Caleb Ji Part 1: Upbringing (1928{1949) Elements of Grothendieck's life and work Part 2: Mathematics (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 Hamburg with the Heydorns, his foster family at the age of 5 Part 1: Upbringing Reunited with parents in Paris 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 Montpellier (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 mathematician. Cartan Seminar Caleb Ji \[Henri Cartan] 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 group 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: Claude Chevalley PhD in Nancy: functional analysis (1949-1953) Caleb Ji Worked with Jean Dieudonn´eand Laurent Schwartz 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 topology 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 algebraic topology." 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) Sheaf" 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 T^ohokupaper Sur quelques points d'alg`ebre homologique, published in Caleb Ji T^ohoku,1957 Part 1: Upbringing Grew from Grothendieck's attempt to reconstruct the (1928{1949) Cartan-Eilenberg book on homological algebra Part 2: Mathematics (1949{1970) Introduced abelian categories Part 3: Gave a general framework for cohomology through derived Freedom (1970{1991) functors, which worked for sheaves Part 4: Hermitage (1991{2014) Return to France 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: geometry 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." { Armand Borel 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 morphism. 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 Mathematicians 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 topos." The 1958 ICM address Caleb Ji Inspired by Serre's FAC, announces the introduction of cohomology into algebraic geometry Part 1: Upbringing (1928{1949) Mentions the Weil conjectures 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 scheme) Upbringing (1928{1949) Given any commutative ring A, define the affine scheme Spec A Part 2: Mathematics to be the locally ringed space 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.