆanS´ınh

Caleb Ji “ During the war her working conditions were particularly Part 1: Upbringing difficult, and her contact with me was restricted to an (1928–1949) intermittent correspondence. She was able to come to France Part 2: Mathematics in 1974/75 [...] and earned her doctoral degree with this thesis (1949–1970) in Paris.” Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) 1970: The great turning point

Caleb Ji

Part 1: Upbringing Dear Sir, Following my letter to the Scientific Committee, I (1928–1949) herewith confirm my departure from the IHES starting 1 Part 2: Mathematics October 1970. Please accept, Sir, the expression of my (1949–1970) distinguished consideration. A. Grothendieck Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Mathematical posts held post-1970

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970) Guest professor at Coll`egede France (1970-1972) Part 3: Freedom Visiting professor at the University of Orsay (1972-1973) (1970–1991)

Part 4: Professor at the University of Montpellier (1973-1988) Hermitage (1991–2014) CNRS researcher (1984-1988) 1970 ICM

Caleb Ji ”This is all the more true since the development of algebraic Part 1: Upbringing geometry, and everything else as well, will come to an end if (1928–1949) our species disappears in the coming decades – a possibility Part 2: Mathematics that today seems more and more likely.” (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Figure: Alexander Grothendieck Survivre et Vivre

Caleb Ji

Part 1: Upbringing Environmental and ecological goals (1928–1949)

Part 2: Born on July 20, 1970 from a conference in Montreal Mathematics (1949–1970) Founding members include AG, Claude Chevalley, Pierre Part 3: Samuel. Freedom (1970–1991)

Part 4: Hermitage (1991–2014) ”Fight for the survival of the human race and for all life, which is endangered due to the ecological imbalance caused by the industrial society of today (pollution and destruction of the environment and natural resources), and by military conflicts and the threat of military conflict.” Survivre et Vivre

Caleb Ji “I was particularly struck by his bold assertion that the most Part 1: Upbringing dangerous people on the planet for the last century or so have (1928–1949) been the scientists, not the generals or the ruthless political Part 2: Mathematics leaders who launch wars. The reason for this is that the (1949–1970) scientist calmly puts into the hands of these powerful people Part 3: Freedom whatever they require in order to exert greater and greater (1970–1991) impact on the planet and on living things.” - Gordon Edwards Part 4: Hermitage (1991–2014)

Figure: Henri Cartan and Jean-Pierre Serre Should we continue scientific research?

Caleb Ji 1) Science [...] delivers devastating forces into the hands of a Part 1: Upbringing minority of leaders [...] (1928–1949)

Part 2: 2) The false views of scientism, allegedly “based on science”, Mathematics (1949–1970) justify [...] our industrialized civilization’s self-destructive drive,

Part 3: known as “progress”, towards unlimited growth in production, Freedom (1970–1991) consumerism [...]

Part 4: 3) Scientific methodology, as practiced today, creates the Hermitage (1991–2014) conditions for alienating relationships [...] 4) Scientific research is not determined by the universal needs of humanity or by the creative drive of the researcher, but rather by social pressures [...] The development of a New Science will play an important role in this process, which will differ from today’s practice in fundamental ways: 1) In the choice of goals [...] 2) In the method [...] 3) In human relationships [...] Survivre et Vivre

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970) “[Although I committed myself with the utmost energy, my Part 3: reflections only touched the periphery of my existence. No Freedom (1970–1991) doubt because I dimly perceived this, I gradually withdrew Part 4: during the year 1972 from anti-militaristic, ecological and Hermitage (1991–2014) “counter-cultural” activities, since I felt that we had reached a point where we were stuck in routine engagement instead of integrating ourselves into a bigger movement [...] ” Proof of the Weil conjectures: 1973

Caleb Ji “If I had done it using motives, he would have been very Part 1: Upbringing interested, because it would have meant that the theory of (1928–1949) motives had been developed. Since the proof used a trick, he Part 2: Mathematics did not care.” - (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Life in Villecun: 1973-1979

Caleb Ji Lived with Justine Bumby, an ex-grad student at Rutgers, Part 1: Upbringing and had a son with her (1928–1949)

Part 2: Mathematics (1949–1970) “All those who were consulted by the author remember

Part 3: Grothendieck in these years as a happy man: he was cheerful, Freedom (1970–1991) he brought wine, fruit, vegetables or nuts to the common

Part 4: meals, he sang, helped with manual jobs, was infinitely Hermitage (1991–2014) generous (in the circle of his friends he was often called “the bank”), and his house was open to everyone; he was (at least for some years) deeply convinced of the value of the alternative lifestyle that he encountered with the region’s “breakaways” and in the rural communes. Later, in hindsight, he wrote that he had experienced this period as a “Sunday”.” - Winfried Scharlau Generosity

Caleb Ji

Part 1: “ my ”relatives” [...] are neither family members nor people Upbringing (1928–1949) of education even of vast culture [...] but the poor among the Part 2: poor” - Alexander Grothendieck Mathematics (1949–1970)

Part 3: Freedom (1970–1991) “His hospitality was startling. Later, when he lived near the Part 4: Hermitage RER stop Massy-Verri‘ere he once invited an entire family who (1991–2014) needed lodging, to stay in his basement and to bring with them their in-laws. He helped them install a taramasalata machine there to give them some economic activity.” -

“ To this very day some of the ex-commune members live on land paid for by Grothendieck,” - Winfried Scharlau Professor at Montpellier

Caleb Ji “I will never forget the evenings spent at Villecun, the two of us Part 1: Upbringing doing math by the light of the old oil lamp. [...] Grothendieck (1928–1949) was completely different, other-worldly. Instead of translating Part 2: Mathematics things into another language, he thought and spoke directly in (1949–1970) the language of modern structural mathematics, which he had Part 3: Freedom contributed greatly to creating.” - Yves Ladegaillerie (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: AG’s house in Villecun Encounter with Buddhism: 1974-1978

Caleb Ji ”Since that memorable day in May when, beneath the midday Part 1: Upbringing sun, I perceived a bizarrely dressed person singing in the road (1928–1949) while accompanying himself on a drum, and aiming (there Part 2: Mathematics could be no mistake about it) directly for my garden where I (1949–1970) was busy with some solitary work - since that day, I have had Part 3: Freedom the privilege and the pleasure of seeing many sympathizers and (1970–1991) adepts of Guruji pass through my home.” Part 4: Hermitage (1991–2014) Nichidatsu Fujii

Founded the activist Buddhist order Nipponzan Myohoji in Caleb Ji 1917 Part 1: Upbringing Travelled throughout promoting peace during WWII (1928–1949) Movement travels the world building peace pagodas, Part 2: Mathematics chanting, and calling for peace and non-violence (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Encounter with Buddhism

Caleb Ji

Part 1: Upbringing (1928–1949) AG seriously considered to be made Fujii’s successor Part 2: Mathematics Continued to engage with Buddhists and their practices (1949–1970) even into his hermitage Part 3: Freedom (1970–1991) Part 4: ”In remembrance of those friends and of their respected Hermitage (1991–2014) Teacher, I still sing the Prayer, whether alone or in a friend’s house, before every meal. This I feel is something of great value that remains from my contacts with Fujii Guruji and his followers.” Turning inward: 1979

Spent a year in total solitude at La Gardette, lived with no Caleb Ji electricity or running water Part 1: Upbringing Delved into parents’ correspondence (1928–1949) Wrote Eloge´ de l’Inceste Part 2: Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: AG at La Gardette Les Aumettes: 1980-1991

House found for him by Y., an adventurous woman with Caleb Ji whom he had a passionate relationship Part 1: Upbringing (1928–1949) “For my part I live a very retired life in a pretty little house, Part 2: Mathematics very secluded and rural, surrounded by hills planted with (1949–1970) vineyards, at the foot of the Mont Ventoux - one of the most Part 3: Freedom remarkable mountains in Europe. I am thoroughly pleased with (1970–1991) my solitude [...] the life of an old eccentric. But an eccentric Part 4: Hermitage who is happy to live [...]” (1991–2014) Mathematical meditations

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970) La Longue Marche `atravers la Th´eorie de Galois (1981) Part 3: Pursuing Stacks (1983) Freedom (1970–1991) Letter to Faltings (1983) Part 4: Hermitage Esquisse d’un Programme (1984) (1991–2014) Les D´erivateurs(1990-1991) Anabelian geometry

Caleb Ji Began with AG’s letter to Faltings

Part 1: Anabelian question: How much information about the Upbringing (1928–1949) isomorphism class of the variety X is contained in the

Part 2: knowledge of the ´etalefundamental group? Mathematics et (1949–1970) Conjecture (proven by Mochizuki): π1 (C) determines C Part 3: where C is an appropriate hyperbolic curve. Freedom (1970–1991)

Part 4: Hermitage (1991–2014)

Figure: Figure: Shinichi Mochizuki Esquisse d’un Programme

Caleb Ji

Part 1: Upbringing Galois-Teichm¨ullerTheory and Deligne-Mumford stacks (1928–1949)

Part 2: dessin d’enfants and Gal(Q/Q) Mathematics (1949–1970) Regular polyhedra, infinite and over finite fields

Part 3: Freedom New foundations for topology (1970–1991)

Part 4: Hermitage (1991–2014) “But today I am no longer, as I used to be, the voluntary prisoner of interminable tasks, which so often prevented me from springing into the unknown, mathematical or not. The time of tasks is over for me. If age has brought me something, it is lightness.” R´ecolteset Semailles

Caleb Ji Written 1983-1986, a personal account of AG’s life as a Part 1: Upbringing mathematician (1928–1949) Harshly critiques many other mathematicians, particularly Part 2: Mathematics Deligne (1949–1970)

Part 3: Claims his work was buried and his vision torn apart Freedom (1970–1991)

Part 4: Hermitage “Yet it is not these gifts, nor the most determined ambition (1991–2014) combined with irresistible will-power, that enables one to surmount the ”invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child play.” La Clef des Songes

Caleb Ji

Part 1: Upbringing A spiritual reflection written in 1987 (1928–1949)

Part 2: Themes: Creativity, repression, dreams, solitude, the Mathematics (1949–1970) Group, creation, love, freedom, self-awareness,

Part 3: self-deception, the nature of God, the spiritual mission Freedom (1970–1991)

Part 4: Hermitage “But above all, the creative path is a solitary path. It is what (1991–2014) frightens. And this great fear of creating, this great fear of being yourself, is none other than the fear of being alone, in a world where the only one who is accepted is the one who merges into the herd or who represents it.” La Clef des Songes

Caleb Ji

Part 1: Explains how society despises and represses true human Upbringing (1928–1949) freedom, physically, intellectually, and spiritually Part 2: Sees social conditioning and the individual’s acceptance of Mathematics (1949–1970) this through self-deception as the root of evil Part 3: Freedom Views all of evolution as a long spiritual journey, led by (1970–1991) God, towards true humanity Part 4: Hermitage (1991–2014) “And again this acquiescence of being, our only and humble contribution to the unknown Work which continues in us, is accomplished and renewed day after day without us even suspecting it, in the shadows and in the silence, in deep places hidden forever from the clumsy gaze of conscience.” Leaving

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom Unexpectedly disappears in 1991 after burning a lot of (1970–1991) papers and leaving the rest of his things in good hands Part 4: Hermitage (1991–2014) Solitude

Caleb Ji

Part 1: Extreme isolation in Lasserre Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Visited by grad students

Caleb Ji ”Hesitantly, I asked him if he would allow me to take a Part 1: Upbringing photograph with him. I said I wanted to keep it as a memento. (1928–1949) In his gentle manner, he told me to come closer. He shook my Part 2: Mathematics hands and hugged me and said that this hug was a better (1949–1970) memento.” - Mohammad Hadi Hedayatzadeh Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Visited by grad students

Caleb Ji

Part 1: ”[...] when he came back out of his house he presented me with Upbringing (1928–1949) a tomato and a packet of almond paste. The tomato was large Part 2: and fresh and came from his garden – impressive for January – Mathematics (1949–1970) and he told me to eat it in good health.” - Katrina Honigs

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Death

Caleb Ji Died on November 13, 2014 Part 1: Upbringing Approximately 60,000 pages of writings left behind from (1928–1949) Lasserre on math, , the psyche, and the problem of Part 2: Mathematics evil (1949–1970)

Part 3: Freedom (1970–1991)

Part 4: Hermitage (1991–2014) Acknowledgments

Caleb Ji Winfired Scharlau and biographies, available Part 1: Upbringing on grothendieckcircle.org (1928–1949)

Part 2: Notices of the AMS articles, particularly Comme Appel´e Mathematics du N´eant—As If Summoned from the Void: The Life of (1949–1970)

Part 3: Alexandre Grothendieck by Allyn Jackson Freedom (1970–1991) and others

Part 4: Hermitage (1991–2014) Some successors

Caleb Ji

Part 1: Upbringing (1928–1949)

Part 2: Mathematics (1949–1970)

Part 3: Freedom (a) Jacob Lurie (b) Shinichi Mochizuki (1970–1991)

Part 4: Hermitage (1991–2014)

(c) Vladimir (d) Voevodsky