Pierre René Deligne

Quick Info

3 October 1944
Etterbeek, Brussels, Belgium

Pierre Deligne is a Belgian mathematician who won a Fields Medal in 1978 for his work on algebraic number theory. He received many other prizes and awards, in particular the Wolf Prize in 2008 and the Abel Prize in 2013.


Pierre Deligne was the son of Albert Deligne, a company administrator, and his wife Renée Bodart. He was born in Etterbeek, one of the nineteen suburban districts that, together with central Brussels, make up Greater Brussels. He was the youngest of his parents three children, having a brother who was seven years older than he was, and also an elder sister. Even before he began his primary education, Pierre was learning mathematics from his brother [33]:-
I was lucky that my brother was seven years older than me. When I looked at the thermometer and realised that there were positive and negative numbers, he would try to explain to me that minus one times minus one is plus one. That was a big surprise.
Pierre attended primary school at Schaerbeek, another of the nineteen suburban districts northeast of central Brussels, from September 1950 to June 1956. An excellent teacher at this school taught him reading, writing and arithmetic. But more than this, he remembered being fascinated by the teacher comparing the surface area of a half-sphere with a disc of the same radius by covering both with rope. Using length to measure area made Pierre think deeply. He continued to learn mathematics from his brother and so when in primary school he was taught the formula for solving quadratic equations.

In September of 1956 he began his secondary schooling at the Athénée Adolphe Max in Brussels. This boys' school was founded in 1909 and later named for Adolphe Max, a famous mayor of Brussels. Pierre continued to learn from his brother and, when in the high school, his brother gave him some notes about solving cubic and quartic equations, including the famous complicated formula for cubics which Pierre found very interesting. His favourite lessons at high school were geometry lessons; learning to write proofs in the style of Euclid he found an excellent exercise. The most important event in his mathematical development came about, not from his brother, but from Jeff Nijs. Pierre was in the Boy Scouts where one of his friends was the son of the high school teacher Jeff Nijs. Hearing about Pierre's passion for mathematics, Jeff Nijs gave him a copy of Bourbaki's Set Theory. This is a difficult text and one which should put off mathematics almost any high school pupil, but not so for Pierre Deligne who was delighted to have his "first real mathematical book." He said [33]:-
I had already read elsewhere how rational numbers, then real numbers, could be defined starting from the integers. But I remember wondering how integers could be defined from set theory, looking a little ahead in Bourbaki, and admiring how one could first define what it means for two sets to have the "same number of elements", and derive from this the notion of integers.
A friend of his family gave the young Deligne a book on complex variable which gave him "tremendous joy." Jeff Nijs realised what a remarkable high school student Deligne was and came up with another excellent idea; he introduced Deligne to Jacques Tits, explained his extraordinary talent, and asked Tits to take good care of him. Although still at high school, Deligne attended Tits's courses and seminars at the university. One day he was absent from a lecture and Tits asked, "Where is Deligne?" When told he could not be there because he had to be on a high school trip, Tits promptly postponed the lecture until the following week. Deligne graduated from secondary school in June 1962 and entered the Free University of Brussels in September of that year.

Although Deligne was an undergraduate at the Free University of Brussels from 1962 to 1966, Tits decided that it would be in Deligne's best interests to learn from Alexander Grothendieck who was lecturing at the École Normale Supérieure in Paris. In November 1964 Tits took Deligne to Paris to attend a Bourbaki seminar and introduce him to Grothendieck. Deligne said [34]:-
I was really taken aback. He was a little strange, with his shaved head, a very tall man. We shook hands but did nothing more until I went to Paris a few months later to attend his seminar
Deligne spent the academic year 1965-66 at the École Normale Supérieure in Paris. He recalls in [34] an incident from the first lecture by Grothendieck that he attended:-
... he used the expression "cohomology object" many times. I knew what cohomology was for abelian groups, but I did not know the meaning of "cohomology object". After the lecture I asked him what he meant by this expression. I think that many other mathematicians would have thought that if you didn't know the answer, there wouldn't be any point to speak to you. This was not his reaction at all. Very patiently he told me that if you have a long exact sequence in an abelian category and you look at the kernel of one map, you divide by the image of the previous one and so on… I recognised quickly that I knew about this in a less general context. He was very open to people who were ignorant. I think that you should not ask him the same stupid question three times, but twice was all right.
Deligne received his Licence en mathématiques in November 1966, the equivalent of a B.A. He continued to study for his doctorate at the Free University of Brussels and in September 1967 he was a junior scientist at the Fond National de la Recherche Scientifique in Brussels, at the same time being a guest at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France where he worked with Alexandre Grothendieck. He was awarded his Doctorat en mathématiques by the Free University of Brussels in November 1968.

After the award of his doctorate, Deligne went to the Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Yvette in France where he was a visiting member until February 1970 after which he became a permanent member of the Institute. At the IHES he worked with Grothendieck initially on the generalisation of Zariski's main theorem. He also worked closely with Jean-Pierre Serre, leading to important results on the ll-adic representations attached to modular forms, and the conjectural functional equations of LL-functions.

Working with both Grothendieck and Serre was important for Deligne, particularly since their approach to mathematics was very different. Grothendieck wanted to understand everything in the utmost generality. He had not much knowledge of the relevant literature, preferring to prove everything for himself. Serre, on the other hand, had an excellent knowledge of the literature and could give precise references. Rather than always looking for the most general situation, he concentrated on beautiful special cases. Deligne thought that these two needed each other and their collaboration was valuable to them both. He also found that balancing their two approaches was a useful learning experience for him. He said Grothendieck's lectures were wonderful but he needed to go to Serre's lectures in order to keep his feet on the ground. During this period at the IHES, Deligne also collaborated with David Mumford on a new description of the moduli spaces for curves: this work has been much used in later developments arising from string theory.

His remarkable mathematical contributions were quickly recognised with the award of major prizes: in 1974 he received the François Deruyts Prize, awarded by the Belgium Royal Academy of Sciences, and the Henri Poincaré Medal, awarded by the French Academy of Sciences. He received the A De Leeuw-Damry-Bourlart Prize in 1975 from the Belgian National Science Foundation.

On 9 September 1980, Deligne married Elena Vladimirovna Alexeeva whom he met on one of his visits to Russia; they have two children, Natalia and Alexis.

Deligne remained based at the Institut des Hautes Études Scientifiques until 1984 when he went to the Institute for Advanced Study at Princeton in the United States, where he was appointed a professor. He did not leave the IHES because he was unhappy there, quite the contrary, but he felt that it was not good to spend the whole of one's career at the same institution and that he would gain much from a move. While at the IHES he gave a seminar each year on a different topic and by going to Princeton he no longer felt it necessary to do this. He talked about the similarities and differences in the two institutions in [34]:-
I would say that the Institute for Advanced Study is older, bigger, and more stable. Both are very similar in the way that there are many young visitors who come there. So they are not places where you can fall asleep since you will always be in contact with young people who will tell you that you are not as good as you think you are. In both places there are physicists, but I think the contact with them was more fruitful for me in Princeton than it was in Bures. In Princeton, there have been common seminars.
As an example of the interaction between mathematicians and physicists, let us quote from the Institute's Report for 1997-98 [48]:-
This was the second year of a three-year interdisciplinary programme in mathematics and physics, led by Professor Pierre Deligne, School of Mathematics, and Professor Edward Witten, School of Natural Sciences, and titled "Algebraic and Geometric Aspects of Quantum Field Theory and Gauge Theory." The programme is evidence of a remarkable commitment to sustained and sophisticated interaction on the part of participating mathematicians and physicists. Over the past few years the theoretical physics community has produced a number of remarkable mathematical conjectures. The aim of this program is for mathematicians to learn the physicists' thought processes that lead to these conjectures. ... Preliminary versions of these lecture notes have been available on the Institute's web site and are being used by mathematicians worldwide as a unique source for this material.
André Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946). This used Zariski's ideas and also made good use of geometric concepts. Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically. Weil's work related questions about integer solutions of polynomial equations to questions in algebraic geometry. He conjectured results about the number of solutions of polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation. The third of his conjectures was a generalisation of the Riemann hypothesis on the zeta function. These problems quickly became major research challenges to mathematicians.

A solution of the three Weil conjectures was given by Deligne in 1974. This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the International Congress of Mathematicians in Helsinki in 1978. A solution to these problems required the development of a new kind of algebraic topology. Jacques Tits said [29]:-
These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of their solution.
Deligne has worked on many other important problems. The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.

In addition to the Fields Medal, Deligne was awarded the Crafoord Prize of the Royal Swedish Academy of Sciences in 1988:-
... for his fundamental research in algebraic geometry.
Deligne has been awarded many other honours for his outstanding contributions. For example he was awarded the Francois Deruyts prize by the Royal Belgium Academy of Sciences in June 1974, the Henri Poincaré medal by the Paris Academy of Sciences in December 1974, and the Doctor A De Leeuw-Damry-Bourlart Prize by the Fond National de la Recherche Scientifique in 1975. He has received honorary doctorates from the Flemish University of Brussels in 1989, and from the École Normale Supérieure in 1995. He has been elected a member of the Paris Academy of Sciences in 1978 and by the American Academy of Arts and Sciences in the same year.

In 2004 Deligne was elected an honorary member of the London Mathematical Society [15]:-
... in recognition of his monumental contributions to algebraic geometry.

Viewed as a whole, Deligne's work concerns many different aspects of the cohomology of algebraic varieties. It has turned Grothendieck's philosophy of motives from a conjectural program into what is the driving force behind many of the most subtle areas of current algebraic geometry and arithmetic. Through an unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity, Deligne has singlehandedly brought about a new understanding of the cohomology of varieties, both classical and in finite characteristic, with numerous applications to deep problems in geometry and number theory.
In the same year Deligne received the 2004 Balzan Prize in Mathematics awarded by the International Balzan Foundation [42]:-
... for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).
Jacques Tits, as a member of the Balzan Prize committee, announced the prize on 7 September 2004 in Milan. He described Deligne's work, then ended by making the following comments [42]:-
A remarkable feature of Pierre Deligne's thinking is that, when confronted with a new problem or a new theory, he understands and, so to speak, makes his own its basic principles at a tremendous speed, and is immediately able to discuss the problem or use the theory as a completely familiar object. Thus, he readily adopts the language of the people he is talking to when engaged in discussions. This flexibility is one of the reasons for the universality of his mathematical work.

Alone or in collaboration, Pierre Deligne has written about a hundred papers, most of them of sizeable length. Because of the conciseness of his style and of his habit of never writing the same thing twice (in fact, quite a few of his best ideas have never been written!), the volume of his publications is a true measure of the richness of his scientific production.
As winner of the Balzan Prize, Deligne received 1 million Swiss francs (about US$800,000), half of which would go to research projects involving young researchers in his field. The prize ceremony took place on 18 November 2004 in the Accademia dei Lincei in Rome. Deligne said, as part of his reply [43]:-
My own research has mainly revolved around algebraic geometry. That this discipline owes so much to the Italian school doubles my pleasure in being in Rome today. Algebraic geometry was born from the realisation, over several centuries, that geometry and algebra are in many cases the expression in two different languages of the same underlying "reality". As a classic example, I will cite the significant identity between "circle of radius rr" and "equation x2+y2=r2x^{2} + y^{2} = r^{2}". The construction of such analogies which, in their finished form, can become dictionaries, and their corollary: the resolution of problems by methods which, at first sight, seem to have nothing to do with the problem posed, are for me one of the great joys that mathematics offers the professional mathematician. These unexpected analogies, dictionaries and comparisons also ensure that mathematics is not scattered over autonomous sub-disciplines, and is a source of its effectiveness when it serves as a tool for other sciences - even if this effectiveness is not the first goal of the mathematician.
A conference on the occasion of the sixty-first birthday of Pierre Deligne took place at the Institute for Advanced Study on 17-20 October 2005. In May 2007 he was elected as an International Member of the Mathematics Section of the National Academy of Sciences. His Research Interests are given on the Academy's website as follows [44]:-
My home is algebraic geometry, a discipline connected to many others; wherever polynomials appear, it can be there to provide a geometric understanding. For instance: Diophantine equations (solutions of equations in integers or rationals), identities between integrals with algebraic integrands, algebraic groups. I am fascinated by the multiplicity of cohomology theories algebraic varieties give rise to, and their interrelation. I constructed one: mixed Hodge theory. Thanks to their properties, spaces and maps coming from algebraic geometry are very special. A grandiose theory of Grothendieck ("motives") makes sense of it, modulo conjectures which remain inaccessible. Some of my works give unconditional variants sufficients for some applications. I have also worked with automorphic forms (related with the arithmetic and the cohomology of algebraic varieties by Langlands' philosophy), with configurations of hyperplanes, tensor categories, multizeta values (a story beginning with Euler).
In February 2008, Deligne was a recipient of the Wolf Prize, given also on this occasion to Phillip Griffiths, and David Mumford. The citation states that the prize is awarded to Deligne [45]:-
... for his work on mixed Hodge theory, the Weil conjectures, the Riemann-Hilbert correspondence, and for his contributions to arithmetic.
Later in 2008 he became Emeritus at the Institute for Advanced Study, Princeton. He continued to receive major honours and awards: he was elected a Foreign Member of the Royal Swedish Academy of Sciences in February 2009 and a member of the American Philosophical Society in April of that year. Perhaps the greatest honour he received was the Abel Prize in May 2013. The citation for the Prize states that the prize is awarded to Deligne ([1] and [24]):-
... for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields. ... Deligne's powerful concepts, ideas, results and methods continue to influence the development of algebraic geometry, as well as mathematics as a whole.
Other honours, not mentioned above, include: elected Membre Associé Étranger, Académie des Sciences, Paris (1978); Doctor honoris causa of the Vrije Universiteit Brussel (1989); elected Membre Associé, Académie Royale de Belgique (1994); Doctor honoris causa of École Normale Supérieure (1995); elected foreign member, Accademia nazionale dei Lincei (2003); ennobled by Albert II, King of the Belgians (2006). This last mentioned honour made him Vicomte Deligne, and he designed his coat of arms which was inspired by the following nursery rhyme [39]:-
Quand trois poules vont aux champs,
La première va devant,
La deuxième suit la première,
La troisième est la dernière.
Quand trois poules vont aux champs,
La première va devant.
This translates as:
As three hens head for the fields,
The first one leads,
The second follows the first,
The third one is last.
As three hens head for the fields,
The first one leads.
Deligne's reasoning is given in [39]:-
This rhyme is intended to teach children the meaning of a few simple words. As such, it appears to adults as a succession of tautologies. Deligne argues that mathematical discourse proceeds in much the same way ...
In 2007 the Belgian post office issued a postage stamp in his honour.

In October 2016 he was elected a Foreign Member of the Russian Academy of Sciences.

In the interview [33] Deligne speaks about teaching:-
I never had to teach. I like very much to speak with people. In the two institutions where I have worked young people come to speak with me. Sometimes I answer their questions, but more often I ask them counter-questions that sometimes are interesting, too. So this aspect of teaching with one-to-one contact, trying to give useful information and learning in the process, is important to me. I suspect it must be very painful to teach people who are not interested, but are forced to learn mathematics because they need the grade to do something else. I would find that repulsive.
Perhaps it would be interesting to record a little about what this mathematical genius does other than mathematics [47]:-
He doesn't own a car and bicycles everywhere, keeping his right pant leg perennially tucked into his sock. He has quite possibly never been seen in a suit, favouring instead well-worn sweaters in earth tones. In Princeton's flash rainstorms he has been known to strip down to the waist to minimise the damage, though he does not take this strategy to the logical extreme.

Deligne loves nature and negotiated to buy the house right next to the Institute woods, even though it had been used for administrative space. In wintertime, he has taken to building an igloo in his backyard and spending a few nights sleeping outside. In summertime, he maintains a garden - potatoes, tomatoes, raspberries, gooseberries, leeks, basil, parsley, tarragon, chives. "Some things I like very much and I try but I am not successful," he says. "I like very much artichoke, not so much to eat them but when they get overripe there is a bud and they open like a flower and they are beautiful. But the climate does not seem to be good for them. Every year I try. I find it very relaxing."
Let us end with two quotes about Deligne's contributions. Peter Sarnak writes [2]:-
There are very few mathematicians whose impact on modern mathematics comes close to that of Deligne. His research in algebraic geometry and arithmetic geometry have shaped these fields and led him to the solution of a number of long standing problems, including the Weil Conjectures (which are the analogues of the notorious Riemann Hypothesis for varieties over finite fields) and the celebrated Ramanujan Conjecture in the theory of modular forms. Deligne's foundational contributions range from the above fields to representation theory of groups, differential equations and monodromy, topology ... Many of the techniques and tools that he developed either in these papers or in response to questions posed to him (he is very approachable and generous) are at the bottom of much of the exciting research that is going in these fields today. Deligne's style is that of wanting to understand in simple and general terms things that are fundamental but appear to be very complex. His development of new insights and methods, as well as the solution of long standing problems, follow naturally from this point of view. Of course his striking success has a lot to do with his tremendous mathematical talent and especially his ability to think abstractly.
Hélène Esnault writes [2]:-
Deligne's work has structured the language in which generations of algebraic geometers and arithmeticians think and write. Many algebraic geometers have slept for years with a few lines of Pierre Deligne's ideas on their night table, to meditate and think further. All algebraic geometers have the deepest admiration for Deligne's work.
Deligne himself, writing about Grothendieck, says [46]:-
I feel extremely fortunate that Grothendieck was my Master. What I learned from him, especially the philosophy of motives, has been a guiding thread in the works of mine I like the most, such as the formalism of mixed Hodge structures. From him and his example, I have also learned not to take glory in the difficulty of a proof: difficulty means we have not understood. The ideal is to be able to paint a landscape in which the proof is obvious.

References (show)

  1. 2013: Pierre Deligne, Institute for Advanced Study, Princeton, USA, The Abel Prize, The Norwegian Academy of Science and Letters.
  2. A conference on the occasion of the sixty-first birthday of Pierre Deligne, The Institute Letter Fall 2005, Institute for Advanced Study (2005).
  3. Abel Prize 2013, London Mathematical Society (2013).
  4. P Ball, Mathematician Wins award for shaping algebra, Nature (20 March 2013).
  5. P Ball, Belgian Mathematician Wins Abel Prize for Shaping Algebraic Geometry, Scientific American (2013).
  6. P Bayer, Pierre Deligne: 2013 Abel Prize, Societat Catalana de Matemàtiques Notícies 35 (2013), 53-64.
  7. P Beazley Cohen and F Hirzebruch, Review: Commensurabilities among lattices in PU (1, n), by Pierre Deligne and G Daniel Mostow, Bulletin of the American Mathematical Society 32 (1) (1995), 88-105.
  8. Belgian-born Pierre Deligne named Abel Prize Winner, The Nordic Page.
  9. Curriculum Vitae: Pierre R Deligne, Institute for Advanced Study (2023).
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  15. Honorary member : Pierre Deligne, Bulletin of the London Mathematical Society 36 (2004), 855-856.
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  17. L Illusie, Pierre Deligne et la géométrie arithmétique, Gazette des Mathématiciens 138 (2013), 49-53.
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  28. Pierre Deligne, in Brigham Nairns (ed.), Notable Scientists from 1900 to the Present: D-H (Gale, Detroit, 2001).
  29. Pierre Deligne, in Chris R Somerville and Elliot M Meyerowitz, Premi Balzan (Fondazione internazionale Balzan, 2004).
  30. Pierre Deligne Honored by Government of Belgium, The Institute Letter Winter 2008, Institute for Advanced Study (2008).
  31. Pierre René, Viscount Deligne, Prabook.com.
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  35. A Rojas León, Pierre Deligne, La Gaceta de la Real Sociedad Matemática Española 16 (3) (2013), 575-592.
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  39. Viscount Pierre Deligne (b. 1944): Belgian Mathematician, Numericana.com.
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  41. Fields Medal Winners.
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  43. P Deligne, Acceptance speech, 2004 Balzan Prize for Mathematics, International Balzan Prize Foundation.
  44. Pierre Deligne. Research Interests, National Academy of Sciences.
  45. Pierre R Deligne, Wolf Prize Laureate in Mathematics 2008/9, Wolf Foundation.
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  47. Pierre Deligne. Video, Simons Foundation (19 June 2012).
  48. Institute for Advanced Study Report for the Academic Year 1997-98, Institute for Advanced Study, Princeton.
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Written by J J O'Connor and E F Robertson
Last Update December 2023