MacTutor
Luc Illusie
Quick Info
Nantes, France
Biography
Luc Illusie is the son of Francis Armand Illusie (20 December 1904 - 14 October 1986) and Amédée Anette Schmitt (22 April 1901 - 3 July 1997). Francis Illusie was born in Vannes, Morbihan, France, the son of François Félix Illusie and Eugénie Augustine Chupin. He trained to became a teacher of history and literature and taught at the Collège Moderne in Nantes. He married Amédée Schmitt who had been born in Ivry-Sur-Seine, Val-De-Marne, France, the daughter of Joseph Théodore Schmitt and Anne Suchéras. She trained to become a teacher of mathematics. Later, after the family moved to Paris, she taught at the Lycée Victor-Hugo and the Lycée Camille-Sée. Francis and Amédée Illusie had two sons: Jean-Paul Francis Theodore Illusie (13 August 1930 - 25 November 2006), who became a teacher of French at a lycée; and Luc Illusie (2 May 1940 -), the subject of this biography.Although Luc Illusie was born in Nantes, he spent the first five years of his childhood in Savenay, a small town about 40 km north-west of Nantes. These first five years were, of course, difficult years. His family suffered the problems of living under German occupation and bombing from Allied Forces. For a detailed description by Illusie of these war years, see THIS LINK.
In the interview [2] Illusie spoke about his parents. He described his father, saying:-
I owe him a lot because he knew so many things. He had an extraordinary memory. He could listen to a speech and then recite it afterwards. He knew so many pieces of the great poets.Speaking about his mother, Illusie said [2]:-
My mother was teaching mathematics. She helped me discover the joys of algebra. When I was nine years old, she taught me how to put concrete problems into equations. She was my mathematics teacher in school when I was between 10 and 12 years old. She taught me geometry and I was fascinated by the beautiful properties of triangles.In the interview [16] he went into more detail about how he first met algebra when his mother showed him how create and solve equations. He was about nine years old and given the following problem.
You have a stack of books on a table, some 2 cm thick, others 3 cm. The stack is 22 cm high, and you know there are four times as many 2 cm books as 3 cm books. How many 2 cm books and 3 cm books are there?When his mother saw he was having difficulty solving the problem, she said: "Suppose there are books 2 cm thick. Then these books make up cm of the stack. In the same way, suppose there are books that are 3 cm thick. Then the whole stack is cm high. So . We also know that there are four times as many 2 cm books as 3 cm books so . Substitute for in to get , so . But so ." Illusie said [16]:-
I was amazed that, in a way, naming a quantity we don't know could, by explaining the constraints, lead to determining this quantity. ... Well, there you have my first discovery with algebra.Illusie attended the Lycée in Nantes until 1956. At this Lycée he found mathematics easy but, because of that, it was not very interesting or exciting. At this time he focussed mostly on the humanities but also liked history and geography; in fact all subjects appealed to him. His French teacher, Henri Lafay, was excellent. For example his class spent two months studying Jean Racine's play Britannicus which is a tragedy in verse in five acts. Illusie enjoyed spending hours at night writing some French texts on literature, such as poems by Victor Hugo [9]:-
The history teacher I had in 1955-56 had the talent of a story teller, speaking without any notes, making us live the campaigns of Napoleon as if we had been watching a movie!In addition to French and history, Illusie studied Latin and classical Greek which he also found fascinating while, as we have indicated, he found mathematics easy but not nearly as exciting as the humanities. The obvious question, then, is, "How did Illusie end up studying science and mathematics?" The Baccalauréat was taken over two years and, in the second of the two, a student had to choose between science and literature. Illusie chose science, not because to found it exciting, but rather because he thought that mathematics and physics would give him a wider choice of jobs. Chemistry did not appeal because he disliked conducting experiments in the laboratory.
The best places to study mathematics and physics were the École Normale or the École Polytechnique in Paris. His parents felt that to maximise his chances of entering one of the Grand École he should prepare by studying at a Lycée in Paris. The whole family moved to Paris and Illusie enrolled in the Lycée Louis-le-Grand. He studied Mathématiques élémentaires in 1957-58 and he was in the Classes préparatoires in 1958-59. In both these years he was taught mathematics by outstanding teachers, but the one who influenced him most was André Magnier (1909-1996). Illusie said in the interview [2]:-
At the lycée I had an extraordinary teacher of mathematics in the second year. His name was André Magnier and in fact he knew Grothendieck very well. He was from Montpellier, where Grothendieck had studied, and he helped him come to Paris. He discovered that he had a special talent and then had him contact Henri Cartan and the Bourbaki group. The rest is history ... Magnier was a very good teacher and it is certainly thanks to him that I also got drawn into mathematics.Illusie entered the École Normale Supérieure (ENS) in 1959. At this stage he was still unsure whether mathematics or physics was the subject in which he would specialise. There were a number of people who influenced him to become a mathematician. One was Adrien Douady who was the senior student assigned to look after him when he was a first year student. Douady (1935-2006), born in La Tronche, Isère, was a student of Henri Cartan working on homological algebra. He was a member of Bourbaki who had a wonderful way of explaining difficult concepts. He gave Illusie some Bourbaki papers, as did Roger Godement, who lectured to Illusie on analysis. In his first year at the ENS Illusie was taught physics by Alfred Kastler and Yves Rocard. Both were brilliant and, in fact, Kastler went on to win the Nobel prize for physics in 1966. The way Rocard presented his material, however, made Illusie think it was not the right area for him so, if he was to become a physicist he would follow Kastler. On the other hand, he attended a course by Henri Cartan on algebra which was a revelation to him. The algebra he had been taught in the lycées had been presented as a mechanical approach to solving problems but now Cartan was presenting the theory which he found fascinating.
By 1962-63, in addition to classes at the ENS, Illusie attended a class by Jean-Pierre Serre at the Collège de France on Galois cohomology. Alexander Grothendieck also attended these lectures and, although Illusie did not meet him then, he was impressed by the probing questions Grothendieck asked. A small number of places were open at the Centre National de la Recherche Scientifique (CNRS) and if Cartan saw he had a really promising student he would recommend them for a place. Illusie was seen as very promising and was appointed as an attaché de recherche at the CNRS in 1963; his contract was renewed each year. At the CNRS Illusie participated in the 16th year of the Henri Cartan Seminar, 1963-64, directed by Henri Cartan and Laurent Schwartz. This seminar was studying the Atiyah-Singer index formula. Illusie said in the interview [2]:-
It was here that I first felt that maybe I could become a mathematician. Sometimes I made a new observation of which Atiyah, Singer and other very impressive people had not thought. Although it was maybe an epsilon, it gave me great confidence that I could continue.The first talk that Cartan asked Illusie to present in this seminar was on the Chern character and the Todd class. After receiving help from Cartan, Illussie delivered the talk and was then told to write it up. Following Cartan's advice, he purchased a German type-writer, taught himself to touch type, and wrote up his talk. Two volumes were published on the work of the Atiyah-Singer index formula seminar in 1965. They contain the following contributions by Illusie: Caractère de Chern. Classe de Todd Ⓣ (9 pages); Compléments de K-théorie Ⓣ (10 pages); Opérateur Ⓣ (6 pages); and Symboles elliptiques Ⓣ (13 pages). The seminar brought Illusie into contact with many leading mathematicians. Alexander Grothendieck was in the audience while Laurent Schwartz and Michael Atiyah gave lectures. Two others who attended Illusie's lectures, and were involved in discussions after them, were Jean-Louis Verdier (1935-1989) and Michel Demazure (1937-); both were doctoral students of Grothendieck.
As a result of his work in the Atiyah-Singer index formula seminar, Cartan suggested a related topic for Illusie's doctoral thesis. His first publication was Complexes quasi-acycliques directs de fibrés banachiques Ⓣ which was transmitted to Comptes Rendus of the Academy of Sciences by Henri Cartan. He explained in [6] how he left Cartan and became one of Grothendieck's students:-
In the course of 1964, I had already obtained a few results, but I had a few questions that I didn't know how to tackle, and Cartan suggested that I should consult the best expert at the time, which was Grothendieck. And then I came to the Institut des Hautes Études Scientifiques (IHES) one afternoon, and I think it was in June of 1964. I met Grothendieck, I explained my questions, I listened to his very long answer, and at the end he said, well, you will work in my seminar in the fall. ... So then I became his student, and I worked in his seminar, he advised my PhD ...Grothendieck's 'Séminaire de géométrie algébrique avec Artin' was on local duality [7]:-
The seminar was on Tuesdays. It started at 2:15 and lasted one hour and a half. After that we had tea. Most of the talks were given by Grothendieck.Grothendieck asked Illusie to make notes of the lectures. This began a series of visits by Illusie to Grothendieck's home which Illusie described in [7]:-
We started at two and worked until maybe four o'clock, then he said, "Maybe we could take a break." Sometimes we took a walk, sometimes we had tea. After that we came back and worked again. Then we had dinner around seven, with his wife, his daughter, and his two sons. The dinner didn't last long. Afterward we met again in his office, and he liked to explain some maths to me.In many ways Grothendieck was a wonderful advisor but often he gave research topics to his students that proved to be difficult problems. These were probably problems Grothendieck himself had failed to solve, and, not surprisingly, often the students made little progress. There was no pressure on Illusie to produce a thesis quickly, however, and he spent much of his time taking notes in seminars and writing them up for publication. Grothendieck made some rather technical suggestions for research problems for Illusie's thesis but these did not work out for him. Early in 1968, however, he suggested [9]:-
... cotangent complex and deformation theory, more precisely finding a common generalisation of his construction of the truncated cotangent complex and that of Quillen in the affine case, and applying this to a bunch of specific global deformation problems. It was indeed a magnificent topic, and I was lucky and happy to be able to work on it ...Illusie quickly made progress and found a simple way to extend Daniel Quillen's construction. He wrote to Quillen who was impressed and invited Illusie to visit him at the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts in the United States. Before making that trip, however, the two met when Quillen visited the IHES. By 1970 Illusie's thesis was essentially completed and in September 1970 he made the visit to MIT where Quillen had invited him to present a lecture course on the results of his thesis. He stayed at MIT until the spring of 1971 but then returned to Paris, earlier than he would have liked due to his mother suffering a stroke. In May 1971 he defended his thesis Complexe Cotangent et Déformations Ⓣ for his doctorate at the University of Paris-Sud. The chair of the examining committee was Henri Cartan and the other members were Alexander Grothendieck, Michel Demazure and Jean-Pierre Serre. The thesis was published in two volumes totalling 560 pages. Larry Breen writes in the review [3]:-
This work should be a basic reference for many people beyond those concerned with deformation theory. To the topologists it offers not only the foundational material on homotopy theory in a topos ..., but also a new point of view on Eilenberg-Mac Lane spectra and a discussion of Mac Lane's canonical resolution of an abelian group, as well as a theory of Chern classes for perfect complexes. For the algebraic-group theorist it provides a thorough discussion of various Lie and Co-Lie complexes. To the algebraic geometer it offers, in addition to the rest, a discussion of formal categories and their associated de Rham complexes. In particular, a new proof is given of the basic fact that the X/S crystalline cohomology of a smooth S-scheme X coincides with its de Rham cohomology, as well as an extension of this result (introducing a derived de Rham complex which generalises both the usual de Rham complex and the cotangent complex!) to the complete intersection case.We noted above that Illusie was appointed as attaché at the CRNS in 1963. He was promoted to chargé de recherche in 1969, and promoted to maître de recherche in 1973. In 1976 he left the CRNS when he was appointed as a professor at the University of Paris-Sud. This university had been founded in 1971, following the splitting up of the University of Paris. (We note that it was renamed Paris-Saclay University in 2019). At the University of Paris-Sud he was the director of the Arithmetic and Algebraic geometry group in the department of mathematics from 1984 to 1995. Illusie retired in 2005 and was made emeritus professor.
Illusie was awarded the Médaille Émile Picard by the Académie des Sciences in 2012. The citation reads [15]:-
The Émile Picard Medal is awarded this year to Luc Illusie for his fundamental work on the cotangent complex, the Picard-Lefschetz formula, Hodge theory, and logarithmic geometry. This work, some of which dates back to his doctoral thesis and other parts of which are much more recent, remains highly relevant, as demonstrated by recent advances in algebraic geometry and arithmetic.In fact Illusie became more active publishing papers after he retired. Up to 2025, 34 of the 77 publications by Illusie listed in MathSciNet were published in 2007 or later.
One of Illusie's interests is travelling. He said in the 2023 interview [2]:-
I enjoy traveling. This is a privilege of mathematicians, that's why it's the greatest job of all. We can travel a lot and meet wonderful people. I was quite frustrated during the COVID period not to be able to have contact in person. But now that we are back to normal, I would like to travel more again, as long as my health enables me.One of Illusie's hobbies is music. He said in the interview [9]:-
I myself am an amateur pianist. For a few years I took lessons with the French pianist Jean Micault, who is now 87. I learnt a lot from him, not only on piano playing, but on teaching, from his talent of obtaining the best from his students, letting them fully express their own personality. You often learn things in one discipline that you can somehow carry over to other disciplines.
References (show)
- Amédée Schmitt, ancestry.com (2025).
- R Bocklandt, Interview with Luc Illusie: Tales from the golden age of algebraic geometry, Nieuw Archief voor Wiskunde (5) 24 (2) (2023), 97-102.
- L Breen, Review: Complexe Cotangent et Déformations (2 volumes), by Luc Illusie, Mathematical Reviews MR0491680 (58 #10886a).
- Entretien avec Luc Illusie (Université Paris-Sud), Institut des Hautes Études Scientifiques (2018).
https://av.tib.eu/media/38617 - Francis Armand Illusie, ancestry.com (2025).
- Francis Armand Illusie, Journal officiel de la République française (26 August 1949).
- L Illusie, Reminiscences of Grothendieck and His School, Notices of the American Mathematical Society 57 (9) (2010), 1106-1115.
https://www.imo.universite-paris-saclay.fr/~luc.illusie/Reminiscences-NoticesAMS.pdf - L Illusie, The Cartier Isomorphism by Luc Illusie, Institut des Hautes Études Scientifiques (23 August 2024).
https://www.ihes.fr/en/cartier-illusie-en/ - Interview of Luc Illusie by Ulf Persson, Laboratoire de Mathématiques d'Orsay (17 January 2012).
https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Persson.pdf - Jean-Paul Francis Theodore Illusie, ancestry.com (2025).
- Luc Illusie: Mathématicien, CNRS Le Journal (2025).
https://lejournal.cnrs.fr/auteurs/luc-illusie - Luc Illusie, Institute for Advanced Study (2025).
https://www.ias.edu/scholars/luc-illusie - Luc Illusie, Mathematics Genealogy Project (2025).
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=13123 - M Jean-Paul Illusie, M Luc Illusie ses enfants, Le Monde (11 July 1997).
https://scholar.lib.vt.edu/InterNews/LeMonde/issues/1997/lm970711.pdf - Médaille Émile Picard (Mathématique): lauréats - Prix de l'Académie des sciences, French Academy of Sciences (3 October 2012).
- F Orgogozo, An interview with Luc Illusie, conducted by Fabrice Orgogozo on 12 June 2021, Institut des Hautes Études Scientifiques (2 November 2021).
https://www.youtube.com/watch?v=0ZoYlwEhH4s
Additional Resources (show)
Other pages about Luc Illusie:
Other websites about Luc Illusie:
Cross-references (show)
Written by J J O'Connor and E F Robertson
Last Update December 2025
Last Update December 2025