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

His Excellency the President of the Republic of Cyprus,
Honorable Ministers,
Honorable Parliament Members,
Distinguished Ambassadors, Ladies and Gentlemen,

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 theater 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 1 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 J¨urgen Ehlers. Ehlers, though himself a physicist, recognized 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 molded 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 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 2 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 traveled 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 3 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 analyzing 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 capitalizes 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 behavior 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 emphasize 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, be- 4 ing 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. 5