## 19 Nov Prof. Alexander Grothendieck – The math genius!

I was reading a math-related post in Facebook when suddenly the message icon in the top right corner of the facebook page turned red and the speakers of my system made a beep sound. It was a message from my friend, Manjil, asking me to write an obituary of the famous mathematician Alexander Grothendieck. I immediately replied “Yes of course!”. Incidentally, the math-related post I mentioned in the first line was a news report on the recent death of the same math genius. I opened an extra tab on my browser and searched for more details about Prof. Grothendieck. And yes, I was shocked to see a huge bunch of info about him on various sites. So, I realised that writing about him is going to be tough! Ultimately I have come up with a brief write up on the life and works of Prof. Grothendieck. I, on behalf of GonitSora, dedicate this write up to the revolutionary mathematician Prof. Alexander Grothendieck. May his soul rest in peace!

Prof. Grothendeick was a French mathematician and the leading figure in creating modern algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations. This new perspective led to revolutionary advances across many areas of pure mathematics. Born in Germany on 28th March 1928, Grothendieck was raised and lived primarily in France. Being a Jew, he did not have a happy childhood and he also had to bear the pathetic sufferings in the Nazi concentration camps. In 1933 Alexander’s parents moved to Paris to escape the Nazis, leaving their five year old son with a family in Hamburg, where he went to school. Unfortunately, Alexander never really knew his father but even then he held him in great esteem. His office at the Institut des Hautes Études Scientifiques (IHES ) (French for “Institute of Advanced Scientific Studies”) had no decoration except a portrait of his father. In 1945, Grothendieck and his mother moved to the village of Maisargues near Montpellier, where he worked in the vineyards and also, with the aid of a small scholarship, studied mathematics at the University of Montpellier. As a student, Alexander became fascinated with mathematics at a young age but he was not ready to take everything for granted. While at school he had felt dissatisfied with some of the mathematics that had been presented to him. He once remarked

What was least satisfying to me in our high school mathematics books was the absence of any serious definition of the notion of length of a curve, of area of a surface, of volume of a solid. I promised myself I would fill this gap when I had the chance.

This clearly reflects his passion for understanding mathematics better. He was very dissatisfied with the teaching he was receiving. He had been told how to compute the volume of a sphere or a pyramid, but no one had explained the definition of volume. It is an unmistakable sign of a mathematical spirit to want to replace the “how” with a “why”. One of his professors assured him that a certain Lebesgue had resolved the last outstanding problems in mathematics, but that his work would be too difficult to teach. Alone, with almost no hints, Grothendieck rediscovered a very general version of the Lebesgue integral. Grothendieck was very hard working. He had very few books of his own. Such was his genius that rather than learning things by reading, he would try to reconstruct them on his own.

After graduating from Montpellier, he spent the year 1948-49 at the École Normale Supérieure in Paris. There he attended Henri Cartan’s seminar on algebraic topology and sheaf theory and he came to contact with leading mathematicians like Claude Chevalley, Jean Delsarte, Jean Dieudonné, Roger Godement, Laurent Schwartz, and André Weil. One of Grothendieck’s fellow students was Jean-Pierre Serre, the first Abel Laureate (2003). Since Grothendieck was at this time more interested in topological vector spaces than he was in algebraic topology, André Weil and Henri Cartan both advised him to go to Nancy where there was a strong team including Jean Dieudonné, Jean Delsarte, Roger Godement and Laurent Schwartz. In 1949 Grothendieck moved to the University of Nancy where he lived with his mother. At Nancy there was an active seminar every Saturday in which all the professors and some of the students used to participate to discuss a variety of different topics. It provided a wonderful environment for the young Grothendieck.

Grothendieck presented his doctoral thesis *Produits tensoriels topologiques et espaces nucleaires* (Topological tensor products and nuclear spaces) in 1953. He spent the years 1953-55 at the University of Sao Paulo and then he spent the following year at the University of Kansas. It was during this period that his research interests changed and they moved towards topology and geometry. During this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. Around this time, he became one of the Bourbaki group of mathematicians which included André Weil, Henri Cartan and Jean Dieudonné. In 1959 he was offered a research position in the newly formed Institut des Hautes Études Scientifiques (IHES) which he accepted. The years 1959-70 that Grothendieck spent at the IHES can be termed as a ‘Golden Age’, during which a whole new school of mathematics flourished under his charismatic leadership. Grothendieck’s Séminaire de Géométrie Algébrique established the IHES as a world centre of algebraic geometry, and him as its driving force. During this period Grothendieck’s work provided unifying themes in geometry, number theory, topology and complex analysis. He introduced the theory of schemes in the 1960s which allowed certain of Weil’s number theory conjectures to be solved. He worked on the theory of topoi which are highly relevant to mathematical logic. He gave an algebraic proof of the Riemann-Roch theorem. He provided an algebraic definition of the fundamental group of a curve. Luc Illusie, one of his PhD students, writes

He spoke with great energy at the board but taking care to recall all the necessary material. He was very precise. The presentation was so neat that even I, who knew nothing of the topic, could understand the formal structure. It was going fast but so clearly that I could take notes.

Grothendieck’s best known contributions are found in a wide variety of topics which include topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann-Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the ‘six operations’), étale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, ‘standard conjectures’, motives and the ‘yoga of weights’, tensor categories and motivic Galois groups. This list reflects his exceptional mathematical talent. Prof. Grothendieck was also a person of great kindness and generosity. He seemed always to be in good spirits, with great mental equilibrium. He possessed a miraculous capacity for work. In 1966, he received the Fields Medal which is the highest award in the mathematical arena. The citation for the Fields Medal read

Grothendieck built on work of Weil and Zariski and effected fundamental advances in algebraic topology. He introduced the idea of K-theory (the Grothendieck groups and rings). He revolutionised homological algebra in his celebrated `Tohoko paper.

The `Tohoko paper’ referred to in this citation is about abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence. The Fields Medal is presented every year at the International Congress of Mathematicians (ICM) and in 1966 the ICM was was held in Moscow, Russia. Mathematicians, in general, keep themselves outside politics but Grothendieck was a bit different. Grothendieck was always strongly pacifist in his views and he had campaigned against the military built-up of the 1960s and hence as a political protest, he refused to travel to Moscow to receive the Fields medal. At the Congress, Léon Motchane, director of IHES, received the Fields Medal on Grothendieck’s behalf. Grothendieck made no public statement about the reasons for not going to Moscow but he declared himself a citizen of the world and requested United Nations citizenship.

Prof. Grothendeick left the IHES in 1970 after he discovered that some of their funding came from military sources. He discovered this in 1969 and, along with the other professors at the IHES, he persuaded the director, Léon Motchane, to take no further funding from the French military. However, a few months later the IHES faced a financial crisis and the director went back on his word. Grothendieck tried to persuade all the professors to resign in protest but the others refused to follow his example. Grothendieck’s letter of resignation was dated 25 May 1970. He also cited some other problems for his resignation. He wrote that he was suffering a “spiritual stagnation”. He abandoned mathematics as the main focus of his energies and turned to political protest, particularly against nuclear proliferation. However, in contrast to the amazing impact of his mathematical work, his political campaigns were rather ineffective. In 1970-72 he held an appointment as visiting professor at the Collège de France, then a similar appointment at Orsay for 1972-73. In 1973 he accepted an appointment as professor at the University of Montpellier. He lived in Villecun near Lodeve from 1973 to 1980, then he moved to live in Mormoiron near Carpentras. He took leave during 1984-88 to direct research at the Centre National de la Recherche Scientifique. He retired at age 60 in 1988. In contrast to his acceptance of the 1966 Fields Medal, Grothendieck declined the Crafoord Prize in 1988.

Between 1980 and 1990, Grothendieck wrote literally thousands of pages some containing his mathematical thoughts and others containing non-mathematical meditations. His mathematical manuscripts are La longue marche à travers la théorie de Galois (The long walk through Galois theory, 1981), A la poursuite des champs (In pursuit of the fields, 1983), Esquisse d’un programme (1983), and Les dérivateurs (1987), while his non-mathematical works are the Eloge (1981, but apparently subsequently lost), Récoltes et Semailles (1983-85), and La clef des songes (1986). In August 1991 he left home suddenly, without informing anyone, for an unknown location. There he spent his time writing an extremely large work on physics as well as philosophical meditations on themes such as free choice, determinism and the existence of evil. He refused practically all human contact. In 2010 he tried to eradicate all trace of his past life, writing a letter to one of his students demanding that his entire back catalogue be removed from libraries and refusing to allow republications. He died in hospital at Saint-Girons in south-west France on 13th November 2014, at the age of 86. Prof. Grothendieck will always be remembered for his contributions to mathematics. May his soul rest in peace!

References:

- http://www-history.mcs.st-andrews.ac.uk/Biographies/Grothendieck.html
- http://www.grothendieckcircle.org/
- http://www.telegraph.co.uk/news/obituaries/11231703/Alexander-Grothendieck-obituary.html