傳播數學知識．促進數學教育

Interview with Prof. Bardos

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewer:** Tai-Ping Liu (TPL)

**Interviewee:** Claude Bardos (CB)

**Date:** August, 2002

**Venue:** Institute of Mathematics, Academia Sinica

Prof. Claude Bardos was born in Paris on April 4, 1940. In 1960, he entered École Normale Supérieure. In 1964 he worked at University of Algiers as a teaching assistant. He was J.-L. Lions’s doctoral student between 1965 and 1969. Since 1970, he was a faculty at Université Paris XIII. Also, he was a Director of Center for Mathematical studies and their Applications at École Normale Supérieure from 1981 to 1987. He is a professor emeritus at Université Paris Diderot - Paris VII. Bardos has worked on a variety of aspects of applied mathematics and makes contributions through research, supervising Ph.D. students, and collaborations. He was awarded Edmond Brun Prize of the French Academy of Sciences (1992), Humboldt Prize (1993), Marcel Dassault Grand Prize of the French Academy of Sciences (2004) and Maxwell Prize, ICIAM in 2019.

TPL: O.k. Let’s get start. Usually we ask from where do you come, how did you get into mathematics.

CB: I have no scientist in my family. My Father and my Mother met and got married in Paris in 1939 just before the war. May be I should add some words about my family. Both my Father and my Mother were Jews from Hungary, but from very different background.

My mother was living in Paris with her family (parents and brother) they were from some type of aristocratic family. Her Father, my Grandfather, was among the few Jews that reached the rank of captain in the army of the Austrian Emperor Francois Joseph. He claimed that once he had dinner with the emperor being the captain of the Imperial Guard. He claimed also that we were related to Von Karman. Since my Grandfather was a very nice person but somehow snobbish I cannot guarantee the exactitude of these facts.

However after the collapse of the Austrian Empire, the economic crisis and the rise of the anti semitism, they were living in very poor conditions in France (my Mother told me that at some point they were genuinely suffering from hunger).

On the other hand my Father had no family. In 1918 he was in the Jewish orphanage in a village in Hungary. The orphanage went bankrupt and had to close and the let the kids on the streets...
In 1939 in Paris he was working as an employee for a Stamp dealer. In fact he was already quite successful in business and it was his boss who had the idea to organize his wedding with my mother with the idea that this active and energetic young men could support the family of my Mother and to make the story short this is what happened.

Very soon my Father became an independent stamp dealer with much bigger success than his former boss. (At the end of his life when he retired, around 1987 he was the first stamp dealer of Paris). During the war he managed to protect his wife, his parents in law (and me of course who was born in 1940) and to be involved in the French resistance.

About becoming a mathematician...
As most of my colleagues I am not able to give a definite answer to the question why did you became a mathematician. May be it is easier to say how this happened.

I was in high school, living with my parents between 1951 and 1957. In the mean time my Grandparents where helping a young Hungarian who had lost his family during the war and who with the French resistance fought in the battle of the Vercor. At the end of the war he took a French name (Jean Bernay) and soon became an engineer in the firm Marcel Dassault which was designing the new generation of French fighter planes "Ouragan" and " Mysteres" and "Mirages". Jean Bernay had a basic training in Calculus so at the age of 13-14, I used to come for diner at least once a week with my Grandparents. They invited Bernay for diner and after diner this guy explained me derivatives, integral and differential equations. I really enjoyed these evenings and that may be one hint of how I became a mathematician. On the other hand besides mathematics in high school I was not specially good (or bad). I was not that much attracted by humanities except history but was very attracted by all kind of techniques or sciences. This was the time when science was very popular with the first space rockets, Gagarin flying around the earth and the beginning (with no concern by that time about environment and waste) of the electrical nuclear industry.

To go further a short crash course on the French educational system in necessary.

The scientific training in France is characterized by the existence of two types of systems.

The Universities and the Grandes Ecoles. The Grandes Ecole share some similarities with the English colleges of Cambridge and Oxford.

However more precisely: 99.5% of the universities are operated by the state in a very centralized system. Some of the grandes ecoles (the most well known are the Ecole Polytechnique, the different Ecoles Normales, the Ecole Centrale and so on...) are supported directly by the state, some by the local communities, some are private, some are partly supported by the state and partly by private funding and so on. Very often faculties usually teach during their career both in the Universities and in the Grandes Ecoles. Some big and prestigious laboratories especially in Physics (Laboratory of Meterologie Dynamique, Laboratory of Physic des Solides) are supported jointly by some ecoles and some universities. So at the level of the researcher and professor there is no much distinction between the two systems. On the other hand at the level of the students especially during the first years the difference is huge.

There is no selection for the entrance in the Universities, any one with the baccalaureate or an equivalent foreign diploma has the right to enter the Universities. The fees are very low (almost symbolic). On the other hand the students receive in general a very poor service. They sit in huge classes, do not have much private tutoring, no good computer equipment and so on....

At variance the entrance in the grandes ecoles is ruled by a very difficult competition with entrance exams for a very limited number of students.

To prepare these exams the students usually spend two or three years in special classes after high school. These classes are also located in high school. They are called classes préparatoires or in slang "taupe". "Taupe " is the French word for mole and the reason for this name is that when you are there you are supposed to work all the time and not to see the exterior world like the mole digging its hole in the earth and never seeing the open air. Next when a student from taupe is admitted to a grande ecole he reaches almost the paradise. In some cases he can also attend lectures at the Universities but he also has special tutoring, special library, good computer equipment and so on. In some grandes ecoles students have to pay for the studies but since they are considered as part of the elite it is easy for them to obtain a loan. And on top of that in the most prestigious ones Ecoles Normales and Ecole Polytechnique which are operated by the Ministry of Education or the Ministry of Defence, the students as soon as their are admitted become civil servants or army commissioned officers with a serious salary and huge housing facilities.

This system has some obvious consequences.

First since the training in the classes preparatoires and the exams are mostly, up to 60%, based on mathematics even for grandes ecoles specialised in Chemistry or Construction or Administration, all the gifted French kids are exposed to two or three years of full time training in mathematics. Furthermore the teachers in the "taupe" are very dedicated and really want to have their students passing the test. I think that this system is one of the main reason of the permanent success of the French mathematical school.

Second at variance with what happens in the universities, in the grande ecoles there exists a genuine system of "alumnis" that means for instance that if a student admitted in the Ecole Polytechnique has used out all his energy in preparing the test and does not want to work any more he can really relax. He is not assured of making an interesting career in research or elsewhere but he can be sure that the system will take care of him. He will have a job for the rest of his life. That means that some good talents in science are lost after their admission to the grandes ecoles. Of course for a brilliant career in research or even in private company one has to go on working very hard even after the admission (but as I pointed out this is not the case for everyone).

Having been good (but not extremely good) in high school I was admitted in a class préparatoire which did not ranked among the most famous one. But then I was lucky enough to succeed at the entrance exam of the Ecole Normale. I think I was 43 from the last to be admitted.

After entering the Ecole Normale I could had the choice to change my field (this is another privilege of this Ecole) and do any other thing. However I think that what attracted me most was not the subject but the atmosphere around the French mathematical community and in particular, on top of everything, the fascinating personality of Laurent Schwartz. In fact I have always enjoyed at most the fact that mathematics is an activity that you share with a well known and well identified community.

TPL: In Paris, every time I go there I find the interaction is very intense.

CB: Ok I find it more subtle, usually see in Paris huge number of excellent mathematicians. But on the other hand I never found a math. lab. so pleasant as Stanford or the Courant Institute. Many people (including me) work at home and don’t show up too often in their labs.

That's tradition in France and especially in Paris. There are not so many offices. Even full professors have to share their offices so they don’t try to come without a precise reason and in the long run the offices remain empty!!

TPL: But because you have such solid training it is easy for them to communicate.

CB: True. The communication takes place among people who share the same training and the same language

TPL: It is always amazing to me that there are so many good mathematicians and you have, sorry to mention that, gone through several decades, in your experience so far you have met so many people, like Laurent Schwartz and Leray. But time is changing all over the world. It must be changing in Paris, you know, in the way people communicate and the type of mathematics they do. Do you find changes or you don't think much about it.

CB: Since you ask many questions about France, I will try a specific answer and insist on the changes that took place by the time I was writing my thesis.

In the sixties the trend in mathematics was really toward abstraction and the concept of "pure mathematics" opposed to "applied mathematics" was really enforced.

May be one should keep in mind that up the thirties there was no real distinction between pure and applied science in France. Here are some striking examples, many other could be quoted:

Marie Curie was the first to introduce x-rays on the battlefield during the first world war to treat the wounded. In 1934 Yves Rocard (the father of one of the prime Minister of the 5 Republic by the time of Mitterand) and Maurice Ponte both members of the physic lab of the Ecole Normale created the first radar. It was used on the ships Oregon and Normandie. Henri Poincaré and Jacques Hadamard worked mostly at the interface of mathematics mechanic and physic.

I think that the distinction between pure and applied mathematics appeared around 1935 in England with the influence of Hardy and in France with the creation of the Bourbaki group. The history of the Bourbaki group is by now very well documented.
For instance one could look at the autobiographies of André Weil "Souvenirs d'apprentissage". I met André Weil in Princeton few years before its death and in the mean time I read his book. I was struck by the fact that one could figure his influence on his collaborators and younger mathematicians not only on scientific basis but also by the jokes, the language, the style that he introduced in the community and which was adopted by many (say of the generation of Verdier, Illusie, Jacquet and others). In the sixties with the publication of their treatise the group was at the peak of its influence and in Paris. Mathematics was mostly algebra and number theory. Laurent Schwartz whom I mentioned earlier was a singular point in this group. Leray both for scientific and may be personal reasons was not a member of the group. In some sense he was isolated in the college de France (an other typical French institution) and was not visible for students. On top of that he was very impressive. For instance we have the tradition to use the "tu " instead of the "vous" among all former students of the same Grande Ecole. Now Leray was the only one to whom everyone addressed with the "vous". On top of that he was always working on very hard topics and, at variance with Laurent Schwartz, was very difficult to understand.

Since the very beginning I was attracted by analysis. As an undergraduate I attended the course of Laurent Schwartz "Methodes Mathématiques de la Physique" it was really great. However next year at the beginning of my post-graduate studies the most "concrete" course I could find in Paris was a course on topology and potential theory given by Gustave Choquet.

Finally one should notice that at the same time the trend in Physic was also toward abstractions. For instance the best students of the classes preparatoires (which I mentioned above) would after their admission to a grande ecole go to abstract mathematics or very theoretical physic.

Now between 60’s and 70’s, things started to move toward more concrete mathematics, not ignoring applications and this was not specific to France.

A good example is harmonic analysis which went back from a very formal point of view to Fourier Analysis and wavelets. The turning points being the book of Stein on singular integrals and later the contribution of Yves Meyer and of his school.

In Paris one of most influential event was the arrival of Jacques Louis Lions. Lions was not (may be by personal taste) a member of the Bourbaki group. He was a student of Laurent Schwarz, became a Professor at the University of Paris in 1962 and at the Ecole Polytechnique in 1966. He was my advisor from 1966 to 1969. Now his career and achievements are very well documented and information can be found in the biography written by Roger Temam (SIAM News July 2001). I just want to pin-point the following issues.

1. Already before 1962 when in was in Nancy, Lions had started a collaboration with the CEA (the French Atomic Energy Commission) about numerical analysis of partial differential equations with computer.

2. He had toward the thesis a completely different approach than the other university professors. At that time the thesis was a very formal affair. Usually the advisor would give a problem to the student, and then just wait for the student to come back with the problem solved, which may take two or three years. At variance Lions organized a real school where the junior could be supported by more mature students (or former students). In my case when I started I was really helped by C. Goulaouic. Lions himself he was a very charming person, spent a lot of time with students reacted extremely fast to the papers presented and had always very positive comments and this way attracted many of them in particular from the Ecole Polytechnique.

3. I already said that at that time (in spite of a prestigious past in field) Physic was even more abstract than mathematics. Lions had the idea first to put directly in contact applied mathematicians and engineers in industry without the contribution of physicists. Contribution by the endeavor of academic physicists came later, say with Clavin or Pommeau.

4. He created for himself (or contributed to the creation) of several institutions that would serve as bridges between industry and university. Probably the most well known is the INRIA but he also contributed to the creation of the SEMA (with Lattès).

5. He was really influenced by the experience of the Courant Institute by Von Neumann and scientists of his own generation like Peter Lax and Louis Nirenberg.

And the result was the rebirth of a large school of applied mathematics in France.

TPL: My impression, knowing you for many years, that even though you are clearly belong to the French school but the way you do mathematics, your interest, and your outlook can not be characterized as a French mathematician. Is that fair to say?

CB: It seems to me that all over time and definitely not only in Mathematics there was a tendency for people more in France than elsewhere to gather in schools or lobbies. (This is even much more obvious for the French psychoanalysis than for the French Mathematics). Now, both by taste and conviction, I have the ambition of being more independent and from the scientific point of view my "model" would be rather MacKean than any of the French "leaders".

TPL: I remember that you, Nishida and Ukai almost in the same period get into kinetic equations. What is the historical background of the subject?

CB: This is a good question and also provides a good example of what was said above.

I think that I was among the first mathematicians in France to become interested in the kinetic equations. There were several contributions from physic and also from mechanic (Cabannes, Guiraud Darrozes) but no mathematical study neither theoretical nor applied. In the mean time the subject has gained a huge momentum with at present a big school consisting researchers that do things much more sophisticates that I would never had been able to do.

As it is well known the subject started with Maxwell and Boltzmann and this is very well described in an excellent historical book written by Cercignani "Boltzmann the man who trusted the atoms". What was at stack at that time was to confirm through mathematical considerations hypothesis about atoms and molecules.

In our time the point of view has changed completely. The fundamental physic which is implied is well known but what peoples want to do is to compute (of course most of the time with a digital computer) the solution.

In short, kinetic equations are needed whenever one considers a media which is rarefied enough so that the velocity of each individual particle has to be taken into account. Such need disappears when the media becomes dense enough and when some averaging process which in most cases is called thermodynamical equilibrium is reached.

Now kinetic equations appear in many fields of application. The most obvious one is the study of the reentry in the atmosphere of a space shuttle which very rapidly moves from a rarefied atmosphere to a dense atmosphere and as it is known that this is the most critical instant of the reentry. Kinetic equation under the name of neutron transport equations were used to analyze the critical size of a piece of uranium (both for civilian and military uses) and at this point people would like to define the convenient approximation (called diffusion approximation) with its convenient boundary condition. Kinetic equation are also used in the design of semiconductor because the devices (chips and so on ) are so small that the current cannot reach a thermodynamical equilibrium in the process. They appear in the design of the reading head of a compact disk because the distance between the disk and the head is so small that the velocity of individual molecules of gazes have to be considered. Finally they turns out to be adapted to the mathematical description of cellular biology.

It seems important to keep in mind that the Boltzmann equation which
involve molecules of air is the from the mathematical point of view the most complicated one because all the effects are in the interaction of the molecules and therefore the nonlinearity is predominant. Other variants involving more sophisticated physic leading phenomena are often more linear and the equations are therefore easier to handle.

May be if Boltzmann had at his disposal theory and experiment of fission the scientific live would have been easier for him.

As Liu said Nishida, Ukai and me we started to work on the subject at the same period and it may be interesting to point out that this renewal of interest did correspond to the time when new applications appeared.

My first contact with transport equation was the essential of my thesis. Following a suggestion of Lions I studied a series of papers of Friedrichs, Lax and Phillips on first order symmetric systems. Then Lions kept asking the question what happen in the very simple scalar case but with singularities on the boundary ( this correspond more or less to the standard boundary value problem for the neutron transport equation). So I worked out the subject. My second contact came much later and was under the influence of Robert Dautray.

Robert Dautray is not very well known in the mathematical community. He is at some point very shy, very discreet in spite of the fact that he was for several years the head (in French "haut commissaire") of the atomic energy commission. However in spite of his administrative charges he was always interested in research both in physic and mathematics. He had been one of the first collaborators of J. L. Lions. Then he came with the idea of writing with Lions a French updated version of the Courant-Hilbert. Inspired by the experience of the Bourbaki and of the Courant-Hilbert, Dautray and Lions decided to involve in the endeavor their regular coworkers and former students. They asked me to join the crowd and to work on a volume devoted to the equation of transport of neutron and the diffusion approximation for such problems.

The final result was a monster book of about 2500 pages in 3 volumes published in French under the title
"Analyse Mathématique pour les sciences et les techniques " published by Masson. Then Dautray came with the idea of splitting the book in 8 volumes with no hard cover saying that each one would be much less expensive and that the collection would sell better. Eventually the book was translated in English ( published by Springer ), in German, Russian and other languages. I don’t know if this book could be very useful for the reader. The ambition toward Encyclopedy and unification of notation which at some points goes again intuition makes it access quite difficult. On the other hand I am convinced that the endeavor had a huge positive effect first on the people that did contributed to it and even to all the community of applied science in France.

At some point Dautray invited almost every week all the participants to his house to discuss the state of the project, to read what has been done and to decide what would be the next chapters and who would be in charge of writing it. He attracted to support his project scientists from different origins, Universities, CEA (Atomic Energy Commission) lab and so on...This really improved the interconnections between different aspects of applied sciences.

Dautray introduced me to several fantastic scientists like for instance Balian. He is very well known and appreciated physicists and claims not to be a mathematician however I found out that most of the motivation for modern microlocal analysis as proposed by Hormander and others were in a series of papers published by Balian and Bloch in the Annals of Physics around 1974.

By that time military duties were still compulsory for everyone in France. But scientists had the opportunity to do most of their time in army research labs and in particular in the CEA (Atomic Energy Commission). That way Julia, Golse and Perthame went for their military duties to the lab of Dautray and joined the project. As I said it is in this setting that I started to work (mostly with Remy Sentis) on the diffusion approximation for the neutron transport equation. The issue on the boundary condition which in this framework implies the notion of extrapolation length lead us to efficient method to compute later the slip boundary condition which is used to improve the validity of the Navier Stokes equations near a space vessel in the reentry phase.

In fact, in one of our papers with Sentis about the use of diffusion approximation for the computation of the critical size we forgot to prove a compactness argument that was compulsory for the validity of proof. Then Dautray and Cessenat (a collaborator of Dautray in the CEA) asked Golse and Perthame to fill this gap. The result was the first French paper on velocity averaging by Golse Perthame and Sentis. Soon after that Pierre Louis Lions improved this lemma with the previous authors. They obtained the version of the velocity averaging lemma to be used later by Di Perna and Lions in their treatment of the Boltzmann equation.

TPL: Now let's taking about proving things. French have been very good to come up very sharp tool in analysis, all the compactness things. And how do you feel? It's quite a success you know. In some sense it defines an important part of French analysis, right? Every time I see it, I am a little bit feel overwhelmed, because it's heavy machinery to me. How do you think but I means I see many inequalities and so forth. May be not so heavy if one get to know it.

CB: What happen I think is we have good tradition in doing with inequality in France and we have also good tradition of functional analysis and the averaging lemma that we just discussed is a good example. However I don’t think that inequalities come out of the blue and they turn out to be "pertinent” if they not only produce proofs but contribute to the understanding of the phenomena. I would even say that a formula (equality or inequality) which do not contribute to our understanding has few chance of being of any use. I am really convinced that for problems like Euler or Navier Stokes what is missing is a better understanding, new idea should go in this direction and not in the genuine use of functional analysis or a prior estimates. For these problems I think all can be achieved by this type of method has already been explored by several generation of mathematician including of course Leray, Caffarelli and Fefferman. I repeat what is needed are new ideas.

TPL: I don't know if this thing has been said in another generation: "That complete new ideas are needed."

CB: I don't know if previous generations had the same type of feeling and along this line talking with you guys I have in mind an even bigger issue.

What in the future, what we going to do.

Comparing mathematics with language (and this is surely not all but one of the aspect of the activity) I make the following remark.

1. Mathematics are with computer and all other kind of technology more and more widely used everywhere all around the world.

2. English is becoming a universal language spoken by all the peoples involving in science business, communication all around the word.

On the other hand, the number of peoples who are genuinely studying English as a language (grammar and so on) is infinitely small compared with the number of peoples that do use the language.

It may be, I don't know, that mathematics will turn that way.

The absolute rigor was needed to construct the corpus but went the construction is done such rigor becomes less important.

Good example would be the Fourier transform and the theory of distribution. These tools are really widely used in particular with the introduction of Fast Fourier Transform and of wavelets. However a complete mastering of the theory is not needed by many people using it.

TPL: Research that is done in the present form started few decades ago.

CB: It started at the beginning of the last century and was extremely successful. It produced tools which are used by many people including non-mathematicians. However in the worst (for us) scenario it may happen that the use of mathematics would go on rocketing while the need for basic research would decrease as it is the case for the use of English language. In any case we have to be prepared to such evolution, which of course doesn't mean that people will not have to be trained in mathematics.

TPL: Going back to the example of speaking English I observed that one do need to learn quite a bit basic language, right?

CB: Sure, and in fact this would partly save the deal for research. Experience has shown that some practice of research is almost compulsory for mathematicians to achieve good teaching at the level classroom

TPL:Yes. however this is not in contradiction with some fundamental change in mathematics research. Have you already observed such change?

CB: Definitely. In particular, in agreement with the previous issues, the notion of research in applied mathematics has appeared. However what is good mathematics is notion more subjective than in other sciences and defining what is good applied mathematics is an even more difficult issue.

TPL: Now, can we return to your personal research, your direct experience in term of your own research. For example, what's are the thing that you most enjoy doing.

CB: In agreement with the previous issues, I confirm what I really like to do is applied mathematics and that mean precisely
that I am much more interested in contributing with the introduction of mathematical analysis to the understanding or
the computation of an issue in physic or engineering than by the construction of new mathematical object.

I already talked about in kinetic equations and I try to give an other example.

TPL: Great. Give an example.

CB: I can describe my contribution to the theory of control. Once again I was introduced in the subject by Jacques Louis Lions. At this moment I think it was around 1979 Jacques Lions was systematically studying the extension to partial differential equations (PDE) of the classical notion of control theory for (ode) and he had the intuition that the best frame for that was the linear wave equation because the Cauchy Kowalewsky theorem is
reasonably close to the classical Cauchy Lipschitz theorem.

At a meeting of the SMAI (French Mathematical Society) he asked to Jeff Rauch and me a specific question about the geometry of the set that should stabilize the solution of the wave equation. Immediately according to "bon sens" we gave the answer saying that this set is characterized by the underlying classical Hamiltonian flow. Lions says ok prove it and it turned out that we needed about 10 year to construct a proof.

Things started to move when I had what I consider now as one of my best contribution in applied math. It was to associate to the project Gilles Lebeau who in this time had just finished a thesis on diffraction theory with Boutet de Monvel. Gilles brought in more sophisticated tools of microlocal analysis. In particular the microlocal propagation of energy as it was done by Melrose and Taylor and the interpretation of diffraction in term of analytical wave front set with the Fourier Bros Igaolnitzer transform. In this way the problem posed by Lions and several extensions were completely solved.

It is not clear that this original problem posed by Lions was really applied. It was an already very abstract version of a question related to the vibrations of the large solar mirrors of satellites and probably too abstract to contribute to the design of the device.

However our program had several outcome. Due to the Heisenberg principle one cannot localize at same time the position of a wave and its direction of propagation (except for plane waves). However this localization becomes valid if some kind of high frequency asymptotic is used. The first and probably most important outcome was a better understanding of this issue. Of course the problem had been the object of the attention of at least two generations of mathematicians including Hadamard and Hormander but recently with not that much link with application. What I claim is that our contribution which did not contained any new mathematical idea (stricto sensu) lead in the meantime to improvement in numerical simulation and in the understanding of the role of Carleman estimates in uniqueness theorem (of Holmgren type). Eventually in the recent year I met a physicist Mathias Fink working in the LOA ("laboratoire ondes et acoustiques " de l'école de physique chimie de Paris, another "grande école") who designs several devices based on what he calls "time reversal method" and the understanding of his process turns out to be much closer to my research (with Gilles Lebeau and Jeff Rauch) than the original satellite mirror problem quoted by Lions.

TPL: In some sense you bring all those people together, right?

CB: OK! Right. Bringing people and idea together is something I really like to do as well as trying to use mathematics to explain phenomena. However that brings me back to this previous issue. Will there be enough clever idea that we may bring to applied problems or will we in a near future completely exhaust the possibility of mathematics.

TPL: I suspect that already 50 years ago or 100 years ago, mathematicians have be asking that same question. And only with honesty and integrity we can move over.

CB: Yes. this was not in contradiction with the rapid expansion of mathematics and mathematics will continue. However the style of research may change and also in the meantime the type or the status of the mathematics researchers.

TPL: I see. Thank you.

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.