傳播數學知識．促進數學教育

Interview with Prof. Tommaso Ruggeri

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

**Interviewers:** Tai-Ping Liu(TPL), Seung Yeal Ha(SYH)

**Interviewee: **Tommaso Ruggeri(TR)

**Date:** December 18th, 2019

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

Prof. Tommaso Ruggeri was born in Messina, Sicily, Italy, July 31, 1947. He obtained a master's degree from the University of Messina in 1969, where he became an assistant professor. Since 1973, he has served at the University of Bologna, and he is now an honorary professor at Bologna. His research interests include rational mechanics and hyperbolic partial differential equations. He and Ingo Müller have written the book Rational Extended Thermodynamics. He is an academician of Accademia Nazionale Dei Lincei.

TPL: I’d like to ask a question I have had in my mind. In Italy, we know that you have Galileo, Fermi, and many other greats in physics. Third-degree polynomials were solved in Italy. And so on, Italy has a glorious tradition in physics and mathematics. Your field is related to theoretical physics or mathematical physics. It has a long tradition, right?

TR: Italy, in a sense, perhaps very special because we have the so-called Rational Mechanics. Rational mechanics is an obligatory subject for students of mathematics, physics, engineering with a long tradition with books that appeared in Italy as early as the end of the 1700s. Probably the best known Italian text of Rational Mechanics abroad is that of Levi- Civita Amaldi whose first edition is from 1923. Rational Mechanics has a mathematical-deductive approach like that of Mathematics and we consider Galileo as the founders of Rational Mechanics in Italy. This subject has reached its peak in more recent times with Lagrange's Analytical Mechanics and Hamilton's Hamiltonian mechanics. Initially, the difference between theoretical physics and mathematical physics is very …

TPL: Subtle.

TR: Subtle, because for me the physical prototype is Landau. Landau was a genius. But if you look at the books of Landau, no proof is rigorous and is often missing. It is physical intuition. Some, they say, oh, this term does not play a very important role and therefore we can delete it. But mathematical physics is completely different in that, like all mathematics, it is rigorous and the proof is theorems. Of course, we, in Italy, there is a large group of mathematical physicists that belong to the GNFM (Gruppo Nazionale di Fisica Matematica) through the National Institute of High Mathematics (INdAM) that is divided into 4 groups: analysis, the numerical analysis which is younger, geometry and algebra and mathematical physics. I was director for more than 20 years of GNFM.

TPL: I see.

TR: But mathematical physics is a little different from what we mean in America because in America mathematical physics means field theory in its properly mathematical aspects. Instead, Italian mathematical physics is a sort of extension of the methodologies of Rational mechanics to other fields where the construction of the mathematical model starting from general axioms is also very important. This tradition of Mathematical Physics deriving from Rational mechanics has also been present in France. Just remember the volumes of Paul Appell, "Traité de Mécanique Rationnelle" of 1921. In the United States, rational mechanics have been exported by Clifford Truesdell. The importance of Rational Mechanics is also evident that in the Accademia dei Lincei, the oldest Academy in the world, there are two sections of Mathematics: one for Pure Mathematics and one for Mechanics and its applications.

TPL: You mentioned Truesdell. Archive for Rational Mechanic and Analysis that Truesdell founded, one of the chosen languages is Latin.

TR: Latin, yes.

TPL: So, Truesdell must feel that Italy is somehow his spiritual home. Why is that?

TR: Yes, Truesdell has been very important to spread the studies of Continuum Mechanics made by Italians. When he arrived in Italy, he became passionate about Italy and its culture and I started studying the Italian language. In this way, he was able to read many articles, in particular, those of Antonio Signorini and his school. Signorini graduated from the Scuola Normale Superiore in 1909 in mathematics, with Gian Antonio Maggi and Luigi Bianchi and Tullio Levi-Civita as supervisors. He and his students gave fundamental contributions to nonlinear elasticity. Truesdell thus discovered that several results that he believed were original had been established by Signorini before and had the merit of recognizing.

TPL: I see.

TR: And therefore, if you look there in the series of volumes of “Handbook of Physics”, it’s full of Italian name, many. And therefore, he like so much Italy, in particular, Bologna, because Bologna is very uniform, it’s the most uniform also because maybe it has the oldest university. And therefore, he decided to buy a house.

TPL: That’s a major commitment.

TR: I had the honor to meet him the first time in Bologna, and I remember one important paper of mine, I give to him, it was in Italian. That time, I don’t write any English. And he liked, he presented this paper to “Annali di Mathematica Pura ed Applicata” which was a very important journal in mathematics in Italy.

TPL: You mentioned Levi-Civita, he’s also a great geometer, right?

TR: Levi-Civita, was a student of Ricci-Curbastro. Ricci-Curbastro built the so-called absolute differential calculus. As you know, Einstein, when he had an intuition of general relativity, did not have the mathematical tool. He, therefore, addresses his mathematical friend Marcel Grossmann, a colleague at the Zurich Polytechnic, who brings to his attention the work of Ricci Curbastro. The theory of general relativity was born in which in the famous equation that governs gravitation a mathematical entity called the Ricci tensor appears and thus Ricci Curbastro and his great student Tullio Levi Civita also become popular, obtaining the deserved recognition. In the mathematical physics community in Italy, there are two substantial groups. One is strongly influenced by analysis. Another is influenced a lot by geometry. These people ware substantially more related to the differential geometry because, in Italy, it has a stronger tradition of geometry.

TPL: I have heard people mentioned the “Great Italian Geometers” in the subject of algebraic geometry before it was formally formulated. The Italian school is more intuitive.

TR: Yes, in algebraic geometry we had a strong Italian school, for example, Castelnuovo, Enriques, Severi and concerning differential geometry we can mention Bianchi for example.

TPL: You mentioned this analytic aspect of mathematical physics.

TR: This a mixture because, of course, analysis and geometry, it’s different. The most famous result of Levi-Civita was the notion of parallel transport in Riemann geometry.

TPL: The field that people like De Giorgi called it the geometric measure theory. It is a combination of geometry and analysis.

TR: Right.

TPL: How did you get into the subject of rational mechanics?

TR: Oh, that's interesting. My life is, as for many, a set of bifurcation points. Everyone's life can be compared to a chaotic dynamic system in which there are bifurcations at all times. There is a famous movie called "Sliding Doors". Do you know this movie?

TPL: Is life-like "Sliding Doors"?

TR: The plot of the film is very interesting. The film alternates between two storylines, showing two paths the central character's life could take depending on whether or not a girl catches a train. She rushes for her train on the Underground but just misses it as the train doors close; but the film then rewinds and the scene is replayed, except that now she just managed to board the train. The film continues, alternating between the two different storylines in which completely different events ensue.

TPL: Okay!

TR: My life was full of sliding doors because when I was a student in high school, also in middle school, I was a very bad student. I’m not so interested in the study. But, I don’t know why, mathematics was always very simple for me, and therefore, I remember that in middle school when there was a discussion in the classroom between the professors concerning the results of the students, I was considered the worst student for all the teachers but the professor of Mathematics, with some embarrassment, he declared instead that I was the best student. Therefore, my father was very worried and when I arrived at the end of high school, my father, who was a military medical doctor, said: “Tommaso, you are intelligent, but you don’t want to study.” Therefore, he suggested to me to go to the military, in the army. This proposal made me desperate. I say to my father, I want to try to go to University, if I am not successful in the first year, I will satisfy you and I am an officer in the army. Right now, I intended to study engineering, electronic engineering. In Messina, there was at that time only the first two years of Engineering and the courses were the same as those of the students of Mathematics and Physics. In these first two years I had no difficulty in passing the exams with the highest score and my father was completely astonished! But to continue studying the other three years of Engineering I needed to move to a University of the North. So, I had planned to go to Padua because there was an uncle of mine there. So, my parents started trying to convince me to stay in Messina. Where are you going? in the North it's cold ... Padua is far from Sicily ... why don't you change and study Mathematics or Physics . This was my physics sliding door. Here, the story is more comical. My mother reluctantly gave me 10 thousand lire which was the money necessary to pay for the transfer from the University of Messina to that of Padua. When I was at the Office ready to pay, I discovered that the transfer from Engineering to Physics cost only 5 thousand lire. That sliding door was to take 5 thousand liras! In this way, therefore I started physics and I liked the subject. Theoretical physics was mainly focused on quantum mechanics which at the axiomatic level is not so clear even if it has had huge successes and therefore I didn't like it very much instead I was very attracted to classical physics such as the continuum mechanics or the relativity that had been taught by Mathematical Physicist. Therefore, in the courses of your choice, I had taken courses in this discipline. Therefore, one of the professors, appreciate me during his exams and he told me that if I could graduate in the first session of June of the fourth year, he could offer me an assistant position in rational mechanics. I was very young because I go out the elementary school in 5 years, usually, it is for 6 years. And therefore, I worked very hard and at 21 years old, I finish my master’s degree, and therefore I become an assistant very young, 22 years old, not yet Ph.D. in 1969. This was a permanent position, but the problem was to start doing research. The professor of Mathematical Physics was a good person, but he was not a great researcher. He had a very large local power and was surrounded by several assistants who were forced to hear his lessons. The first problem with him came when I wanted to publish my first paper which was, in fact, my thesis that I had done completely on my own. After I handed him the manuscript, after a few months he had only concerned me with the introduction. The practice at the time was that all the assistants had to have the professor's approval and then finally, in the positive case, the paper was published in the local Academy in a Journal that nobody read outside of Messina. Going to the library, I discovered various international journals and thus getting help to translate the work into English, I decided to send it to a Mathematical Physics journal and luckily, the paper was accepted.

TPL: Great.

TR: The professor got very angry. Furthermore, I had broken the rules, and this had created a revolution in our group and therefore I was isolated. Therefore, I suffered a lot. And then, I had the second sliding doors, deciding to go to a Summer School in 1970. The school was held in Bressanone, a small town in South Tyrol in the province of Bolzano on the border with Austria. It had been organized by Carlo Cattaneo and had the title "Relativistic Fluid Dynamics". The lecturers were A. Lichnerowicz a famous french mathematician who spoke of shock waves using distribution theory, A.H. Taub who presented variational problems and J. Ehlers who carried out a course in relativistic kinetic theory. I did not understand much but I realized what it meant to do research and I was very interested in the topic and subsequently, I made several contributions to these subjects. Upon my arrival at the Bressanone station, there was only a taxi. At the same time, I entered the taxi on one side and the other, a young French man, on the other side, and we reported that we were both going to the same place. This was a very important moment in my career because this guy, Guy Boillat who had studied in Paris under the supervision of Lichnerowicz, was very original. He had started his doctoral thesis by traveling to Northern countries including Norway and Denmark and had been in contact with an applied Japanese mathematician Tosiya Taniuti. Taniuti had studied at the Courant Institute and had introduced Boillat to the works of Peter Lax. Boillat was the first to study the evolution of discontinuity waves in a general way for a generic hyperbolic system of the first order. These studies of his doctoral thesis were collected in a small volume "La Propagation des ondes" (1965). I started studying this little book and soon we became good friends and after publishing many works on non-linear wave propagation. On this occasion, I understood the importance of going to the Conferences. After 2 years, I go to the same summer school and this summer school, in this case, was organized by Dario Graffi. who was a very famous professor in Bologna, and one of his previous students was Luigi Caprioli who was also a professor in Bologna. This year, I make a presentation, a short talk. At this time, Caprioli was the first full professor in Bologna in the engineering faculty. The Faculty of Engineering was not happy that the professors of Mathematics who found a job later tried to move to the Science Faculty and wanted to create an Institute of Applied Mathematics that would provide a stable group of mathematicians. Caprioli, therefore, was looking for mathematicians available to form the first group and therefore asked me if I was available to come to Bologna. I was immensely happy and once again a favorable sliding door for me. But when I arrived in Bologna, I had a moment of great difficulty. Caprioli was no longer active in research and the young assistants of my age had a preparation infinitely superior to mine, knowing the functional analysis, the differential geometry and other mathematical tools that I did not know. Once again, I felt alone. And then, there was summer school again. Yes, I met Yvonne Choquet-Bruhat who, as you know, was the first woman admitted to the Academy of France and the first to prove an existence theorem for Einstein's equations. On this occasion, I presented a talk on the symmetrization of hyperbolic systems compatible with an entropy principle that I told you about in these days. Choquet Bruhat appreciated this work of mine which she presented to Ann. Inst. H. Poincarè. Subsequently, we started a collaboration first in Bologna and then in Paris and together we obtained a very good result concerning Einstein's equations of general relativity. As you know, in Einstein's system of equations, some equations are evolutionary, others are constraints that limit the initial data. If the initial data are compatible, then the constraints are satisfied for all subsequent times. However, free quantities remain. One is g00. which is called the "lapse" which is a measure of time and g0i (i = 1,2,3) the so-called "shift". Usually, physicists and numerical analysts pose g0i = 0 and g00 = 1 but in this way, the Cauchy problem is not well-posed. The idea we had was to not fix the lapse in advance and to combine the evolution and constraint equations so that for a suitable choice of the lapse the system became symmetric with a well position of the Cauchy problem. We published this article in Communications in Mathematical Physics. This is one of the papers that has been important in my career and is widely cited. The third moment was when I met Ingo Müller, and this was also an occasional case. A colleague of mine from Ferrara had invited Müller as a visiting professor and to give a series of lectures on Thermodynamics and asked me to come to Ferrara to listen to the lectures. During the first conference, Ingo Müller presents the entropy principle exploited through a technique that uses Lagrange multipliers and I immediately made it clear that this technique was completely similar to what I had found to symmetrize the hyperbolic systems and the multipliers coincided with the field which I had called the “main field” which made the system symmetric. Immediately after the conference, I discussed it with him who was enthusiastic about this similarity and for this reason almost every day Ingo left Ferrara to come to Bologna to discuss with me. And therefore, this was my third important moment in life in a certain sense because we started practically from these discussions to introduce the modern Extended Thermodynamics field which I had my greatest recognition thanks also to the book we wrote together, and which has become popular. Therefore, I tried to find the best of the situation, but I was lucky in a way because I found important people who helped me on the one hand, but on the other hand, I was complementary, therefore, we benefited as a whole. My career has been very beautiful, and I was very happy when at the age of 32 year I become full professor and at the age of 52 I was nominated as a member of the Accademia dei Lincei.

TPL: That’s very interesting, because basically, you’re self-propelled. You don’t have someone to protect, to guide you. You just go by yourself.

TR: I am optimistic; therefore, I like to enjoy life. I try not to stay and sit down. I try to grow, to be curious. For a long time, I like also to listen to people that work in different fields. On the other hand, I don't know exactly what kind of researcher I am because, in a certain sense, my research is at the frontier of three different topics: continuum mechanics, hyperbolic conservation laws and kinetic theory. I know many people from the three groups who often invite me to their conferences, but I don't belong to any of the three families. Probably for a pure mathematician I appear a physicist and for physicists I am mathematician. I remember once talking to Costas Dafermos about this and he very ironically said to me: "Tommaso, don't worry. You are the last, natural philosopher!". Although, he said it jokingly, I liked it a lot because actually for me a true Mathematical Physicist is a Philosopher of Nature.

TPL: Talking about philosophy, you make a point in your talk that there are people who try to derive the Boltzmann equation from Newton particle system and so on, and it’s extremely difficult and to this day, it still has not succeeded. But your point on that is maybe we should not have a unified theory. For a given situation, there’s an appropriate model and, for another situation, different scale, another model. There’s no reason one should link one to the other.

TR: This is my opinion. Because there is no perfect mathematical model. All have validity in a certain range and on a certain scale. Therefore, declaring that continuum mechanics or Boltzmann equation can be obtained from Newton's particle mechanics would be like admitting that Newtonian mechanics is a perfect model! Of course, it is interesting to study singular limits to try to find connections between theories on different scales, however declaring that a model is exact is, in my opinion, a contradiction. The model is not physical reality even if, as Galileo said, the language of nature is mathematics. Therefore, it is fantastic, when I teach my students, the possibility for us to build a mathematical model that describes nature and that foresees the future before it happens. We launch an object such as a satellite and from the initial conditions only we know perfectly well how it will move later.

TPL: There’s a prediction power.

TR: Yes, predict the future. Unfortunately, the discovery of the deterministic chaos due to Lorenz inflicted a severe blow to the Enlightenment belief that man was able to uniquely predict the development of future events for all times after the initial instant. Now, we know that unfortunately, our prediction is only valid for a limited time. However, as we know, each model has its validity under certain conditions. For example, Newtonian Mechanics is perfect for bodies that move at not too high speeds and that are not too small. In the case of particles that have velocities very close to that of light, Newton's mechanics are replaced by Einstein's theory of relativity and if the particles are small, it is necessary to use the quantum mechanics model. Moreover, Einstein's dream of having a unitary theory never came true. Every day there are situations in which it is necessary to build a more valid mathematical model than the previous one, and I like to remember a guide for the researcher following Paul Dirac's teachings. Dirac said: "The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still consider simplicity in a subordinate way to beauty … It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence".

TR: Seung is another example of another sliding door for me.

TPL: I see, I see. Oh, wow.

TR: This due to your fault. Because one time, Professor Liu invited Professor Ha and me to Stanford together with other scientists. As soon as I arrived at Stanford, I immediately understood why this university is very famous. In the morning when we all arrived at the Department there was no program of our talks contrary to what is usual. Liu begins to say: Who wants to talk? Here is the blackboard. But it's fantastic because some people talk for 3 hours, others talk for 10 minutes, so everything was very strange to me. But this stay at Stanford was truly fruitful for me. Seung Ha presents the flocking model of Cucker-Smale and I immediately realize that there is an analogy between this model and the isothermal model for a mixture of fluids. Hence a discussion with him taking into account the analogy with the mixtures we were able to construct a new model for flocking which also takes into account the internal energy (temperature) of the particles. This fruitful collaboration of ours has resulted in several common papers that were published in excellent international journals. Another benefit that I had at Stanford was that by speaking with Laurent Desvillettes I learned that there is a kinetic theory for polyatomic gases and this allowed me to solve a problem that had been open for over more than 20 years realizing together with Masaru Sugiyama and collaborators an extended thermodynamics for polyatomic gases which is bringing new and important contributions to the thermodynamics of non-equilibrium. Therefore, Stanford and Tai-Ping Liu are other examples of my sliding doors. Now I am retired but I still work hard, and I have still a lot of collaborations. Some people try to apply for me for the Emeritus professor. To become Emeritus in Bologna is very difficult.

TPL: I see.

TR: Since every year only very few professors (7-8) in all disciplines of the University are chosen through a complex procedure which consists of 4 steps: The first step, you need 3 letters of recommendation from professors who are still in service of which at least one must be a foreigner. In the second step, there must be at least 50 signatures of professors in your sector who propose your appointment. In the third step, the Department of membership must vote unanimously for the proposal and finally, in the fourth and final step, there is a commission made by the rector and by emeritus professors who must select the emeritus professors. So, the Minister has to sign the appointment.

TPL: Okay.

TR: Therefore, on February 5, 2020, there will be a formal ceremony at the university where the Rector will give me the diploma of Professor Emeritus. Therefore, this is beautiful.

SYH: Okay, so let me ask you a question. So, I know you studied physics in undergraduate and then after that, you somehow, more or less, you studied mathematical physics. Is this a natural track for a physics student?

TR: No, not really. But in mathematical physics would be, in Italy, maybe 50% come for physics because physics is a motivation. And then, they maybe learn some mathematical tools. It’s not so simple for a mathematician to become mathematical physicists because they don’t know the physical motivation. Therefore, in some, by tradition, mathematical physics belongs to mathematics. Therefore, we have 50% that come for mathematics and 50% come for physics. I’m an example.

SYH: Right.

TR: But many people was an example. You know Gallavotti, Pulvirenti, Marchioro, Presutti, and many others come from physics.

SYH: So, in Italy, mathematical physics is one of the strong traditions…

TR: Yes, Rational mechanics was obligatory for all mathematicians in Italy, for all the physicists, for all engineers. Now, unfortunately, many courses in engineering have removed rational mechanics and above all analysts have started to deal with applications too and so the group of mathematical physicists has reduced considerably even if it remains of good quality.

SYH: Yeah.

TR: In the past due to the influence of Bourbakism mathematicians have concentrated mainly on pure mathematics.

SYH: Yeah.

TR: Application was considered not so good.

SYH: Yeah, right.

TR: Now, the computer changes it.

SYH: Right.

TR: Now everything is changed because if you want to receive grants you have to make applications and so now many analysts have become applied mathematicians.

SYH: I see.

TR: Therefore, we lost some powers.

SYH: Right.

TPL: This is a very smart guy called Parisi.

TR: Parisi.

TPL: Yeah.

TR: Parisi now is president of Accademia dei Lincei. He’s a theoretical physicist. He was a candidate in 2002 to the Nobel Prize in physics. You know, the Nobel Prize, if you are not American, the probability is…

SYH: Very low.

TR: Very few. He have an h-index of 118 according to Google. He is a very good physicist, but he had a very strong mathematical interest. For example, he writes a very important paper for flocking using statistical physics model.

SYH: Right.

TR: Another field in which Parisi give important contributions was the so-called spin glass. Another well-known researcher who straddles theoretical physics and mathematical physics is Giovanni Jona-Lasinio. Pioneer of theoretical research concerning spontaneous symmetry breaking, it gave its name to the model of Nambu - Jona-Lasinio. Nambu was known awarded Nobel Prize in Physics in 2008. Mathematical physics was initially only on the subjects of rigid body, continuum mechanics, relativity and the theory of diffusion. Only in recent times has Mathematical Physics been enriched with other sectors thanks to theoretical physicists who have a mathematical interest. For example, statistical mechanics become part of mathematical physics with the arrival of Giovanni Gallavotti. The kinetic theory thanks to Carlo Cercignani, the quantum mechanics with Sandro Graffi.

TPL: How is the state of rational mechanics now?

TR: Now? For the mathematician, it is still mandatory. It is also true for physicists, but the presentation is quite different, let's say, for Landau it starts with the minimum of action. Of course, the students don't have the tools to understand the variational method but from the physical point of view, they like this presentation. In mathematics, mechanics are often only dynamic systems. A book adopted in several universities is the Arnold book which although it is a good book, in my opinion, is not the Rational Mechanics that I know.

SYH: Dynamical system, yeah.

TR: For us, rational mechanics consists of the first part in the geometric models of bodies (point, rigid body, systems with a finite number of degrees of freedom) then the axioms of Mechanics come and then the big chapters of dynamics and statics. Then we move on to Analytical Mechanics and Hamilton's equations. There are differences according to the students. For example, students of civil engineering are more interested in statics than in dynamics. I wrote together with three other colleagues a book on rational mechanics (Springer) mainly aimed at students of Engineering. The book is adopted in various universities and on average we sell more than a thousand copies a year.

TPL: I was exposed to this when I was in Maryland, Stuart Antman, he was a one-time editor in chief of Archive for Rational Mechanic and Analysis.

TR: But this journal, unfortunately, it changes a lot because as you know, it’s founded by Truesdell. And a lot of paper was also in the spirit of rational mechanics and modelization. Now, it’s more analysis maybe.

TPL: Yeah, talking about that I just received email today, the editors in chief for Rational Mechanic and Analysis are John Ball and Dick James and now, it’s changed.

TR: It is changed.

TPL: Yeah, it changed to Sverak.

TR: What is his name?

SYH: Sverak. Navier-Stokes guy.

TR: Ah, I see.

TPL: Yeah, so you say more analysis, and another one is Otto.

TR: Yes, Felix Otto. I was in the committee for a prize of Accademia dei Lincei and we give it to him. He’s an analyst.

TPL: Yeah, so now, these are the editors in chief.

TR: It's a pity because in a certain sense it has lost part of the value that the Journal had in the past. Initially, Truesdell himself, Noll, Ericksen, Müller, Gurtin, Joseph and several others publish papers that were mathematically rigorous but had the main purpose of presenting sound physical models in the spirit of the Rational Mechanics. Now it is difficult for many of us to find an appropriate journal. Continuum Mechanics and Thermodynamics (CMT) was invented by Ingo Müller but has now moved a lot towards engineering. I now publish rather in good Physics journals that also accept works with a mathematical formalism such as Physical Review, Annals of Physics. Journal of statistical physics or Physics of Fluids.

TPL: Well, but somehow, we move along. You are doing very well. Yeah.

TR: This situation makes science poor. Some good mathematicians know little about the physical model. Many people publish in leading journals such as Archive Rational Mechanics and Analysis or Communication in mathematical physics making assumptions that are very weak from a physical point of view. I know several works on some mathematical problems of relativistic fluids in which inappropriate constitutive equations are used, taking them as an example from non-relativistic theory. For this reason, I think collaboration for example between mathematical physicists and analysis is very fruitful. The collaboration between Seung and me is an example. He knows the analysis much better than I do and I know the model better than he does.

TPL: But I guess, we have a dilemma because human capacity is finite, and so, you say that we need to have a balance, need to collaborate with someone with other expertises.

TR: It’s much more power in my way.

TPL: To be able to talk to other guys is a quality not everyone has.

TR: In my generation, the preparation was very vast but not so deep as to start doing quality research. I remember that after graduation I knew a little bit of everything from relativity to continuum mechanics, I was familiar with classical analysis and differential geometry, but I did not know where the state of the art of research was. On the other hand, there are now young people who know only one topic in-depth that allows them to immediately publish in a good journal. However, as soon as you move them from their sector, they know nothing and are not interested in anything other than their field.

SYH: So was paper. Yeah. Book.

TR: Yeah, another current problem is the issue of bibliometrics indexes such as h-index or the IF of journals. If today I propose to one of my students to publish on the Accademia dei Lincei he immediately says no better a more important journal because this is not useful for my career. As we know, the same reviewer system is bugged by these parameters today. For example, if you refuse a paper in which you are widely cited, it is not convenient for you and therefore you think twice before refusing it. If, on the other hand, you submit a paper, perhaps the reviewer asks to quote him several times and blackmails you that if you don't do it, the paper is rejected. The same system for qualification in Italy is regulated by the bibliometrics index.

TPL: We were talking to the Japanese senior professor, a number theorist, and he said that at that time when he was young, you don’t even need a PhD to be a professor.

TR: Yes, even in Italy the PhD did not exist and is relatively recent. Usually, you could become an assistant even without papers but with the only master thesis if the professor saw talent in you.

TPL: Okay, okay.

TR: In my day there were only two positions at the university: assistant and a full professor. Due to the post-war economic boom, there were many possibilities for positions.

TPL: A lot of job opening. In 1973, OPEC was formed, so it was very hard to find a job, and I found a job without a paper submitted. So, it was not normal at that time. That was the mindset at the time, people feel, okay, the dissertation is good and we look at it and agree, okay, we offer him the job.

TR: Well, the only advantage, at least, in my generation was that very good person have talent, all of them become a professor. But also, maybe people not so good become a professor!

TPL: I see. Seung you have a final question to ask?

SYH: So, what’s your future expectation for your subject?

TR: As far as Mathematical Physics is concerned, I am not so optimistic because, probably by analogy with the principle of entropy, all culture tends towards a sort of homogenization and equilibrium. I'll give you an example of what I mean. Italy was formed in the past from many regions and each had its language, a very rich dialect. for example, the Sicilian I know before I learned Italian is a very rich language, the result of mixtures of many dominations that have followed in Sicily. If I speak Sicilian with a Bolognese, he doesn't understand anything to tell you how different they were. However now we all speak Italian, but we have forgotten our original language. The same thing is now happening in English. Every one of us speaks a modest common English which is certainly not Shakespeare's English. Well, in my opinion, this also happens in mathematics. We are losing the skills that everyone had in his field by creating a sort of average applied mathematician. Furthermore, the language of mathematics often becomes not communicable externally, and then unsuitable examples are used. For example, after Alessio Figalli won the fields medal to explain the event in the newspapers, they described him as the one who studied how clouds move (sigh).

TPL: Yeah, let me tell you a story. After Andrew Wiles solved Fermat’s last theorem, he comes to Stanford to give the public lecture. And then, someone from the audience asked a question and the question is that Mr. Wiles, now that you have solved the most difficult mathematical problem, maybe you have time to think about how the brain works. He said: I only know how to solve this equation. Well, the importance of mathematics, the centrality of mathematics is now clear and recognized by the general public. So, any serious topic, if not continued as a research subject, it should serve as a general educational object.

TR: On the other hand, there is a difficulty, which is as follows: the description of nature is also more complex and requires increasingly complicated models. Look, for example, at a topic we know very well, the Riemann problem. We know the theory only in the case of one space dimension. In more dimensions we know almost nothing, however, the physical reality is three-dimensional, and the numeric analyst makes simulations of Riemann-like problems daily. There is often a great distance between the model maker and the analysis. Once I confess by talking to Ingo Müller of Tai-Ping Liu he told me I have great respect for him, who is certainly a great mathematician but unfortunately, he only knows how to study Burgers' equation! On the other hand, if we give an analyst the complicated equations of Extended Thermodynamics, he immediately abandons the idea of being able to say something serious in terms of qualitative analysis. We must take into account that nature is often described by non-linear systems in which very little is known at a mathematical level. Take for example the Navier-Stokes equations in which the 1 million $ problem of proof of existence and uniqueness in the same functional class is still unsolved. However, engineers and computer programming use this equation every day. Therefore, in a sense, in my opinion, there is still a gap, on the one hand, the description of the realistic model and the mathematical, analytical, geometric instrument which are not sufficient to study it rigorously. This is my point. On the other hand, I agree with you. Mathematics becomes popular because now people understand that mathematics is involved in normal life. You drive and you have GPS that shows you the way and there is some math, you do a CT scan and there is math. Type a password and there is math ... then math starts to understand that it is not just an abstruse and abstract discipline. On the other hand, as the philosopher Giovanbattista Vico said, the science and history have a periodic character. In a certain era, Mathematics was at the center of Science, then in 1900 the period of physics came, biology is currently in vogue and artificial intelligence will probably be soon.

TPL: About the artificial intelligence and numerical computation, let me quote something from my old colleague in Maryland, Babushka. He’s a really important guy in finite elements. He once said that engineers do not mind if only 20% of the computation is correct. However, the difficulty is to know which 20%. So, we need some mathematics to have a sense of that.

TR: Yes, yes. Exactly.

TPL: And AI also. The reason AlphaGo is important and AI sudden flourishes after AlphaGo is that it can be tested against the Go master. This Go master, Lee Sedol, was considered a legend because he has won so many games and so on. So, he represents a benchmark. And the AlphaGo beats him, and so people will say that, okay, now we know that’s a real thing.

TR: Yes. Look at the statistics for example. For a long time, it was not considered important. Now, everyone wants statisticians in particular big data experts who, among other things, require solid and strong mathematics. This field is now truly popular in all countries. My son who took his master's in computer engineering is now doing the PhD in Artificial Intelligence and machine learning and is also studying a lot of statistics.

TPL: That’s great; I learn things from you. Thank you.

- Liu Tai-Ping is a faculty member at the Institute of Mathematics, Academia Sinica.
- Seung Yeal Ha is a faculty member at the Department of Mathematics at the Seoul National University.