1.1. The Otto Hahn Medal.

The Otto Hahn Medal is awarded by the Max Planck Society. Every year since 1978, the Max Planck Society has awarded the Otto Hahn Medal to young researchers for outstanding scientific achievements, mostly in connection with their doctorate. The Medal is endowed with 7,500 euros of prize money. The aim of the award is to motivate particularly talented individuals to pursue a university or research career.

2.1. Basilis Xanthopoulos.

Basilis Xanthopoulos was born in Drama, Greece in 1951. After majoring in Mathematics at the University of Thessaloniki he moved to the University of Chicago, where he earned his Ph.D. in 1978 under the supervision of Robert Geroch. During this time, he also collaborated with Abhay Ashtekar and especially Subrahmanyan Chandrasekhar with whom he wrote a series of influential papers on colliding gravitational waves. After a post-doctoral position at Syracuse he returned to Greece, serving on physics faculties, first in Thessaloniki and then in Crete. Xanthopoulos made important contributions to several branches of mathematical general relativity, especially the areas of exact solutions, global issues associated with asymptotic flatness and black holes, as well as gauge theories. He was widely admired for his clarity of thought, energy and selflessness. His promising career was tragically cut short during his seminar on 27 November 1990 when he was shot to death by a deranged student.

2.2. The "Basilis Xanthopoulos" Award.

The "Basilis Xanthopoulos" Award is made by the Foundation for Research and Technology, Greece. It is given for research on Gravitational Physics and was established by the Foundation for Research and Technology in memory of Professor Basilis Xanthopoulos. It is given every three years during the conference of the International Society on General Relativity and Gravitation.

2.3. The 1991 Basilis Xanthopoulos Award.

The 1991 Basilis Xanthopoulos Award was given to Demetrios Christodoulou, Professor of Physics at the Courant Institute of the New York University:-

... for his many basic contributions to rigorous mathematical results in the general theory of relativity. They include: i) the boost theorems which establish the existence of a large class of solutions to Einstein's (and Yang-Mills) equations in a region obtained by boosting the initial Cauchy surface a finite amount; ii) a complete mathematical analysis of the spherical collapse of a scalar field including the structure of the horizons and singularities; and iii) his work with Klainermann on the global existence of solutions to Einstein's equations with weak initial data.

3.1. The MacArthur Fellows Program.

The MacArthur Fellows Program is intended to encourage people of outstanding talent to pursue their own creative, intellectual, and professional inclinations. In keeping with this purpose, the Foundation awards fellowships directly to individuals rather than through institutions. Recipients may be writers, scientists, artists, social scientists, humanists, teachers, entrepreneurs, or those in other fields, with or without institutional affiliations. They may use their fellowship to advance their expertise, engage in bold new work, or, if they wish, to change fields or alter the direction of their careers. Demetrios Christodoulou received the MacArthur Fellows Award from the MacArthur Foundation in June 1993:-

... for his work in mathematics and physics.

3.2. Christodoulou receives the MacArthur Award.

Demetrios Christodoulou of Princeton University has been awarded a fellowship from the John D and Catherine T MacArthur Foundation. A total of thirty-one new MacArthur Fellows were named this year. Christodoulou will receive $260,000 over five years. The no-strings-attached fellowships support creative work in a wide variety of fields.

Christodoulou was born 19 October 1951, in Greece. Now a professor of mathematics at Princeton, Christodoulou works in the field of general relativity. His recent research has centered on global properties of solutions to Einstein's equations of general relativity. Christodoulou received his M.A. in 1970 and Ph.D. in 1971 from Princeton. He was a research fellow at the California Institute of Technology (1971-1972), a professor at the University of Athens (1972-1973), and a visiting scientist at CERN in Geneva (1973-1974). He spent 1974-1976 at the International Centre for Theoretical Physics in Trieste, and during 1976-1978 was a Humboldt Fellow. He has also been a professor at the University of Syracuse (1983-1987) and at the Courant Institute of New York University (1988-1992). His honours include the Otto Hahn Medal (1980) and the Xanthopolous Prize in Relativity (1991).

In March 1996 Demetrios Christodoulou was presented with the Excellence in the Sciences Award from the Academy of Athens.

5.1. The Bôcher Memorial Prize.

The Maxime Bôcher Memorial Prize is awarded by the American Mathematical Society every five years for a notable research memoir in analysis which has appeared in the previous five years. The prize honours the memory of Maxime Bôcher (1867-1918), who was the Society's second Colloquium Lecturer (1896) and tenth president (1909-1910) and was also one of the founding editors of Transactions of the AMS. The recipient must be a member of the Society, or the memoir for which the prize is given must be published in a recognised North American journal. The prize carries a cash award of $4,000.

5.2. The eighteenth Bôcher Prize in 1999.

The eighteenth Bôcher Prize was awarded to Demetrios Christodoulou, Sergiu Klainerman, and Thomas Wolff at the 105th Annual Meeting of the American Mathematical Society in January 1999 in San Antonio.

The prize was awarded by the AMS Council acting on the recommendation of a selection committee, whose members at the time the 1999 prize was awarded consisted of James Glimm (chair), Peter Sarnak, and Leon Simon.

5.3. Citation for Demetrios Christodoulou.

The Bôcher Prize is awarded to Demetrios Christodoulou for his contributions to the mathematical theory of general relativity. In particular, the prize is awarded for his remarkable memoir with S Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton Math Ser., vol. 41, Princeton University Press, 1993, which establishes the nonlinear stability of the Minkowski metric; and for his fundamental papers "Examples of naked singularity formation in gravitational collapse of a scalar field", Ann. Math. 140 (1994), 607-665, and "The instability of naked singularities in the gravitational collapse of a scalar field", Ann. Math. 149 (Jan. 1999), 183-217, which show, contrary to the widely held view, that naked singularities occur in the gravitational collapse of a scalar field; and for his analysis of the instability of these singularities.

5.4. Response by Demetrios Christodoulou.

It is a great honour for me to be awarded the 1999 Bôcher Prize. I would like to express my gratitude to the Bôcher Prize Committee and the American Mathematical Society for recognising my work.

My mathematical research has centred on the study of global problems associated to nonlinear systems of partial differential equations of hyperbolic type, i.e., the global existence and regularity of solutions to the initial value problem, the formation and structure of singularities, and the long-time asymptotic behaviour. I have given particular attention to the Einstein equations of general relativity where the solution of problems requires a combination of purely analytic and differential-geometric methods.

The joint work with Sergiu Klainerman demonstrated that any asymptotically flat initial data for the vacuum Einstein equations which is suitably close to trivial data gives rise to a global solution, a geodesically complete spacetime tending to flatness at infinity along any geodesic, thus establishing the stability of the Minkowski space of special relativity in the framework of the general theory and providing the basis for a rigorous theory of gravitational radiation.

Another work, involving several steps completed over the years, concerns the initial value problem in the large under the assumption of spherical symmetry, for the Einstein equations with matter, the energy-momentum-stress tensor being that corresponding to a scalar field. One of the principal results of that work was that if the initial data verifies a certain largeness condition, then catastrophic gravitational collapse must occur. This is signalled by the formation of a trapped region, that is, a spacetime region where the future light cones have cross-sectional areas decreasing with time. The formation of a trapped region is preceded by that of an event horizon, namely, of a future boundary of the set of points which are causally connected to infinity. The trapped region ends at a singular boundary, whose structure was analysed in detail. Another, unexpected result of the work was that naked singularities, namely, singular points which are not preceded by a trapped region and which are causally connected to infinity, also occur. The work culminated in a paper which shows that in the space of initial data the subset leading to the formation of naked singularities has positive codimension, hence must be viewed as being exceptional, and initial data belonging to the complement of this subset leads either to a complete regular solution or to the formation of a trapped region.

Despite the fact that some progress has been achieved, our subject is still very much in its infancy. The global stability of the Kerr family of non-trivial stationary solutions of the vacuum Einstein equations has not yet been investigated. The simultaneous removal of symmetry assumptions as well as restrictions on the size of the initial data leads us to a territory which is at present almost completely unexplored. Moreover, the general framework of nonlinear systems of partial differential equations of hyperbolic type offers many additional problems of fundamental significance, such as the development of a general theory of shock formation and evolution in 3-dimensional continuous media.

5.5. Biographical sketch.

Demetrios Christodoulou was born October 19, 1951, in Athens, Greece. He received his Ph.D. in physics from Princeton University in 1971. He held positions as a Humboldt fellow at the Max Planck Institute (1976-81), as a visiting member at the Courant Institute (1981-83), as an associate professor (1983-85) and professor (1985-87) at Syracuse University, and as professor of mathematics (1988-92) at the Courant Institute. Since 1992 he has been professor of mathematics at Princeton University.

Christodoulou was awarded the Otto Hahn Medal in mathematical physics in 1981, the Xanthopoulos Award in relativity in 1991, and a MacArthur Fellowship in 1993. He also received the Excellence in the Sciences Award from the Academy of Athens in 1996, an Honorary Doctorate in the Sciences from the University of Athens in 1996, and a Guggenheim Fellowship in 1998.

6.1. The Cyprus Mathematical Society.

The Cyprus Mathematical Society is a non-for-profit organisation established in Cyprus since 1983. The Cyprus Mathematics Society is managed and developed by volunteers. Its purposes are to promote Mathematics Education and Science. It is a member of the European Mathematical Society and a founding member of the Mathematical Society of South Eastern Europe .

The activities of the Society include the organisation of all national Mathematics competitions and it is in charge for training and organising the Cypriot National team representations for the participation of Cyprus in all levels of the International and Balkan Mathematical Olympiads.

Other activities include the Summer Mathematical School, the annual conference on Mathematical Science and Education, the annual Pupil Mathematics Conference, the Mediterranean Conference on Mathematics and other international conferences such as the EUROMATH.

In January 2000 Demetrios Christodoulou was awarded the Zenon Prize by the Mathematical Society of Cyprus.

The President of the Hellenic Republic awards five classes of the Order of Phoenix. The Order was established on 13 May 1926, by the republican government of the Second Hellenic Republic. It was retained after the restoration of the monarchy in 1935 and continues to be awarded by the current Third Republic.

In July 2000, Demetrios Christodoulou was made a Commander of the Order of Phoenix. He received the badge of the Order, which is a white-enamelled cross in gold for the Commander Class, with the Phoenix (symbolising the rebirth of the Hellenic nation) at the centre.

8.1. The 2006 award.

Professor Demetrios Christodoulou. Professor of Mathematics and Physics at the Federal Institute of Technology in Zürich (ETH Zürich). He was awarded the Bodossaki Excellence Award for his important contribution to the Theory of General Relativity and Gravity. Year of award: 2006.

8.2. Greek mathematicians share the Aristeio prize.

The Aristeio Bodossaki prize goes this year to two Greek mathematicians who have received international recognition in their fields. The award ceremony is to be held this coming Wednesday, 14 June 2006. It is the third time the prize has been awarded to Greek scientists who have earned international renown in science, technology and medicine, who are truly worthy of emulation, and are a source of great national pride. This year's awardees are Athanassios Fokas, 52, a member of the Athens Academy, and Dimitris Christodoulou, a professor at the University of Zurich, for his considerable contribution to the theory of general relativity and gravity. The two scientists will share the prize money of 150,000 euros. The Aristeio Bodossaki award for excellence is awarded to scientists who have made decisive contributions toward furthering their scientific field by means of original and completed mainstream work of the highest possible quality, which has shaped important parts of their field, and has gained international recognition. It was awarded at the Bodossakis Foundation, where the hosts were the rector of Athens University, Professor Georgios D Babiniotis, and chairman of the board of the Bodossakis Foundation Stamatis Mantzavinos. The guest of honour was President Karolos Papoulias. A reception followed. A dinner was held for the two recipients and their families on Tuesday afternoon at the Aigli, Zappeion.

9.1. The Tomalla Prize of the Tomalla Foundation.

The Tomalla Foundation for Gravity Research promotes research into gravity in Switzerland and throughout the world. It was founded in 1982 according to wishes of Dr Walter Tomalla, an engineer from Basel, Switzerland. Every third year, the Tomalla Foundation awards prizes for exceptional research in gravitation and/or cosmology, and funds research fellows and visitors for gravity research at Swiss universities.

9.2. The 2008 Tomalla Prize.

The 2008 Tomalla Prize was awarded to Demetrios Christodoulou:-

... for his important contributions to general relativity, especially for his rigorous demonstration of global non-linear stability of Minkowski spacetime.

9.3. Laudatio by Ruth Durrer: Tomalla Prize 2008 for Demetrios Christodoulou.

Dear Demetri, dear Dean, dear colleagues, students and friends

I am very happy to greet you here today to the Tomalla Prize 2008 ceremony.

The Tomalla Foundation for Gravity Research has been founded in 1982 according to the testamentary wishes of Dr Walter Tomalla, an Engineer from Berlin, who moved to Switzerland after the second war. With his foundation, Dr Tomalla wanted to promote gravity research in Switzerland and in the world.

The foundation periodically awards prizes for exceptional research in gravitation and it funds research fellows and visitors for gravity research at Swiss Universities. It can also promote research on gravity by other activities. It participated, for example in the costs of the edition of the collected papers of Albert Einstein, in an experiment to measure the gravitational constant $G$ and in other scientific activities around gravity.

Two Tomalla Prize laureates (Subrahmanyan Chandrasekhar and Joe Taylor) have afterwards been awarded the Nobel Prize in physics and one (Andrei Sakharov) had obtained the Nobel Prize for peace.

The Tomalla Prize is awarded to a leading scientist in his field which may concern any aspect of gravity research be it experimental or theoretical cosmology, relativistic astrophysics, gravitational waves or mathematical research in general relativity.

This year, we are most proud to attribute the Prize to Professor Demetrios Christodoulou for his important contributions to general relativity, especially for his rigorous demonstration of global non-linear stability of Minkowski spacetime.

For the non-experts: global non-linear stability means that if at some initial time, space is nearly flat and slowly changing, this remains true as time goes on. This means that each observer, which moves freely (along a time-like geodesic) sees the spacetime around it remain nearly flat. Of course this is true in full generality only if spacetime is 'empty'. If there is matter present, even very mild initial conditions can lead to black hole formation via gravitational collapse. Gravitational collapse is actually another of Demetrios' research topic on which he made very significant progress and which is the subject of his talk today.

Maybe some of you think "stability of Minkowski space, this is trivial, empty space cannot be anything else than flat". This is of course not true, there are gravitational waves, waves of spacetime curvature which have themselves energy and momentum and which can propagate freely through empty space much like electromagnetic waves. Hence the 'gravitational collapse of such waves' could in principle lead to singularities, to large deviations from Minkowski spacetime. Demetrios has shown that this does not happen if the waves are sufficiently weak initially. The proof of this statement is very difficult and long. I think an important point is also that Demetrios had to develop several very useful mathematical techniques to perform the proof. This is the reason that whenever you listen to a talk on hyperbolic differential equations or mathematical relativity you hear the name Christodoulou mentioned several times. The techniques he has developed now belong to the main tools when studying non-linear hyperbolic differential equations. I think it is fair to say that Demetrios is the most influential mathematical relativist of our generation.

Even though Demetrios first became really famous for his proof of global non-linear stability he subsequently made other very important contributions to general relativity, most notably on gravitational collapse. (I was fortunate to participate in a course by him on this subject when I was a post-doc in Princeton. There I learned that Demetrios is not only a brilliant scientist but also an excellent teacher.)

In a fundamental paper which appeared in the Annals of Mathematics 1994, Demetrios showed the existence of what we call 'naked singularities' by presenting several examples. Before this, there was a so called 'cosmic censorship' postulate, namely that all singularities, which we know do form in general relativity, are hidden behind a horizon and do, in this sense not affect an observer which is not falling into them. This is very important since it means that the world-lines of observers which do not fall in the singularity can be continued even after the formation of the singularity. The latter is never inside the 'domain of dependence' of such world-lines. Demetrios now showed that 'cosmic censorship' is not always obeyed and naked singularities can form in the gravitational collapse of a scalar field (a hypothetical, not quite ordinary form of matter which offers itself to a simplified treatment). But then he went on, analysed these singularities and showed that they are unstable (and therefore physically irrelevant).

Another of his results which I find extremely fascinating is known as 'Christodoulou's memory effect' or the 'Christodoulou drift'. Demetrios has shown that a non-linear effect can built up during the passage of a gravitational wave. After the passage of a gravity wave burst in a laser interferometric gravity wave experiment, this effect leads to a permanent displacement of the two mirrors which are bouncing back and forth the laser light. Even though this effect is even more difficult to measure than ordinary gravitational waves, since it is not oscillating, it may not be out of range of future experiments.

Demetrios was born in Athens, Greece. His exceptional mathematical talent was discovered early, and at the age of 19 he terminated his undergraduate studies with a master at Princeton University. The main result of his master thesis on reversible and irreversible transformations in black hole physics, is published as a single author PRL. Only a year later, at the age of 20, he finished his PhD also at Princeton University. After that he had several positions at CalTech, at CERN, at the Courant Institute in New York, at Princeton University and since 2001 he is professor for mathematics and physics at ETHZ. (This is only the second time that we can honour a 'Swiss' researcher with the Tomalla Prize.)

Of course, the Tomalla Prize is not the first honour for Demetrios. He obtained other prestigious awards. Among them let me just mention the Otto Hahn Medal, the MacArthur Fellows Award, the Bôcher Memorial Prize and several honorary doctorates.

But let me stop here. Demetrios is with us today and he will take us on a journey from the geometrical basis of general relativity to his results on gravitational collapse and the formation of unstable naked singularities. His colloquium is entitled "The Fabric of Spacetime".

10.1. The Shaw Prize.

The Shaw Prize was established under the vision and generosity of the late Mr Run Run Shaw in 2002. It is an international award to honour individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The award is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.

The Shaw Prize consists of three disciplines, namely, Astronomy, Life Science and Medicine, and Mathematical Sciences and is presented annually by the Shaw Prize Foundation since 2004.

10.2. The 2011 Shaw Prize in Mathematical Sciences.

The 2011 Prize was awarded to Demetrios Christodoulou and Richard S Hamilton:-

... for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

10.3. Contribution of Demetrios Christodoulou.

Since Riemann's invention of a geometry to describe higher dimensional curved spaces and Einstein's introduction of his equations to describe gravity, the theory of the associated nonlinear partial differential equations has been a central one. These equations are elegant but in general they are notoriously difficult to study. One of the key issues is whether the solutions develop singularities.

Demetrios Christodoulou has made fundamental contributions to mathematical physics and especially in general relativity. His recent striking dynamical proof of the existence of trapped surfaces in the setting of Einstein's equations in a vacuum demonstrates that black holes can be formed solely by the interaction of gravitational waves. Prior to that he made a deep study of this phenomenon in symmetrically reduced cases showing that unexpected naked singularities can occur but that they are unstable. In joint work with Klainerman he established the nonlinear stability of the Minkowski spacetime. His work is characterised by a profound understanding of the physics connected with these equations and brilliant mathematical technique.

10.4. An Essay on the Prize.

After Newton's introduction of calculus and in particular differential equations to describe the motion of the planets, classical physics and geometry developed with more complex phenomena naturally being formulated in terms of partial differential equations. The Einstein equations in general relativity and the Ricci Flow equation in Riemannian geometry are two celebrated geometric partial differential equations. The first describes the geometry of four dimensional spacetime and it relates gravitation to curvature. The second gives an evolution of Riemannian geometries in which the flow at a given time is dictated by the curvature of the space at that time. Both of these equations are very elegant in their formulation. They are nonlinear partial differential equations in several unknown quantities which in turn depend on several variables. While they are of quite different characteristics in terms of the classification of such equations, they share the feature that they are notoriously difficult to study rigorously (even on a computer). Central to the understanding of the solutions, is whether they form singularities or not, and if so what is their nature. In the spacetime setting, examples of singularities are black holes and more generally gravitational collapse. In the Ricci Flow, should singularities arise in the course of the evolution, then for certain applications they need to be resolved. Christodoulou, in the case of Einstein's equations, and Hamilton in the case of the Ricci Flow, have made many of the fundamental breakthroughs in the theory of these geometric equations and especially in understanding their singularities. Their works have spectacular applications both to mathematics and to physics.

Demetrios Christodoulou

Christodoulou works in mathematical physics and in particular the differential equations that describe classical physical phenomena. His continued and profound study of the global behaviour of solutions to the Einstein equations has been instrumental in shaping our present day understanding of the critical features of these solutions. In particular, his striking recent proof of the dynamical development of trapped surfaces in the setting of Einstein's equations in a vacuum shows that black holes can be formed solely by the interaction of gravitational waves. Earlier, he had made a breakthrough study of the reduced spherically symmetric Einstein equations, showing that naked singularities can occur for these but that they are rare and unstable. This work resolved the much debated problem of "weak cosmic censorship" for these symmetrically reduced cases. His joint work with S Klainerman establishing the long sought after nonlinear stability of Minkowski spacetime, is by now one of the classical theorems in the theory. In other very novel works, Christodoulou has given the first detailed treatment of the delicate problem of the formation of singularities, called shocks, in the equations for fluid flow in three dimensions. Christodoulou's work combines a deep understanding of the underlying physics with brilliant mathematical technique. This has allowed him to resolve central problems that have resisted progress for generations.

10.5. Autobiography: Demetrios Christodoulou.

I was born in Athens in 1951 to a lower middle class family. My father was born in Alexandria to Greek parents from Cyprus who had immigrated to Egypt. My mother was born in Athens to a family of Greek refugees from Asia Minor. Neither of my parents had higher education, but my father inspired me in childhood with stories from a distant past when ancient Greece had made outstanding contributions to human civilisation. A problem in Euclidean geometry was the spark which initiated in me, in the summer of 1966, a burning interest in mathematics and theoretical physics. My case was brought to the attention of Achilles Papapetrou, a Greek physicist at IHP, who in turn contacted Princeton physics professor John Wheeler, on leave in Paris at that time. So, at the beginning of 1968 I came to Paris and was examined by them. This led to my admission as a graduate student in the Princeton physics department in the fall of 1968.

A decisive turn in my career came in 1977, at a time when I was a postdoctoral fellow at the Max Planck Institute for Astrophysics in Munich. There, Jurgen Ehlers, the leader of the group in which I was working, although himself a physicist, realised that I had a talent in mathematics and gave me an unlimited leave of absence with pay to study mathematics in Paris under the guidance of Yvonne Choquet-Bruhat. Thus, I finally found my true calling and in the period 1977-1981 I studied mathematical analysis in the French school.

In 1981, I returned to the U.S. and one of the first scientists I met was the famous Chinese mathematician Shing-Tung Yau. I became closely associated with him for a period of five years, an association which played a decisive role in my mathematical make-up. From Yau I learned geometry and how to effectively combine geometry with analysis in what is today called geometric analysis, a field which Yau pioneered. I can summarise my scientific contribution since, as the extension of geometric analysis from the initial field of elliptic equations to the field of hyperbolic equations.

The first work of geometric analysis of hyperbolic equations was my work with Sergiu Klainerman on the stability of the Minkowski space-time, the fruit of an intensive effort in the period 1984-1991. This work demonstrated the stability of the flat spacetime of special relativity in the framework of the general theory and gave a detailed description of the asymptotic behaviour of the solutions. Basically, an initial disturbance in the fabric of space-time propagates, like the disturbance in a quiet lake caused by the throwing of a stone, in waves, the so-called gravitational waves. However, as I showed in a further 1991 paper, there is a subtle difference from the lake paradigm. For whereas spacetime becomes, again, like the lake, flat after the passage of the waves, the final flat spacetime is related in a non-trivial manner to the initial flat spacetime and this leads to an observable effect: the permanent displacement of the test masses of a gravitational wave detector.

Roger Penrose had introduced, in 1965, the concept of a trapped surface and had proved that a spacetime containing such a surface cannot be complete. A little later, it was shown that under the same assumption there is a region of spacetime which is inaccessible to observation from infinity: the black hole. However, the available mathematical methods were incapable of investigating how trapped surfaces form in evolution and of revealing the nature of the spacetime boundary. Penrose conjectured that the boundary is always contained in a black hole, a conjecture called cosmic censorship. Seeking to answer these questions in a simpler setting I studied, in a series of papers completed in the period 1984-1997, the spherically symmetric Einstein equations with a scalar field as the matter model. An unexpected result was that naked singularities, that is, singularities not contained in a black hole, can also form. Nevertheless, I proved that these are unstable, thus establishing a generic version of cosmic censorship in this framework.

As professor of mathematics at the Courant Institute 1988-1992 and at Princeton 1992-2001, I enjoyed a very stimulating scientific environment. In 2001, I returned to Europe, taking up my present position as professor of mathematics and physics at ETH in Zurich.

The period 2001-2008 was, for me, one of most intense intellectual effort. I turned to the study of the formation of shocks in compressible fluids in the physical case of three spatial dimensions. Here the aim was to carry out the analysis up to the singular boundary. The Eulerian equations of fluid mechanics have affinity with the Einstein equations of general relativity, both constituting nonlinear systems of hyperbolic type. At the same time, I turned to the study of the formation of trapped surfaces in general relativity, in vacuum and without any symmetry assumptions, through the focusing of incoming gravitational waves. The breakthroughs came in 2004, and the two works were completed in 2006 and 2008 respectively. In the case of the second work, the breakthrough took the form of a new method which exploits the assumption that the initial data contain somewhere an abrupt change and allows us to attack problems which had seemed unapproachable.

28 September 2011, Hong Kong

10.6. ETH Zurich researcher wins "Asia's Nobel Prize".

Demetrios Christodoulou, a mathematician and physicist at ETH Zurich, wins the illustrious "Shaw Prize in Mathematical Sciences". The award of "Asia's Nobel Prize" honours Christodoulou's contributions to differential geometry.

Demetrios Christodoulou, Professor of Mathematics and Physics at ETH Zurich, has won the "Shaw Prize in Mathematical Sciences". According to the Shaw Prize Foundation in Hong Kong, Demetrios Christodoulou is being honoured for his highly innovative research work on non-linear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

Since 2004, the Shaw Prize Foundation has honoured astronomers, life scientists and mathematicians who have achieved a significant scientific breakthrough that has resulted in a positive impact on mankind. The Foundation was established by Run Run Shaw, the founder of a major film company in Hong Kong.

In the distinguished company of the most renowned geometricians.

A happy Demetrios Christodoulou says, "I feel very honoured, because the Shaw Prize is "Asia's Nobel Prize" and, together with the Abel Prize is the highest award available for mathematicians. It has always been awarded exclusively to outstanding mathematicians." According to Christodoulou, China's Shiing-Shen Chen, the first winner of the Shaw Prize in Mathematical Sciences, was "the most important geometrician of the 20th century". The second winner, Andrew John Wiles, became well-known far beyond the world of mathematics when, in 1994, he succeeded in proving what is known as Fermat's conjecture. Prior to that, mathematicians had laboured in vain for about 350 years on a solution of this equation by the French mathematician Pierre de Fermat.

Until 2006, the so-called Poincaré conjecture was one of the seven most important unsolved problems in mathematics. Then the Russian mathematician Grigori Yakovlevich Perelman managed to prove this theorem of the famous French mathematician Henri Poincaré.

The decisive foundation for Perelman's proof was laid by the American Richard S Hamilton, Professor of Mathematics at Columbia University in New York.

Hamilton and Christodoulou share the 2011 Shaw Prize in Mathematical Sciences, endowed with one million dollars. Demetrios Christodoulou says, "I am particularly pleased because I have the highest esteem for his work in differential geometry. Hamilton is also one of my closest friends."

Demetrios Christodoulou has taught and researched at ETH Zurich since 2001. His research focuses on partial differential equations and differential geometry in conjunction with the development of general relativity and fluid mechanics. His papers have already been honoured several times in the past: for example he received the Otto Hahn Medal in 1981, the Bôcher Memorial Prize of the American Mathematical Society in 1999 and the Tomalla Prize for gravity research in 2008.

11.1. The Nemitsas Prize 2016 in Mathematics.

Professor Demetrios Christodoulou was selected by the International Experts' Committee for the Nemitsas Prize 2016 in Mathematics and the Foundation's Board of Directors unanimously confirmed their decision. Professor D Christodoulou is one of the most accomplished mathematicians worldwide, and it is an honour for Cyprus to have such a distinguished scientist.

11.2. Presentation of the Award.

The 7th annual award ceremony was dedicated this year to mathematics and it was a great success. It took place on 3 October 2016, at the Presidential Palace of the Republic of Cyprus. The Prize was awarded by the President of the Republic of Cyprus Mr Nicos Anastasiades to Professor Demetrios Christodoulou.

11.3. Speech by Marios Mavronicolas at the Award Ceremony.

It is with great pleasure that I have the honour and privilege to present, on behalf of the Takis and Louki Nemitsas Foundation, the work and personality of Professor Demetrius Christodoulou, who tonight receives the 2016 Nemitsas Prize in Mathematics.

Demetrius Christodoulou was born in Athens, in October 1951. His father Lambros was born in Alexandria to Greek Cypriot parents who had immigrated to Egypt. His grandfather Miltiadis came from Agios Theodoros and his grandmother Eleni from Khirokitia. His mother Maria was born in Athens to refugee parents from Asia Minor, Greeks who hid Greece within themselves. Little Demetrius was nurtured in the greatness of the race and the contribution of Greece to world culture. Since he was a small child, his Alexandrian father Lambros inspired him with all the adoration for Greek culture, taking him to ancient monuments, museums and archaeological sites and characteristically telling him that the ancient Greeks were invincible.

The young Demetrius studied at the "Experimental Lyceum of Athens", today the Moraitis school. His interest in mathematics and theoretical physics was sparked at the age of fourteen by a problem in Euclidean geometry. A student then in the third grade of high school, he was furiously trying to solve an unsolvable problem in mathematics, the problem of trisecting an angle, not knowing at the time that the solution with a ruler and compass had been proved impossible. He himself confesses today that it was his futile attempt to find the non-existent solution that caused him to suspect the immense depth of mathematics.

The focus of Demetrius Christodoulou's interests in physics and mathematics soon turned to Riemannian geometry, to Einstein's general theory of relativity and to space-time. In the 1960s he was a child prodigy in mathematics and physics. He was a student at the High School in 1968 when the newspapers of Athens wrote with bold headlines: "Student-Einstein: High School Student Reads Higher Mathematics and Solves Difficult Problems". When he was a 16-year-old teenager, he worked on Laplace transforms, and the fourth-year students of the Polytechnic would give him their own problems and he would solve the problems for them. Young Demetrius' fame spread quickly and reached important Greek scientists abroad, such as the theoretical physicist Achille Papapetrou and the internationally renowned mathematician Christos Dimitriou Papakyriakopoulos. The news spread and crossed the Atlantic.

At the age of 17, in 1968, Demetrius Christodoulou left Greece without finishing school to continue his studies as a graduate student in physics at Princeton University in the United States. In the autumn of 1970, one month after his nineteenth birthday, the young Demetrius published his first scientific paper entitled "Reversible and irreversible transformations in black hole physics", which opened a new chapter in the thermodynamics of black holes. At the age of 20, in 1971, he received his PhD in physics from Princeton University. Under the guidance of Professor J A Wheeler, his PhD thesis is a landmark in the field of the physics of black holes.

The development of Demetrius Christodoulou's brilliant research and academic career has been rapid since then. After a year, as a postdoctoral fellow at the California Institute of Technology, he returned in 1972 to Greece as a Professor at the National and Kapodistrian University of Athens. Soon, however, he resigned, and continued his career as a researcher at CERN (in Geneva), the International Centre for Theoretical Physics in Trieste, the Max Planck Institute (in Munich) and the famous Courant Institute in New York. In 1983 he became Associate Professor of Physics at Syracuse University in New York where two years later, in 1985, he was promoted to full Professor. In 1988 he returned to the Courant Institute as Professor of Mathematics. Between 1992 and 2001 he was Professor of Mathematics at Princeton University in the USA, which has flourished as a world's leading centre of excellence in Mathematics for the last 80 years. From 2001 he moved to the Eidgenössische Technische Hochschule Zürich, one of the largest and most important in Europe, as a Professor of Mathematics and Physics.

A significant part of Demetrius Christodoulou's scientific work focuses on Einstein's General Theory of Relativity, the stability of the Minkowski space, the creation of black holes in the vacuum under strong gravitational waves, and on three-dimensional fluids. A section of his work is the solution of difficult differential equations which arise in these studies.

The main work for which the laureate is known worldwide has to do with the equations of Einstein's General Theory of Relativity. Demetrius Christodoulou developed new mathematical methods for solving or understanding the solutions to Einstein's equations, which, while elegant, were notoriously difficult. Processing them to produce a meaningful result had encountered insurmountable mathematical difficulties for many years, since 1915 when they were stated.

In the context of the General Theory of Relativity, Demetrius Christodoulou has proved the stability of the Minkowski space-time in which we live. Let me explain that Minkowski space is the mathematical space in which Einstein's Special Theory of Relativity is best suited to represent. In this space there are the usual three spatial dimensions, which combine with the dimension of time to form a four-dimensional topological manifold to represent spacetime. The laureate has demonstrated the infinitely nonlinear nature of gravitational waves and formulated rigorous theorems about the creation of black holes. Already his doctoral thesis contained the central idea that led other great researchers, such as Hawking and Bekenstein, to the final formulation of the thermodynamics of black holes and Hawking radiation. Finally, the works of Professor Christodoulou in the theory of differential equations, in addition to their general importance for mathematics, are also important for the study of fluid dynamics. Throughout all his work, the laureate reveals in an elegant and unique way the mysterious and, perhaps, romantic relationship between mathematics and physics.

Professor Christodoulou's illustrious career is full of many important international awards, distinctions and scholarships. In 1981 he received the Otto Hahn Medal in Mathematical Physics from the Max Planck Society. In 1991 he received the Vassilios Xanthopoulos Award from the International Association for General Relativity and Gravitation. In 1993 he received the MacArthur Foundation Honorary Fellowship in Mathematics and Physics. In 1996 he received the Science Award from the Academy of Athens. In 1998 he received the honorary John Simon Guggenheim Fellowship. The most important prize of the American Mathematical Association in the field of analysis is the Bôcher Memorial Prize, which was awarded to him in 1999. In 2006, together with Professor Athanasios Fokas, he received the Aristhio Bodosakis Award. In 2008 he received the Tomalla Prize from the Tomalla Foundation for the application of General Relativity in Astronomy. In 2009, the President of the Hellenic Republic awarded Professor Demetrius Christodoulou with the Commander Grade of the Order of Phoenix.

In 2011 the Shaw Foundation decided to award Professor Demetrius Christodoulou, together with another giant of modern Mathematics, Professor Richard Hamilton, the Shaw prize in Mathematics. The Shaw Prize, also called the Nobel Prize of Asia, is considered by many to be, together with the Abel Prize of Norway, one of the two most important international prizes in mathematics. The two professors were awarded for their work in nonlinear differential equations and Lorentz and Riemannian geometries, and their applications to general relativity and topology.

Demetrius Christodoulou has been elected a member of the American Academy of Arts and Sciences, the European Academy of Sciences, the US National Academy of Sciences since 2012 and the Academia Europaea from this year. He has also been awarded an Honorary Doctorate by a number of universities such as, for example, Brown University, the National and Kapodistrian University of Athens, the National Metsovian Polytechnic, the Aristotle University of Thessaloniki and the University of Cyprus in 2003. He has given numerous honorary lectures as keynote and plenary speaker at universities, conferences and research institutes internationally. For example, he gave the Keynote Address at the International Congress of Mathematical Physics in 2009 and at the International Congress of Mathematics in 2014.

Demetrius Christodoulou has a characteristic simplicity in his lectures; an emotionality and an intimacy characterise them, while he always speaks without text and enchants the audience. When he was once asked in an interview to give an explanation for this, he said that he does not prepare a text in order not to spoil his spontaneity. He then wondered if Alexander the Great had a text and noticed that some politicians have a text because they read, perhaps, what their advisers write for them. His speech is elliptical, full of parentheses, brackets and cross-references, as, perhaps, is the thought of mathematicians.

The laureate is the author of several established and classic monographs. He has also supervised a significant number of PhD theses and has excelled in his keen interest in supporting and promoting young scientists in mathematics.

Greece and Cyprus are well represented by notable scientists in the international community of mathematics. An essential part of this success is due to the influence that Demetrius Christodoulou exerts on mathematicians from Greece and Cyprus, either directly through teaching and research guidance or indirectly, being himself a brilliant example to follow.

Although his academic seat is in Switzerland, Demetrius Christodoulou maintains a keen interest in academic developments in Greece, which he visits to give lectures at every possible opportunity. The lecture he gave in the Old Parliament of Athens on "Mathematics in Ancient Alexandria, Euclid - Archimedes" held under the auspices of the Association "Friends of the Library of Alexandria" remains historic. His lecture, which took him back to Alexandria, is developed in a book of the same name published by Eurasia Press - it is, perhaps, his first book in Greek. His eyes sparkle whenever he talks about the mathematics of the ancient Greeks, his second great passion. At every opportunity he points out that the contribution of the ancient Greeks to modern mathematics is not taught as much as it should be in Greece, despite the enormous global interest.

Allow me to quote verbatim a clear statement by the laureate about the social crisis in Greece: "I remember the time when I was a small child, Greece had a very low standard of living, but a very high intellectual level. It is a fact that the image that our country transmits abroad is not the best. Gone are the days when there were Greeks with international influence. We show the image of a state where, unfortunately, the brazen and frivolous are the ones who are projected and survive."

Demetrius Christodoulou is internationally recognised as a communicator of scientific ethics and the selfless search for truth. The first contact you will have with him will convince you beyond any doubt of his outstanding intellect and charismatic personality. He is warm, cultured, generous and above all simple and humane.

The laureate is automatically ranked as one of the leading international scientists in mathematics. Summarising his work and personality, I feel proud to state that he is a leading, talented and complex human being who, with his scientific work, honours Hellenism internationally. Allow me, in fact, to dare to say that in my opinion Demetrius Christodoulou is a modern Karatheodoris.

We hope, Professors Christodoulou, that today's award of the Nemitsas Prize 2016 will strengthen your ties with your distant and beloved homeland, the divided Cyprus, which we know you always think of and love.

Ladies and gentlemen,

In closing, please allow me to state, with no trace of exaggeration, that today, through the award of the 2016 Nemitsas Prize to Cypriot origin's Professor Demetrius Christodoulou, which will be given to him in a few minutes by the President of the Republic of Cyprus in this modest ceremony, that the Nemitsas Foundation is honouring Demetrius Christodoulou for his contributions to mathematics. But also Demetrius Christodoulou is honouring, through his presence here, the Nemitsas Foundation, and all of Cyprus.

Prominent Greek, originating from Saint Theodoros and Khirokitia of Cyprus, charismatic and unsurpassed scientist, scholar, mentor, colleague and friend Demetrius, we thank you very humbly for everything you have contributed and for the honour you are bringing to all of us tonight. I welcome you to the recipients of the Takis and Louki Nemitsas Prize Laureates.

11.4. Talk by Demetrios Christodoulou during the Award of the Nemitsas Foundation Prize in Mathematics.

I was born in Athens 65 years ago. My father Lambros was born in Alexandria to Greek parents from Cyprus who had immigrated to Egypt. My grandfather Miltiades was from Agios Theodoros and my grandmother Eleni was from Choirokoitia. My mother Maria was born in Athens to a family of Greek refugees from Asia Minor. My paternal grandparents left Egypt and settled in Greece in the 1950's, so I also had the opportunity to listen to many wonderful stories from my grandfather about his adventures as a youngster in the Public School of Cyprus during the last part of the 19th century. As a child, I used to take long walks with my father, often in the environs of the ancient monuments and he used to inspire me with stories from the glorious distant past of Hellenism. My father also used to take me to a movie theatre for children which showed documentaries. I still remember how impressed I was on seeing a documentary about Einstein. When I was 14, a burning interest in mathematics and theoretical physics was, all of a sudden, born in me. Within a couple of years I became fascinated with the concepts of space and time, with Riemann's geometry and Einstein's relativity. My case was brought to the attention of Achiles Papapetrou, a Greek physicist at the Institute Henri Poincare, who in turn contacted Princeton physics professor John Wheeler, on leave in Paris at that time. So in the beginning of 1968 I came to Paris and was examined by them. This led to my admission as a graduate student in the Princeton physics department in the fall of 1968, a month before my 17th birthday.

In the fall of 1970, one month before my 19th birthday, I wrote my first scientific paper "Reversible and irreversible transformations in black hole physics". When I first presented it, Wheeler, seeing that a new chapter was opening, the thermodynamics of black holes, became so excited that he set off firecrackers. He had, you know, a particular love of explosions. The fall of 1977 was very important for my course in life because at that time my scientific outlook was radically transformed. I had been since the previous year as a postdoctoral fellow at the Max Planck Institute in Munich, in the group of Jürgen Ehlers. Ehlers, though himself a physicist, recognised that I had mathematical talent which had not yet manifested itself. My knowledge of mathematics at that time was only at the undergraduate level. Ehlers was extraordinarily generous with me. He gave me a leave of absence with pay for an indefinite period in order to go to Paris to study mathematics under the guidance of Yvonne Choquet-Bruhat and in the period 1977-1981 I studied mathematical analysis in the French school. So with the encouragement of Ehlers I finally found my own path in science, the development of mathematics for the solution of physical problems.

In 1981 I returned to the U.S. and one of the first scientists I met was the famous Chinese mathematician Shing-Tung Yau. I became closely associated with him for a period of 5 years, an association which moulded my character as a mathematician. From Yau I learned geometry and how to effectively combine geometry with analysis in what is called geometric analysis, a field which Yau pioneered. My own contribution since has been the extension of geometric analysis from the initial field of elliptic equations to the field of hyperbolic equations, and from the geometry of space to the geometry of spacetime. My motivation for this extension was the study of the dynamical problems of continuum physics.

The first work of geometric analysis of hyperbolic equations was my work with Sergiu Klainerman on the stability of the Minkowski spacetime, the fruit of an intensive effort in the period 1984-1991. This work demonstrated the stability of the flat spacetime of special relativity in the framework of the general theory. In general relativity spacetime is curved and its curvature, which corresponds to gravitation, satisfies Einstein's equations. The main conclusion of the work is that an initial disturbance in the fabric of spacetime propagates, like the disturbance in a quiet lake caused by the throwing of a stone, in waves, the gravitational waves. However, as I showed in a further 1991 paper entitled "The nonlinear nature of gravitation and gravitational wave experiments", there is subtle difference from the lake paradigm. For, whereas spacetime, becomes again, like the lake, flat after the passage of the waves, the final flat spacetime is related in a non-trivial manner to the initial flat space-time and this leads to an observable effect, called "nonlinear memory effect", the permanent displacement of the test masses of a gravitational wave detector. There are currently ongoing efforts to detect this effect.

In the period 1988-1992 I was professor of mathematics at the Courant Institute. In 1992 I returned to my alma mater, Princeton University, as professor of mathematics. In 2001 I returned to Europe taking up my present position as professor of mathematics and physics at the ETH in Zurich. This position, which had teaching duties only during the fall semester, allowed me to be in Greece for the rest of the year focusing in complete isolation on research. And of course at the same time enjoying the beauty of life in the home country.

The period 2001-2008 was for me one of most intense intellectual effort. I turned to the study of the formation of shocks in compressible fluids in the physical case of 3 spatial dimensions. My work in this topic resulted in a monograph "The Formation of Shocks in 3-Dimensional Fluids", which studies what happens after a long time when we have an arbitrary initial disturbance in any fluid. After a suitably long time, depending on the size of the initial disturbance, certain surfaces in spacetime appear, where the rate of change of the physical quantities, such as the temperature, pressure and velocity of the fluid, blow up. From these surfaces discontinuities in the physical quantities develop, the shocks. This problem was first studied by Riemann himself in 1860, however only in the simplified form it takes when we restrict the dimensions of space to one. My monograph treated the real physical problem. The concept of spacetime played a central role here as well, however not the real spacetime but rather what I called "acoustical spacetime", which corresponds, so to say, to the experiences of a blind person, who can only hear. In 2004, when I had already worked on the problem for 5 years, I came to a dead end. So I put the problem aside for the period of Easter, and soon afterwards travelled to America. At Princeton I met my friend Andrew Wiles who had particular experience with mathematical difficulties, as he had solved a problem which had been open for 350 years. Returning in the evening to my hotel in New York, I fell asleep. And during the night, in my sleep, not only did the solution of this problem come to me, but also that of another problem, which I shall discuss in the following, with which I had not occupied myself at all. On waking up in the morning, I wrote down notes, which I completed on returning to Greece. Those notes that concerned the other problem, I placed in a drawer, waiting for 2 years to be exploited, because in the meantime I had to complete the monograph on shocks, which I dedicated to the memory of my beloved father.

Now, in regard to the other problem, Penrose had introduced in 1965 the concept of a trapped surface on the basis of which he proved a remarkable theorem asserting that a spacetime containing such a surface must come to an end. A little afterwards it was shown that, under the same hypothesis of the presence of a trapped surface, there is a region of spacetime which cannot be observed from infinity, the black hole. A major challenge since that time had been to find out how trapped surfaces form by analysing the dynamics of gravitational collapse. In that magical night of 2004 the idea came to me which would enable me eventually to meet this challenge in May 2008 when I completed the monograph "The Formation of Black Holes in General Relativity". This monograph studies the formation of trapped surfaces in pure general relativity, that is, in the absence of matter, through the focusing of gravitational waves. My old physics professor, John Wheeler, had mentioned to me this problem back in 1968. Fortunately, he did not assign it to me as my Ph.D. thesis topic, otherwise instead of getting my doctorate at 19 I would only have gotten it at 56. The idea constitutes a new method which capitalises on the hypothesis that the initial data contain somewhere an abrupt change, like the ground when we are on a plateau and we reach the edge beyond which a plain extends. This method allows us to study the long time behaviour of the corresponding solution, illuminating a region of knowledge which had previously been considered inaccessible.

It is said of us mathematicians that we only need paper and pencil. However the truth is that these are only needed in recording the ideas and the results which follow. The ideas themselves do not even need pencil and paper. To me inspiration always comes in the night when I am at that intermediate state between deep sleep and awakedness, when concentration in the world of ideas is at a maximum.

Closing, I would like to emphasise the contribution of the Nemitsas Foundation in raising the intellectual and spiritual level of our people. The importance of medicine and of engineering is appreciated by all, since health is the greatest good and the development of technology that which allowed the human race first to survive and eventually to dominate the earth. The natural sciences, physical and biological, being the basis of engineering and medicine respectively, also contribute. At the same time however, as part of what Aristotle called philosophy, they participate in the intellectual and spiritual life of man. Music and the other fine arts likewise participate. In all these fields the Nemistas Foundation has given awards in preceding years. And this year proceeds with an award in mathematics, which support the physical sciences, and which according to Aristotle are also part of philosophy.

12.1. The Henri Poincaré Prize in Mathematical Physics.

The Henri Poincaré Prize in Mathematical Physics is sponsored by the Daniel Iagolnitzer Foundation and awarded by the International Association of Mathematical Physics. The Prize was created in 1997 to recognise outstanding contributions in mathematical physics, and contributions which lay the groundwork for novel developments in this broad field. The prize is awarded every three years at the International Mathematical Physics Congress and in each case, is an award to three individuals.

12.2. Citation for Demetrios Christodoulou.

Demetrios Christodoulou awarded the Henri Poincaré Prize in Mathematical Physics 2021:-

... for pathbreaking contributions to mathematical understanding of the Einstein equations, including fundamental results on black hole formation and the discovery of a nonlinear memory effect in the theory of gravitational radiation, and for introducing a powerful geometric point of view for the problem of shock formation for compressible fluids.

12.3. Laudatio for Demetrios Christodoulou by Igor Rodnianski.

It is a great honour for me to introduce Demetri Christodoulou - a laureate of the 2021 Poincaré Prize - awarded "for pathbreaking contributions to mathematical understanding of the Einstein equations, including fundamental results on black hole formation and the discovery of a nonlinear memory effect in the theory of gravitational radiation, and for introducing a powerful geometric point of view for the problem of shock formation for compressible fluids."

Demetri Christodoulou is a singular mathematician whose work has had a profound impact on the fields of General Relativity, hyperbolic partial differential equations and fluid dynamics.

Even though I described him as a mathematician, Demetri started his journey in the physics department. He received his PhD in physics in Princeton at the age of 19 under the direction of Johny Wheeler. His thesis showed existence of an irreducible mass of a black hole and played a key role in the development of black hole thermodynamics. It has been reported that Demetri's original thesis problem, which he was unable to solve at the time, was a bit more challenging and its solution required a 40 year detour.

This detour took Demetri first to mathematics and then to the point of view, his point of view, which sees general relativity as the arena of partial differential equations and geometry.

One of the early successes of this philosophy was his work throughout the 80's and the early 90's on the spherically symmetric Einstein-scalar field model. Here, Demetri proved a slew of remarkable results, giving an almost complete description of large data dynamics and establishing a very satisfying dichotomy: for generic initial data, gravitational and scalar waves either disperse and the spacetime converges to the flat Minkowski space, or a black hole with the exterior which converges to Schwarzschild forms. The generic caveat is crucial, for he also found exotic solutions containing so called naked singularities which, mercifully, turned out to be unstable. This circle of ideas was also an inspiration behind the discovery and the study of the so called critical phenomena in numerical relativity, in which one probes the universality of behaviour on the boundary in transition from "regular" to "singular" regime.

A truly watershed moment was his proof, in 1993, jointly with Klainerman, of stability of Minkowski space. This was not just a fundamental result but it gave birth to the synthesis of Lorentzian geometry and hyperbolic PDE's. Its lasting impact is still felt today. This work, among other things, established the laws of gravitational radiation and led Demetri to the discovery of the nonlinear gravitational memory effect - a measurable phenomenon in which a wave train causes a permanent relative displacements of test masses. This is now known as the Christodoulou memory effect.

In 2009 Demetri returned to his original thesis problem, his pièce de résistance - the problem of black hole formation. Here, the story begins with the Penrose's incompleteness theorem from 1965 which guarantees geodesic incompleteness of any spacetime satisfying a null energy condition, possessing a non-compact Cauchy hypersurface, and also, crucially, a 2-dimensional trapped surface. For all the incredible importance and influence of this result, it fails to address if this type of geodesic incompleteness (colloquially referred to as singularity) can develop in evolution. Yet, this result lies at the foundation of all of our current understanding of the predicted theory of gravitational collapse and, in particular, its mechanism of black hole formation. With trapped surfaces being a characteristic feature of black holes, this boils down to the question of evolutionary formation of trapped surfaces.

The problem lay dormant for 40 years until it was solved in 2009 by Christodoulou for the problem without matter (note that matter makes black hole formation easier) and without symmetry. It was a remarkable tour de force. The basis of it was an astonishing insight identifying a whole class of initial data which, on one hand are sufficiently large, since this phenomenon requires a strong gravitational field regime, while, on the other, still allows one to control the dynamics all the way to the eventual formation of a trapped surface. The appropriate data turned out to correspond to sharp directional bursts of curvature. It takes a painstaking analysis and a deep understanding algebraic structure of the Einstein equations to construct the necessary large enough portion of spacetime. It turned out to be an even more powerful idea when viewed in a more general PDE context.

After that or, actually, already slightly before, Demetri turned to an even older subject - dynamics of 3-dimensional compressible fluids governed by the Euler equations. A fundamental feature of these equations is that its solutions develop shock singularities even when generated by smooth and, even more remarkably, small data. This is a classical phenomena much studied in the literature going back to Riemann and Stokes. The 20th century development of the subject was focused on creating a framework accommodating global solutions containing shocks. This became tractable only for the 1-dimensional equations and was based on analysis associated with the functions of bounded variation with the pioneering work by Glimm, Lax, Oleynik, Kruzkov and others. In higher dimensions, including the physical 3-dimensional case, conceptual and analytical difficulties were very high and progress very slow.

In 2007 Christodoulou published a monograph containing a proof of shock formation for the 3-dimensional relativistic Euler equations. There, he identified a precise class of initial data for which he constructed a maximal Cauchy development with a singular boundary and gave a complete geometric description of both the boundary and of the singular behaviour of solutions. The result was a consequence of a completely novel geometric point of view on the problem. The corresponding results for the Newtonian case were given a self-contained adaption by Christodoulou-Miao in 2012.

It turns out, that part of the constructed Cauchy development becomes unphysical and has to be replaced by a physical solution containing a shock which emanates from the first singular surface. This is the problem of shock development, which, once again, was solved by Demetri in his monumental work in 2017, albeit in the restricted irrotational context.

These works signify the dawn of new era in the study of higher dimensional compressible fluid dynamics, or more general classes of equations admitting shocks, whose ultimate horizon is perhaps to supersede the global theory of weak shock admitting solutions, developed in the 1-dimensional case.

Let me conclude by saying that Christodoulou's impact on General Relativity, fluid dynamics and PDE's can not be overstated. It will be absorbed, internalised, and felt for many years to come. Demetri's career took him from Princeton, to Caltech, to Munich, to Syracuse, to New York, back to Princeton and then to Zurich. It was said about Feynman that the productivity of his colleagues was inversely proportionate to the distance from their offices to his. I think a bigger compliment might be the opposite statement. For Demetri, this can be attested by generations of mathematicians who have read and tried to digest Demetri's papers and books. I, personally, was fortunate enough to meet Demetri when I came to Princeton and then be influenced by his work over the years, so I was lucky to benefit from both. I would like to offer Demetri again my warmest congratulations on his outstanding achievement in winning the 2021 Henri Poincaré Prize.

Igor Rodnianski, Princeton University.

13.1. The Marcel Grossmann Award.

The Marcel Grossmann Award is made by the International Centre for Relativistic Astrophysics. Each recipient is presented with a silver casting of the T.E.S.T. sculpture by the artist A Pierelli. The original casting was presented to His Holiness Pope John Paul II on the first occasion of the Marcel Grossmann Awards in 1985.

13.2. The 2021 Marcel Grossmann Award.

Demetrios Christodoulou was awarded a 2021 Marcel Grossmann Award by the International Centre for Relativistic Astrophysics:-

... for his many lasting contributions to the foundation of mathematical physics including the dynamics of relativistic gravitational fields. Notably for: contributing in 1971, at the age of 19, to derive with Remo Ruffini the mass-energy formula of black holes as a function of their angular momentum, charge and irreducible mass. Christodoulou turned then to the study of partial differential equations and mathematical physics, to which he remained dedicated for the rest of his career. Highlights in this area include the theoretical discovery of the nonlinear memory effect of gravitational waves (Phys. Rev. Letters 1991), the monograph (1993) in collaboration with Sergiu Klainerman on the global nonlinear stability of the Minkowski spacetime, the monograph (2009) on the formation of black holes in pure general relativity by imploding gravitational waves, and the monographs (2007 and 2019) on the formation and further development of shocks in fluids.

13.3. Remo Ruffini on Demetrios Christodoulou.

Remo Ruffini is the Director of the International Centre for Relativistic Astrophysics Network and one of the founders of the International Centre for Relativistic Astrophysics. He writes about Demetrios Christodoulou and why he has been awarded the 2021 Marcel Grossmann Award.

It was back in 1967 that Achille Papapetrou mentioned the case of the 16 year old Demetrios Christodoulou to John Archibald Wheeler. Wheeler interviewed Demetrios in Paris and brought him immediately to Princeton where he was registered as an undergraduate at the university. After one year he entered the graduate school and started collaborating with me. At the time I was working with Wheeler on the effective potential approach to geodesics co-rotating and counter-rotating in the Kerr metric (later renamed as ISCO). In parallel, Frank Zerilli was working on the gravitational radiation emitted by the fall of a test particle in a Schwarzschild black hole. From these limited conceptual arena Charles Misner and later Kip Thorne launched a programme for the detection of gravitational waves on the Earth.

A new approach started with the arrival of Demetrios: he was just creating mathematics following his needs. We identified the reversible and irreversible transformations of a Kerr black hole. Wheeler advanced a thermodynamic analogy. I addressed the need of identifying the concept of irreducible mass (from the Italian "irriducibile"), and Demetrios's contribution was to integrate, overnight, the differential equation for infinitesimal reversible transformations which led to the finite mass-energy formula of a Kerr black hole. That evening, while walking back home through the Institute for Advanced Study woods, I expressed to Wheeler the great relevance of the newly found formula by Demetrios and proposed to let Demetrios be the single author of this article, admiring his great mathematical talent. Wheeler agreed. The Editor of the Physical Review Letters objected since in that two pages article the Fig. 2 by Wheeler and myself was still unpublished. Actually that Fig. 2 followed a discussion I previously had with Penrose in Florence which allowed us to present there, for the first time, a "Penrose Process". Some difficulties in achieving this process were obvious from the example in Fig. 2, which Roger later recognised himself. The Editor finally agreed on our written request and the paper appeared on 17 September 1970 [D Christodoulou, Reversible and irreversible transformations in black hole physics (1970)]. On January 1971 appeared my article with Johnny introducing the Black Hole, with the new physics we were developing in Princeton, including the concept of the "ergosphere". On 1 March 1971 we submitted the mass formula of the Kerr-Newman metric, including the relation between the surface area of the horizon and the irreducible [D Christodoulou and R Ruffini, Reversible transformations of a charged black hole (1971)]. On 11 March 1971 the same results were independently confirmed by Steven Hawking, extending further the applicability of our equation.

The thesis was successfully discussed by a committee including Eugene Wigner, one of the closest collaborators of Albert Einstein and David Wilkinson, the head of the NASA Wilkinson Microwave Anisotropy Probe mission, and Johnny and myself as supervisors. The new message was clear: Black Holes, far from being a sink of energy, were energy sources emitting "in principle" 50% of their mass energy, being extractable [D Christodoulou and R Ruffini, Reversible transformations of a charged black hole (1971)].

Demetrios turned soon to the study of partial differential equations and mathematical physics, to which he dedicated for the rest of his career and results were published in four monographs: [D Christodoulou and S Klainerman, The Global Nonlinear Stability of the Minkowski Space (1994), D Christodoulou, The Formation of Shocks in 3-Dimensional Fluids (2007), D Christodoulou, The Formation of Black Holes in General Relativity (2009), and D Christodoulou, The Shock Development Problem (2019)]. In 1968, Johnny proposed to Demetrios the collapse of a "geon" composed of massless scalar field as a second topic for his thesis. It took almost forty years for him to solve this problem, extended by Demetrios to the focusing of gravitational waves leading to black hole formation [D Christodoulou, The Formation of Black Holes in General Relativity (2009)].

A "long march" started on 12 December 1970 with the launch of the Uhuru satellite by Riccardo Giacconi. Early in 1971 an almost daily conversation with him and Herb Gursky at the Smithsonian Astrophysical Observatory, leading to the discovery of binary X-ray sources. This was soon followed by the announcement of Cygnus X1 identified as the first black hole in our galaxy.

Today, after fifty years, this "long march" has reached a definite result: through the grandest observational multi-wavelength effort in the history of mankind, from space, ground and underground observatories, we are finally finding evidence that black holes are "alive" and their "extractable energy" in our mass formula [D Christodoulou and R Ruffini, Reversible transformations of a charged black hole (1971)], is the energy source of the most energetic cosmological sources: gamma ray bursts (GRBs), the active galactic nuclei (AGNs) as well as the ultra-high energy cosmic rays (UHECRs). Their "inner engine", has three independent components: 1) a Kerr black hole which is neither in a stationary state nor in vacuum, 2) a background magnetic field aligned with the black hole rotation axis, and 3) an extremely diluted fully ionised plasma. There is no role in this inner engine for the innermost stable circular orbit (ISCO). Indeed a new electro dynamical field equations describe the synchrotron radiation emitted close to the black hole horizon, they point to a discrete and repetitive emission of "blackholic quanta" in the MeV and in the GeV. The magnitudes and the emission time scales of these quanta, for M87 and GRB 130427A, are expressed as a function of the above three parameters. A long lasting GeV emission with a luminosity decreasing as a temporal power law, allows for the first time in GRBs, the determination of the black hole mass and spin as well as their time evolution perfectly fulfilling our mass energy formula [D Christodoulou and R Ruffini, Reversible transformations of a charged black hole (1971)]: a long lasting emission process profoundly different from the traditional process of continued gravitational contraction.