Plant your math and let it grow
A few days ago I was lecturing in Tohoku University for the Takagi lectures and for that occasion I wrote a survey of the synthetic theory of Ricci curvature, a subject which has blossomed over the past years.
What does synthetic mean? You know, we learn in school that there are various ways to do elementary geometry in the plane. It can be done with equations and Cartesian coordinates (lines are described by equations of the form ax + by + c =0 ; circles by equations like a(x^2 + y^2) + bx + cy + d = 0; etc.). But it can also be done in the style of the ancient Greeks, using axioms and properties involving triangles, lines… never writing down any equation. The first approach is analytic (let’s compute and use equations!), the second one is synthetic (let’s use concepts and properties). There are advantages and drawbakcs in each approach, but often the analytical one is more systematic and the synthetic one more elegant. The analytic approach also comes with more quantitative results, while the synthetic one often provides a better understanding. It is a major conceptual progress in the mathematical education of a high school students when he or she understands that any problem in plane geometry can be tackled equivalently by the two methods.
What does Ricci curvature mean? Obviously it is a notion of curvature, and obviously it was introduced by somebody named Ricci. Indeed, Gregorio Ricci-Curbastro was one of the major Italian geometers of the beginning of the 20th century; and the curvature which has been named after him is one of the most important notions of curvature, those mathematical concepts which have been used since Gauss and Riemann ways to quantify how much a geometry departs from being Euclidean. This particular curvature is all about distortion of volume by the non-Euclideanity, and it is most famous for the role that it plays in Einstein’s theory of general relativity. To state things a bit abruptly, if you live in a geometry whose Ricci curvature is positive, light deflection (due to light rays being curved and not straight) results in the fact that bright bodies always appear to you to be larger (in the sense of bigger volume) than they really are.
Ricci curvature is a favorite of probabilists working in non-Euclidean geometry, and all along the 20th century it has been treated mostly in an analytical way — with equations! But, starting from the nineties, experts were wondering how to understand Ricci curvature inequalities in a synthetic way, all the more that the synthetic treatment of sectional curvature was encountering many successes. To solve this problem, it took a few beautiful concidences, and years of combined efforts.
I was involved in the beginning of this theory some 15 years ago; at the time it was just a couple of manuscripts. Now it is spread over dozens and dozens of papers, and fills up thousands of pages. Still, I find it interesting to remember how it all started.
Back in 1998, just after the defense of my PhD, I was participating in a workshop organized by my tutor Yann Brenier, on the subject of optimal transport. This is the theory of rearranging a certain amount of material from some initial configuration to some final configuration. Participants in the workshop were specialized in various topics: statistical physics, isoperimetry, fluid mechanics; but they were all linked by an interest for optimal transport.
The optimal transport problem was first mentioned in 1781, in a famous manuscript by Gaspard Monge about “déblais” and “remblais” — finding the best way to transport and rearrange a certain amount of matter, coming in a prescribed distribution, into another one. For more information on Monge, the father of descriptive geometry and École Polytechnique, friend of Napoléon, and visionary specialist of geometric analysis, one may consult the beautiful article by Étienne Ghys, as well as the companion article on the birth of optimal transport. By the way, specialists of operational research say that this is the oldest problem in operational research which has been`solved” in some sense!
Since Monge’s problem is of mathematical nature, it can be recast as the transport of any kind of material. In my beloved book Optimal Transport, old and new I decided to do it with croissants, as a tribute to the Boulanger de Monge which was once my cherished bakery (alas, it is not what it used to be!). So here is the problem: Suppose you have a bunch of bakeries, spread across Paris, producing croissants every day, and a bunch of cafés, also disseminated here and there, who have to provide consumers with fresh croissants every day. Both the amounts of production at the bakeries and of consumption at the cafés are known. Each time one basket of croissants is moved from one bakery to one café, this has a certain transport cost, depending on local geography. Now, how to match bakeries and cafés in such a way that the total transport cost is as low as possible?
From this question also arise problems of economical interest: for instance, assume that the price charged to a café by a bakery should depend on the transport cost; then which is the best way to set prices? The Russian mathematician and economist, Leonid Kantorovich, one of the heroes of the unclassifiable novel Red Plenty, worked on the Monge minimization problem through this angle. He actually dreamed of devising a rational theory of prices — which, given the time and place where he worked, nearly amounted to reserving a place in Goulag or execution list. However, Kantorovich was saved by his use in key governmental programs, and later received the Nobel Prize in economics in 1975.
Nowadays, this optmization problem is called the Monge-Kantorovich problem, and a large community is familiar with it. The stuff which is transported may represent matter, molecules of gas, or any other thing. In the 1998 workshop, I presented the older work of Hiroshi Tanaka, where the transported matter is a (model of) dilute gas. Tanaka showed in the seventies that if two gas distributions are given, both of them evolving according to a certain model from physics (to be precise, the spatially homogeneous Boltzmann equation with Maxwellian molecules), then the total transport cost needed to transport one gas distribution into the other one is always nonincreasing as time goes. I also wrote on this contribution in Section 7.5 of my first book on optimal transport, and also explain it in Section 4.2 of my survey on collisional kinetic theory.
Among the participants in the workshop was Felix Otto, a young German mathematician. At the time he was getting a position in University of California Santa Barbara; later he would have positions in Bonn and Leipzig. He had already done a very original work together with Richard Jordan and David Kinderlehrer, which was a new interpretation of the heat equation, based on its properties with respect to Boltzmann’s entropy and optimal transport. Now he was generalizing this theory to consider some further nonlinear equations of diffusion type. I listened carefully, as nice PhD students should, and in the evening Felix and I had a good discussion.
A few weeks later, at home in Summer, I found myself reading a set of lecture notes from Michel Ledoux, who at the time was working on his book on concentration theory. Eric Carlen, who sat of my PhD jury, had advised me to go and meet Michel in Toulouse; I remember vividly the hard time which I had finding my way for the first time in the big University of Toulouse. Michel had been very excited to hear about my work on the Boltzmann equation, and he had given me these lecture notes to satisfy my curiosity about his work.
As I was going through Michel’s notes, amazed by the beauty and elegance of his account, something caught my attention: several keywords and key concepts which I was familiar with were present, and there was optimal transport arising here again. For some reason I thought that this should have something to do with the work of Felix. It took just two attempts, and a few minutes, to find this link. This was one of those precious sudden enlightenment moments which can change one’s researcher’s life…
I wrote down a sketch and sent the first conclusions to Michel, who had an enthusiastic response. I decided to write a paper with Felix on this subject; for me this collaboration was all the more precious that I, as a beginning mathematician, had standards of rigor and writing skills which were still quite perfectible, and teaming up with Felix was a great opportunity to progress in these skills. It took just a couple of weeks to write up our paper. (Incidentally, another collaborative work which we started at the same time took nearly 10 years, and two additional co-authors, to be completed.)
In our manuscript, Felix and I obtained several results (let’s use a few big words):
show that the logarithmic Sobolev inequality always implies a Talagrand inequality, thus providing a functional basis for the Herbst-Ledoux principle that logarithmic Sobolev inequalities imply Gaussian concentration bounds;
deduce simple sufficient conditions according to which a probability measure would satisfy a Talagrand inequality;
provide new proofs, based on optimal transport, of a couple of already known theorems in the area;
find a new information-theoretical interpolation inequality, according to which the relative entropy is controlled by a bit of the Boltzmann entropy and a bit of the optimal transport;
develop Otto’s formalism, explaining how to consider the space of probability measures as a non-Euclidean geometry;
make the link between McCann’s notion of “displacement convexity” and Ricci curvature, with the idea that the convexity of Boltzmann’s relative information function would be convex along geodesics of optimal transport only if the Ricci curvature is nonnegative. This was stated as a conjecture: we gave arguments to support the idea, but were lacking some key ingredients to complete the proof.
I did not fully realize it at the time, but it was a huge harvest for so little effort. The manuscript was submitted to the Journal of Functional Analysis, and accepted the next day (!) by Paul Malliavin, who obviously felt the potential of our work even better than me. Indeed, this has become my most quoted paper, and in some sense the starting point of a whole theory which would lead to a new point of view on Ricci curvature, and a synthetic approach to Ricci curvature bounds.
The story went on with a lot of surprises and collaboration; it involved researchers from France, Canada, Germany, Italy, Japan, America, Russia, China… Another defining event, for me, was the encounter with John Lott in 2004 in Berkeley; together we started to work explicitly on the synthetic theory of Ricci curvature, at the same time as Karl-Theo Sturm in Bonn. The geometries which we defined are now often called the LSV spaces (LSV = Lott-Sturm-Villani). It is always moving to see your name being transformed into a definition! This story was also the opportunity to write up my book Optimal transport, old and new, which went on to win the Doob Prize of the American Mathematical Society.
I shall say no more about these further developments, and refer to my synthesis notes for a synthetic account. Let us however go back to the story above, and try to squeeze out some conclusions and advice, for whoever will take them.
First: It is not always the effort that you put in a research contribution which matters; sometimes it is just about being at the right place with the right idea, in contact with the right persons (just as prices should not be just determined by the amount of labour put in production, in spite of what the Soviet government would tell Kantorovich).
Second: It is always (or almost always) good to be curious, even about things which are not directly in your area. Neither the talk by Felix, nor the course by Michel, were dealing with kinetic theory of gases, my PhD subject; but I was very enthusiastic anyway.
Third: A great progress is not always a new theorem or conclusion. It may be a new proof, or a new point of view. Our job as mathematicians is not just to prove things; more generally, it is to provide better understanding, and this may also depend on a new viewpoint.
Fourth: Research, especially fundamental research, is unpredictable: nobody expected this encountering between non-Euclidean geometry, optimal transport and entropy (I did not expect it either!) But now it has turned into a very fruitful field, and solved a few big problems in the area.
The last comment is about the nature of research. Yes, a research system is about publishing, and fundings, and positions, and science agencies, and fundings, and research strategies, and networks of laboratories, and spending billions of dollars, euros, renmibi or whatever. But in the end, the key moment will be one in which somewhere, in some brain, some spark of a new idea is lit. A significant goal of all those complicated research ecosystems is to increase the opportunity of this precious, fragile and unpredictable seed, anxious to grow into a fully developed theory.Posted in category: Uncategorized