Celebratio Mathematica

Alain Connes

Interview with Alain Connes

by Allyn Jackson

Alain Connes walking near the country home where he now spends most of his time.
Photo courtesy of Alain Connes.

Alain Connes was born on 1 April 1947 in Draguig­nan, in the south of France. After at­tend­ing sec­ond­ary school in Mar­seille, he entered the Ecole Nor­male Supérieure in Par­is in 1966. He earned his doc­tor­ate in 1973, un­der the dir­ec­tion of Jacques Dixmi­er.

In the early part of his ca­reer, Connes held po­s­i­tions in the CNRS (Centre Na­tion­al de la Recher­che Sci­en­ti­fique) and the Uni­versity of Par­is VI and was also as a vis­it­or at Queen’s Uni­versity in Ontario and the In­sti­tute for Ad­vanced Study in Prin­ceton. He was ap­poin­ted to the Léon Motchane Chair at the In­sti­tut des Hautes Et­udes Sci­en­ti­fiques in Bûres-sur-Yvette in 1979, and, in par­al­lel, to the Chair of Ana­lys­is and Geo­metry at the Collège de France in 1984. In 2017 he re­tired from both po­s­i­tions. He has also held dis­tin­guished pro­fess­or po­s­i­tions at Vander­bilt Uni­versity and at the Ohio State Uni­versity.

In ad­di­tion to the Fields Medal in 1982, his hon­ors in­clude the Ampère Prize of the French Academy of Sci­ences (1980), the Cra­foord Prize (2001), and the CNRS Gold Medal (2004).

Connes made ma­jor ad­vances in op­er­at­or al­geb­ras that re­vo­lu­tion­ized the sub­ject and stim­u­lated a great deal of fur­ther re­search. This led him to de­vel­op, start­ing in the late 1970s, an en­tirely new branch of math­em­at­ics, non­com­mut­at­ive geo­metry, which turned out to have deep con­nec­tions to many fun­da­ment­al ques­tions in math­em­at­ics and phys­ics. Through his dozens of col­lab­or­at­ors and more than 250 pub­lic­a­tions, Connes has had a ma­jor im­pact on math­em­at­ics over the past sev­er­al dec­ades.

What fol­lows is the ed­ited text of an ex­tens­ive in­ter­view with Connes, held in Ju­ly 2020.

Early life in the South of France

Connes (right) with his father and two brothers.
Photo courtesy of Alain Connes.

Jack­son: You were born in Draguig­nan, which is in the south of France. Can you tell me about your child­hood grow­ing up there?

Connes: Let me tell you how my grand­par­ents got there. My grand­par­ents on my moth­er’s side were both born in Con­stantine, Al­ger­ia. They came to France in 1918, after my moth­er was born; she was born in Tangi­ers, Mo­rocco. When they ar­rived in France they looked in a ref­er­ence book for the city that was the health­i­est in France. And they found Draguig­nan!

My grand­fath­er was an en­gin­eer, but he re­tired quite early in his life. They then bought an es­tate in Draguig­nan. My par­ents met in Draguig­nan in 1944, when the Amer­ic­ans came to the south of France at the end of the war, in what we call the débar­que­ment. There were per­haps 1000 glider planes, people jump­ing out with para­chutes — com­ing to Draguig­nan!

My moth­er died last year at age 101. She and my fath­er both died (he also lived un­til age 101) in the house in Draguig­nan where I was born. It is heart­break­ing be­cause now we are selling the house, which is where all of my sub­con­scious is mapped. It’s not just a house, it’s an es­tate with a big garden, a small wood, vine­yards, and very big, old ce­dar trees. I am at­tached to that place, be­cause of the quiet­ness. It’s a place that has serenity built-in.

The three brothers; Alain is on the left.
Photo courtesy of Alain Connes.

When I was eight years old, my fath­er de­cided that edu­ca­tion in Draguig­nan was not good enough for us. So he took a very dan­ger­ous job in Mar­seille, as the lead­er of a po­lice squad fo­cused on the traf­fick­ing of al­co­hol. From time to time, he would dis­ap­pear dur­ing the night when he was ar­rest­ing the ban­dits.

Jack­son: Was al­co­hol il­leg­al at the time?

Connes: Al­co­hol was not il­leg­al, but it was very much taxed, so there was a lot of il­leg­al traf­fick­ing. For­tu­nately, he left this job just at the be­gin­ning of the Mafia’s ar­rival in 1966. So he didn’t have to deal with the Mafia, but it was still quite dan­ger­ous. He had a car with a ra­cing en­gine and of­ficers on mo­tor­bikes with him. It was nev­er a simple thing to ar­rest the people, be­cause they had to ar­rest them while they were traf­fick­ing. We were all in stress whenev­er he would dis­ap­pear like that dur­ing the night.

Jack­son: Your fath­er was French, is that right?

Connes: Yes. His par­ents were from the south­w­est of France.

Jack­son: What kind of work did your fath­er do when you lived in Draguig­nan?

Connes: At that time, my fath­er was em­ployed by the tax of­fice in a job also re­lated to al­co­hol, over­see­ing the pro­duc­tion of wine in the south. His own fath­er had been killed in the first World War when my fath­er was one year old, so he nev­er knew his own fath­er. He wanted to give us an edu­ca­tion as a real fath­er. That was quite im­port­ant. I have two broth­ers. My older broth­er cre­ated his own soft­ware com­pany, and he is quite rich. My young­er broth­er is a doc­tor and a well known spe­cial­ist for Crohn’s dis­ease; he is re­tired now.

We had a very tough edu­ca­tion. For in­stance, dur­ing hol­i­days we would get ad­di­tion­al work from my fath­er, and when school was in ses­sion he would add to the work as­signed by the school — his ad­di­tion­al con­tri­bu­tion!

Jack­son: How did you feel about that?

Connes: It was ter­rible! We would find all pos­sible means to es­cape from that, of course.

Jack­son: Do you think it ul­ti­mately helped you?

Connes: It’s dif­fi­cult to say. We three broth­ers were quite dif­fer­ent. My older broth­er would fight with my fath­er. I was not fight­ing. I was sort of cool, tak­ing it easy. It’s not clear at all that it’s a good re­cipe, but it in­flu­enced us a lot. For in­stance, once I came home, and my fath­er asked about the res­ults of a math com­pet­i­tion I had been in. I said, “I was second.” Then he slammed me in the face, be­cause I was not first.

Jack­son: That’s pretty heavy pres­sure.

Connes: There was pres­sure. But be­cause we were three broth­ers and were united, we would find all pos­sible ways to es­cape. For in­stance, we knew where my fath­er hid the trans­la­tions of the Lat­in as­sign­ments he gave us. That made life a lot easi­er! He would ask us to re­cite the les­sons twice: on the day when we were giv­en the les­sons and a second time on the day be­fore we were to re­cite them in school. We would re­cite them to our moth­er be­cause then it was pos­sible to look at the book while my moth­er was cook­ing! So we found all kinds of tricks to es­cape. But it was a very tough edu­ca­tion. There was really a fear in all of us, dur­ing our whole child­hood.

One thing I should add is that I really loved my ma­ter­nal grand­moth­er. She brought in the sooth­ing, emo­tion­al side, which my moth­er also brought in. My grand­moth­er was a pi­an­ist.

Jack­son: Your grand­moth­er was a pi­an­ist and your grand­fath­er an en­gin­eer. So they were well off when they were in Al­ger­ia.

Connes: Yes, they were very well off when they ar­rived in France, but they lost everything be­cause they bought ex­actly the wrong type of bonds. They lost their whole for­tune, everything.

Jack­son: But they had their home, the es­tate in Draguig­nan.

Connes: Yes, they had their home, which was ex­tremely help­ful of course.

From lycée to Ecole Normale

Jack­son: When you moved to Mar­seille when you were eight years old, what kind of school did your fath­er put you in?

Connes: We were put in the Lycée St Charles, which was very close to our home. It was a stand­ard type of school, but good, with good teach­ers. At that time it was com­pletely egal­it­ari­an. If you got in­to the school, you would have a good edu­ca­tion.

Jack­son: Were you in­ter­ested in math­em­at­ics at that time?

Connes: Yes, but I was not in­ter­ested in com­pet­i­tion. I was really in­ter­ested in my own think­ing, and this went on un­til I was in pre­par­at­ory school for en­ter­ing the Ecole Nor­male. At that time I had already de­veloped a the­ory that I liked very much. If the prob­lems that the teach­er as­signed were re­lated to my own think­ing, then I could do very well. But if they were not re­lated, then they didn’t ap­peal to me.

I had one teach­er when I was in pre­par­at­ory school who was very good and who was in­ter­ested in what I was de­vel­op­ing. This was very nice, to feel that there was some­body who cared about my idea.

Jack­son: What was this idea?

Connes: Later when I went to the Ecole Nor­male I dis­covered that it was already known. The idea was to re­place dif­fer­en­ti­ation by fi­nite dif­fer­ence op­er­a­tions. I had de­veloped a whole sys­tem for this. What was really im­port­ant was that it was my own. It was not something that I had picked up in books.

When I entered Ecole Nor­male, I was much more pre­oc­cu­pied by try­ing to find a girl­friend than work­ing. This is the truth! Also, at that age I had a very bad com­plex about my phys­ic­al ap­pear­ance. For in­stance, I would walk only on the right-hand side of the street be­cause I hated my right pro­file and didn’t want people to see it.

Jack­son: But your right side looks fine.

Connes: It looks fine now! It was strange, this com­plex. Also, when I would try to work, I would sing and be dis­turbed by my own singing. That was pe­cu­li­ar, but that’s how I was!

Jack­son: This was a stage of grow­ing up.

Connes: Ex­actly. I was not so much ob­sessed by math­em­at­ics. I was really in­ter­ested in my own do­ings, not so much in passing ex­ams or things like that.

Jack­son: Did your broth­ers also do the pre­par­at­ory classes and go to the Ecole Nor­male?

Connes: Yes. Two years be­fore me, my older broth­er had entered the Ecole Nor­male. So he opened the way. My young­er broth­er did his stud­ies in medi­cine, be­cause there was a tra­di­tion in the fam­ily of my moth­er of med­ic­al doc­tors. She was a med­ic­al doc­tor, and her grand­fath­er also.

Jack­son: Did you learn to play an in­stru­ment?

Connes: When I was five years old I began pi­ano les­sons, and I really loved it. But when we moved to Mar­seille, we could not have a pi­ano in the house. My fath­er told me I had to choose between mu­sic and stud­ies. So I dropped the pi­ano then. I of course al­ways re­gret­ted im­mensely to have done that. When I was twenty, I star­ted again to play the pi­ano, but of course I had missed the most im­port­ant years for learn­ing. I have done a lot of work to re­cov­er from that, but I nev­er re­covered to the point that I would have been at. But okay — this is life.

Jack­son: You can’t do everything.

Connes: You can’t do everything. I now see very well that I have a part of the brain that is mu­sic­al. In fact I just wrote a pa­per for the Journ­al of Math­em­at­ics and Mu­sic. But I know that the part of the brain that is oc­cu­pied by mu­sic is sort of com­pet­ing with the part that is oc­cu­pied by math­em­at­ics. Of course, they are ex­tremely close. This might sound strange, but of­ten I learn a lot in math­em­at­ics by study­ing scores of mu­sic.

Jack­son: How does that hap­pen?

Connes: In math­em­at­ics, you might in some cases have the im­pres­sion that you have reached the highest level of soph­ist­ic­a­tion. But then you study a great mu­sic­al score, and you find that the com­poser has a level of soph­ist­ic­a­tion that is about twice the level of soph­ist­ic­a­tion of the best math­em­at­ics. This is what I have in mind. There are com­posers, es­pe­cially of the Ro­mantic peri­od, who have reached a level of mu­sic­al pre­ci­sion that I al­ways find com­fort­ing and a source of en­ergy to do math­em­at­ics. So I use mu­sic­al scores as a source of soph­ist­ic­a­tion, but I also like to im­pro­vise and to let things out.

Jack­son: And the singing when you were at the Ecole Nor­male?

Connes: That was poor singing, Cor­sic­an songs! It was just for fun. I had a happy tem­pera­ment, es­pe­cially com­ing from the south of France and find­ing my­self in Par­is, where people were much more in­tel­lec­tu­al.

Jack­son: Was it a big cul­ture shock to go to Par­is?

Connes: Oh, yes. Without be­ing dis­par­aging, it’s true that the south, in par­tic­u­lar Mar­seille, doesn’t have at all the kind of in­tel­lec­tu­al back­ground that Par­is has. I was amazed when I was in Par­is be­cause it was very ac­cept­able to be totally im­mersed in in­tel­lec­tu­al stuff. In Mar­seille you had to be well dressed. People were judged on their phys­ic­al ap­pear­ance much more than in Par­is. In Par­is you could see in the streets people who were not dressed cor­rectly and looked like tramps. They didn’t care, it was not im­port­ant. In Mar­seille, it was im­port­ant. I don’t know if it has changed by now, but this was how it was at the time.

Jack­son: Was your fath­er sat­is­fied when you got in­to the Ecole Nor­male?

Connes: Oh, sure. My moth­er would have pre­ferred us to go to Ecole Poly­tech­nique be­cause they had a beau­ti­ful uni­form! Oth­er­wise, my par­ents were very sat­is­fied.

Freedom to think and grow

Jack­son: It was 1966 when you went to the Ecole Nor­male. What was that like?

Connes: We had a mar­velous pro­mo­tion of young guys, and many be­came ex­cel­lent math­em­aticians. In that spe­cif­ic year, at this spe­cif­ic time at the Ecole Nor­male, we had no pres­sure. We had the op­por­tun­ity to stop do­ing routine work of pre­par­at­ory school and to try to think. I have beau­ti­ful memor­ies from that year. A friend of mine would ask me a prob­lem, and then for the whole week­end I would think only about this prob­lem. That was great. We were genu­inely in­ter­ested in prob­lems in math­em­at­ics. This was our daily bread. But we were not work­ing on as­sign­ments; we would not fol­low classes. We had some minor ex­ams to pass at the end of the year, but we were free to think about math­em­at­ics.

Nowadays people in Ecole Nor­male are much more treated like chil­dren. They have to pass ex­ams and do this and that. They are not giv­en this fun­da­ment­al bless­ing, which is time to think and de­vel­op on their own. All the friends I made then did ex­tremely well ex­actly be­cause we were treated in a way that al­lowed us to grow.

This is the time when I learned that if for in­stance you have a very com­plic­ated cal­cu­la­tion to do, the best way is to put things in your head first, and then take a walk. No pa­per, no pen­cil. When you take a walk, your mind will learn to build a men­tal pic­ture. To con­struct this men­tal pic­ture, to make it ex­ist — this is the most dif­fi­cult part of math­em­at­ics. In or­der to do it, you have to be fight­ing with a prob­lem for a while — not read­ing a book, not be­liev­ing a res­ult is true be­cause some­body says it is. No, this doesn’t mat­ter. What really mat­ters is you fight with it by your­self, alone. Then gradu­ally the men­tal pic­ture will ex­ist in your mind.

Jack­son: What is this men­tal pic­ture? It is a geo­met­ric pic­ture?

Connes: I don’t know how it is ma­ter­i­al­ized in the brain, but it is something that, when you think about it, lights up and sends you sig­nals. Even more strik­ing is that it will con­tin­ue to send you sig­nals even when you are not think­ing about it. It’s ex­actly like when you leave your home and five minutes later you say, “Oh shit, I for­got to turn off the stove.” These things ex­ist in the brain, and they send you sig­nals. Sim­il­arly in mu­sic, you can have something that ex­ists in your mind, a tune or a theme. This is something amaz­ing and very hard to define.

Jack­son: With mu­sic you can go over a piece in your mind as the piece pro­ceeds in time. Is it like that with the men­tal pic­ture of math­em­at­ics?

Connes: It de­pends wheth­er it’s al­gebra or geo­metry. If the prob­lem is geo­met­ric and there is a solu­tion, it will ap­pear in one stroke, with no time de­pend­ence. It will be one shot. But not in al­gebra. Al­gebra is much more time-de­pend­ent and evolving. In al­gebra, when you are do­ing com­pu­ta­tions, there is a def­in­ite ana­logy with the time de­pend­ence in mu­sic, which is ex­tremely strik­ing.

Noncommutativity generates time

Connes: In fact, it goes much fur­ther. One of the things to which I con­trib­uted in 1972 was the fact that, when you take a non­com­mut­at­ive al­gebra, you have a ca­non­ic­al time evol­u­tion.

Two Ja­pan­ese math­em­aticians, [Minoru] Tomita and [Masami­chi] Take­saki, had dis­covered that if you have a state on a cer­tain kind of al­gebra, then there is a time evol­u­tion. What I dis­covered by do­ing ex­tremely com­plic­ated cal­cu­la­tions over many months is that this time evol­u­tion is in fact in­de­pend­ent of the state, when you look at it in the right way, mean­ing that you for­get about the trivi­al auto­morph­isms. The proof when I wrote it down was in­cred­ibly simple, but it came from do­ing lots of cal­cu­la­tions. The fruit was ex­tremely simple, but the pre­par­a­tion was ex­tremely com­plic­ated.

The out­come of this still fas­cin­ates me now: The non­com­mut­ativ­ity, which was dis­covered by people in quantum mech­an­ics, in fact is a gen­er­at­or of time. I am still think­ing about the fact that the pas­sage of time, or the way we feel that time is go­ing on and we can­not stop it, is in fact ex­actly the con­sequence of the non­com­mut­ativ­ity of quantum mech­an­ics, or more ex­pli­citly of the in­her­ent ran­dom­ness of quantum mech­an­ics.

Something Heis­en­berg dis­covered, which is ab­so­lutely amaz­ing, is that when you re­peat cer­tain mi­cro­scop­ic ex­per­i­ments, the res­ults will nev­er be the same. You send a photon through a very small slit and look where it lands on a tar­get. If you re­peat the ex­per­i­ment, you will nev­er be able to pre­dict where the photon will land. One can use this fact to con­coct ran­dom num­bers, and, un­like gen­er­at­ing ran­dom num­bers by com­puter, one could cre­ate a se­cur­ity sys­tem that would be per­fectly safe. Even if an at­tack­er knew all the devices you are us­ing, the at­tack­er would nev­er be able to re­pro­duce it. This is the most strik­ing fact of quantum mech­an­ics. The philo­soph­ic­al is­sue that has fas­cin­ated me for all these years, is that I be­lieve it is pre­cisely this type of ran­dom­ness that is at the ori­gin of the passing of time.

Left to right: Jacques Dixmier, Danye Chéreau and Alain Connes at the Salon du Livre de Paris in 2013.
Photo by Jean-François Dars.

I wrote a book with my wife and with my teach­er, Jacques Dixmi­er, Le Théâtre Quantique.1 The pur­pose of the book is to ex­plain this idea, which is much more a philo­soph­ic­al is­sue than a math­em­at­ic­al is­sue.

Jack­son: Why does the ran­dom­ness in quantum mech­an­ics pro­duce time?

Connes: The non­com­mut­ativ­ity is at the ori­gin of this ran­dom­ness. What Heis­en­berg dis­covered is that if you try very hard to know the po­s­i­tion of the photon, to­geth­er with its mo­mentum, you can­not do it. This is pre­ven­ted by the fact that po­s­i­tion and mo­mentum do not com­mute. Why does this non­com­mut­ativ­ity gen­er­ate time? In the rel­ev­ant equa­tion, even though two things do not com­mute, you can still in­ter­change their or­der, so you re­place AB by BA. This changes the mean­ing, just as “mel­on” is not the same thing as “lem­on,” even though the let­ters are the same. But there is a price to pay to in­ter­change the or­der: When you per­mute A and B, and you make the A pass on the oth­er side, you have to make it evolve with time. And the time in which it has to evolve is in fact the purely ima­gin­ary num­ber \( i \). This is what is be­hind the scenes.

Heis­en­berg made his dis­cov­ery at a time when he was sick with hay fever in the spring­time. He was sent to an is­land called Heligo­land, which is in the North Sea. He stayed there for a few weeks, do­ing his own com­pu­ta­tions. One night, at I think 4 in the morn­ing, he had in front of his eyes the whole scene. And he was scared, be­cause what he saw was quantum mech­an­ics, later called mat­rix mech­an­ics. He had dis­covered non­com­mut­ativ­ity of phys­ic­al quant­it­ies.

In the non­com­mut­at­ive world there is something that is totally ori­gin­al that does not ex­ist in the com­mut­at­ive world, where “mel­on” would be the same thing as “lem­on”, and that is this God-giv­en time evol­u­tion. It makes things much more in­ter­est­ing than if they were stat­ic. When you pass to the com­mut­at­ive, you lose a lot of in­form­a­tion that, if you keep it, will al­low you to com­press the ex­tern­al world in a much sim­pler man­ner.

Finding your own garden

Jack­son: I would like to re­turn to your early days at the Ecole Nor­male. When you went there in 1966, the IHES [In­sti­tut des Hautes Et­udes Sci­en­ti­fiques, foun­ded in 1958] was go­ing strong. Al­ex­an­der Grothen­dieck and his school were there. Were you a part of that at all?

Connes: No. At that time, the way I per­ceived the de­vel­op­ment around Grothen­dieck was: I have only one way to be my­self, which is to stay as far apart as I can from this group. But I have to add that now I have read Grothen­dieck’s book Récoltes et Se­mailles,2 and I have of course read many of his pa­pers. I have come to love those de­vel­op­ments. I am also now in­volved in try­ing to have some texts of Grothen­dieck pub­lished and re­vived.

When you be­gin to do math­em­at­ics, you have to have your own garden, even if it is re­mote from the very fash­ion­able things. And you have to be­gin to ex­ist there. It doesn’t mat­ter if it’s a small garden. What mat­ters is that it’s yours. What mat­ters is that you have been think­ing a lot about it and you like it, and you take it as a start­ing point. This is the way I felt.

Jack­son: At the time a lot of math­em­at­ics was very much dom­in­ated by Grothen­dieck and his stu­dents.

Connes: Not only that, but I heard some people say­ing, “Why are you do­ing math­em­at­ics? It will all be done by these people.”

Jack­son: That there was no math­em­at­ics out­side what they were do­ing at IHES?

Connes: Yes. Much later Grothen­dieck un­der­stood that this was the wrong at­ti­tude. The title of his book, Récoltes et Se­mailles [“reap­ing and sow­ing”], in a way can be un­der­stood as say­ing that he com­pre­hen­ded that, by be­ing too force­ful, he had had a neg­at­ive ef­fect. For­tu­nately math­em­at­ics is such an im­mense sub­ject that there is room for every­body. Still, so­ci­olo­gic­ally, when you are a be­gin­ner, it’s very hard.

Jack­son: You were in Par­is dur­ing the big up­heavals in 1968.

Connes: Yes. In 1968, my older broth­er Bern­ard was fight­ing on the bar­ri­cades. On my side, not at all. I was hav­ing a love af­fair, so I didn’t care. I didn’t get in­volved at all. I was dis­tant. I was not in­volved polit­ic­ally. I didn’t want to be.

Jack­son: What were you think­ing about math­em­at­ic­ally at this time?

Connes: When I was at the Ecole Nor­male, I de­veloped something quite spe­cial about zer­os of poly­no­mi­als in the com­plex plane. I was asked by [Charles] Pisot, a num­ber the­or­ist, to talk in his sem­in­ar about what I did. It was a rather ori­gin­al ap­proach, but it was about a mar­gin­al top­ic. I wrote a Comptes Ren­dus note about this.3

I was also par­ti­cip­at­ing in the sem­in­ar of [Gust­ave] Cho­quet. Cho­quet was a very bright, very witty math­em­atician. His sem­in­ar was very en­joy­able. He de­cided I should learn phys­ics, so he sent me to a sum­mer school in phys­ics in Les Houches in 1970. I was there with my fu­ture wife. This was the first time I learned about op­er­at­or al­geb­ras. That was great. I met a lot of people. Then the year after, some people I had met in­vited me to a meet­ing in Seattle.

A story of serendipity

Connes: This is how I star­ted to work on the Tomita–Take­saki the­ory. It is a story of serendip­ity. Be­fore I went to Seattle I got mar­ried with my wife, in 1971. Neither of us had traveled to the US be­fore. I de­cided to ac­cept the in­vit­a­tion just be­cause I wanted to vis­it the US! I didn’t look at all at the top­ic of the con­fer­ence. We flew to New York to vis­it my broth­er, who was in Prin­ceton at the time. This was in Ju­ly, and it was so warm that the only place that was sort of ac­cept­able was the book­store. We spent a lot of time in the book­store. We were go­ing to travel by train from Montreal to Van­couver and then to Seattle. We had something like five days in the train, with the Great Plains to be crossed — rather bor­ing. So I said, Why don’t I buy a math book to read dur­ing the trip? I hes­it­ated between sev­er­al books that looked quite in­ter­est­ing. Fi­nally I bought a small book of lec­ture notes.4

When we were on the train, I opened the book, and it looked fas­cin­at­ing. Fi­nally we ar­rive in Seattle, I go to the con­fer­ence, and I look at the pro­gram. Oh my God — the au­thor of the book, Take­saki, is one of the lec­tur­ers! This is a sign! I de­cided to go to no lec­tures ex­cept his lec­tures and to study this stuff.

We were in Seattle for a few weeks and had a won­der­ful time. When we came back I looked for who in France was do­ing this kind of math and found it was Jacques Dixmi­er. I de­cided that in Septem­ber I should go to the sem­in­ar of Dixmi­er.

He opened the sem­in­ar by bring­ing in sev­er­al pa­pers and ask­ing who wanted to speak about which pa­per. I raised my hand and took one pa­per,5 just ran­domly. It was on a totally dif­fer­ent top­ic from the Tomita–Take­saki the­ory. I went back home by train, and in the train I found that what the au­thors, [Huzi­hiro] Araki and [Ed­ward James] Woods, were do­ing was in fact deeply re­lated to the Tomita–Take­saki the­ory.

The same day I wrote a let­ter to Dixmi­er, and soon after that I had an ap­point­ment with him. The only thing he told me is: “Fon­cez!” “Go, go, go — go fast.” I wrote im­me­di­ately a Comptes Ren­dus note6 to ex­plain that the in­vari­ants of Araki and Woods could be com­puted us­ing the Tomita–Take­saki the­ory. That was the be­gin­ning of my work.

Jack­son: Dixmi­er un­der­stood ex­actly that you were on to something.

Connes: He un­der­stood com­pletely. And of course he has been my friend since then.

Jack­son: But this is all quite ran­dom, isn’t it, that you picked up that book in Prin­ceton and found Take­saki in Seattle?

Connes: Yes, it was totally ran­dom. Some people said things that were not so nice; they said I was lucky. But serendip­ity is not be­ing lucky. It’s trans­form­ing what you are giv­en in­to luck. As you say, there is a def­in­ite ele­ment of ran­dom­ness, and then one has to do an enorm­ous amount of work. But some­how, it’s work that is guided by the idea that there is something there. In math­em­at­ics this mat­ters more than any­thing else, the gut feel­ing that there is something. It’s not at the level of ra­tion­al think­ing; it’s at the level of in­tu­ition. It is something that is hard to trans­mit to some­body else but that in­hab­its you and al­lows you to go for­ward. And Dixmi­er per­ceived this com­pletely.

Jack­son: He is 96 years old now.

Connes: Yes, and re­cently we wrote, with my col­lab­or­at­or [Ca­ter­ina] Con­sani, a very tech­nic­al pa­per. He is the only one I know who really un­der­stood what we are do­ing there! He is an amaz­ing man. At 96, he had com­ments that were per­fect.

From factors to foliations

Jack­son: In 1973 you fin­ished your thes­is, un­der Dixmi­er’s dir­ec­tion. Can you tell me con­cep­tu­ally what you did in your thes­is?

Connes: I did two fun­da­ment­al things. The first was to show that this time evol­u­tion was in fact in­de­pend­ent of the state, which gives many in­vari­ants of von Neu­mann al­geb­ras, of factors. Factors were in­tro­duced by von Neu­mann to ex­plore non­trivi­al fac­tor­iz­a­tions of the Hil­bert space in quantum mech­an­ics. The second was the main thing, which was to re­duce the Type III factors, which were the ones von Neu­mann had left out com­pletely, to the Type II and auto­morph­isms.

Jack­son: At the time that you star­ted work­ing on this, the Type III factors were not well un­der­stood.

Connes: They were not at all un­der­stood. What I proved in my thes­is is that, first of all, they are clas­si­fied in Type III\( _{\lambda} \), where lambda is between 0 and 1. Then I gave a com­plete re­duc­tion, ex­cept for the Type III\( _1 \), to Type II and auto­morph­isms. Much later, Take­saki did the case of Type III\( _1 \).

After I did this work in June 1972, I went for hol­i­days with my wife. I was not wor­ry­ing at all about pri­or­ity. Dixmi­er had to call me dur­ing the hol­i­day and tell me that I should pub­lish something, be­cause oth­er­wise it would be lost. I was kind of na­ive.

Jack­son: Some­body else was work­ing on the same thing?

Connes: Sure. There was a group of people in King­ston, Ontario, work­ing later on the same thing. But I was the first to dis­cov­er the most im­port­ant res­ults.7 This prob­lem of pri­or­ity would re­cur many times in my ca­reer. But we don’t work to have our name on something. We work for the pleas­ure of dis­cov­ery. And this pleas­ure is something that no one can take away from us. I re­mem­ber I made the dis­cov­ery when I was vis­it­ing Er­ling Størmer in Nor­way, dur­ing those long days in June when the sun doesn’t ac­tu­ally set. I have won­der­ful memor­ies from this time.

Also in my thes­is I found that there are factors that are hy­per­fin­ite but are not in­fin­ite tensor products. This was a res­ult I an­nounced in Ju­ly of the same year, 1972, and that used the whole power of my the­ory. It was not just an ab­stract res­ult. It had many con­sequences that were sur­pris­ing to people at the time.

In 1976 I was ad­mit­ted to IHES as a vis­it­or. I was a well known spe­cial­ist in my area, but the area was not as well known as those of the people at IHES. So at that time I felt like a stranger. I felt that what I was do­ing was very nice, but people did not know about it.

Then I met a fant­ast­ic per­son, Den­nis Sul­li­van, who was at IHES at the time. He has this in­cred­ible So­crat­ic power. He would sit with a new­comer and ask, What are you do­ing in math­em­at­ics? The new­comer would think, This guy is an idi­ot, he is ask­ing such simple ques­tions. You think you know everything and he knows noth­ing. But after a while you real­ize, my God, this is something that I did not un­der­stand in my own work!

With Sul­li­van ex­plain­ing a lot of things to me, I found that, while the sub­ject I had been work­ing on was not fa­mil­i­ar to so many people, there was a way to fab­ric­ate factors in a well known geo­met­ric con­text, the con­text of fo­li­ations. So I made con­tact with dif­fer­en­tial geo­metry. I dis­covered that their fa­mil­i­ar ob­jects, fo­li­ations, im­me­di­ately gives rise to factors, and the most exot­ic factors were ap­pear­ing from the most nat­ur­al fo­li­ations.

An ex­ample is the Anosov fo­li­ation, a well stud­ied fo­li­ation that comes from the geodes­ic flow on a Riemann sur­face. It turns out that the Anosov fo­li­ation gives rise ex­actly to the hy­per­fin­ite Type III\( _1 \) factor, which is a very dif­fi­cult and exot­ic factor in the clas­si­fic­a­tion.

This oc­curred between 1976 and 1978, when Sul­li­van and I were dis­cuss­ing a lot to­geth­er.

Jack­son: This was also around the time that Vaughan Jones be­came a stu­dent of [An­dré] Hae­fli­ger in Geneva. Hae­fli­ger was a ma­jor fig­ure in fo­li­ations at that time. Did you have con­tact with Hae­fli­ger be­cause of the con­nec­tion of your work to fo­li­ations?

Connes: No, only mar­gin­ally. The way of think­ing of Sul­li­van was much closer to my own way. I don’t like to read pa­pers, and neither does Sul­li­van. He has a way of com­mu­nic­at­ing that is or­al but is also ges­tur­al. This fit­ted me per­fectly. He would ex­plain no­tions that, if I were try­ing to learn them from books, it would have taken forever, and I wouldn’t have got it. But he would just make some ges­tures and ex­plain something, and I got it.

You see there the enorm­ous in­flu­ence of in­sti­tu­tions like IHES. Just giv­ing talks is not the same. You have to live with these people, you have to be around, you have to have leis­ure time, time for lunch, time for tea. And pro­gress oc­curs by ac­ci­dent. You could nev­er plan it.

Jack­son: Go­ing back to Vaughan Jones — he was es­sen­tially your PhD stu­dent, even though he was a stu­dent in Geneva and Hae­fli­ger was form­ally his ad­viser.

Connes: That’s true. Vaughan is a very good friend. He picked up on something I had done when I was in King­ston in 1975, about auto­morph­isms of fi­nite factors, and then he de­veloped a beau­ti­ful gen­er­al the­ory of sub­factors. In the 1980s he made a mag­ni­fi­cent dis­cov­ery, the dis­cov­ery of the con­nec­tion with knot the­ory. That was fant­ast­ic.

It’s a strange story in a way, be­cause after Vaughan dis­covered his new knot in­vari­ant, which came from factors, it was re­cast in a dif­fer­ent man­ner by [Ed­ward] Wit­ten, un­der the in­flu­ence of [Mi­chael] Atiyah also. I had to put my foot in the door so that Vaughan Jones would get the Fields Medal. His dis­cov­ery was dressed up in terms of func­tion­al in­teg­rals and things of this type, while the real in­put, the real strength of the dis­cov­ery, was from his own work on sub­factors. I was a little bit put off by this.

Jack­son: What do you mean you had to put your foot in the door about his Fields Medal?

Connes: What I am say­ing is that the trend of that time was to put more em­phas­is on the func­tion­al in­teg­ral as­pect of the knot the­ory, than on the true ori­gin of the in­vari­ant, which was com­ing from the factors. Of course, when you write things that are more geo­met­ric, it’s easi­er to un­der­stand. On the oth­er hand, it’s ab­so­lutely amaz­ing that the the­ory of factors, which looks rather exot­ic, turns out to be re­lated to knot the­ory, which is very con­crete, very ba­sic. And Vaughan dis­covered a real in­vari­ant in knot the­ory. This is an amaz­ing dis­cov­ery. I don’t know many dis­cov­er­ies that can com­pete with it. You need an open­ness of mind to do that. He was in Switzer­land with people who were geo­met­ers, but I don’t know the role that this might have played in his dis­cov­ery. One would have to ask him.8

Complementary ideas

Jack­son: You talked about how im­port­ant it is to have your own ideas, “your own garden,” as you put it. But what is amaz­ing in math­em­at­ics is that you go any­where in the world, and oth­er math­em­aticians have those same ideas, and you can talk to them about them.

Connes: Well, the ideas are not ex­actly the same. Yes, we can com­mu­nic­ate, but what is really in­ter­est­ing is to meet math­em­aticians with com­ple­ment­ary ideas.

In 1978 I spent a year at the IAS [In­sti­tute for Ad­vanced Study] in Prin­ceton, and I met the per­son who would be­come my greatest col­lab­or­at­or, the rep­res­ent­a­tion the­or­ist Henri Mo­scov­ici. The stay in Prin­ceton was im­port­ant to me be­cause I met him there. Oth­er­wise, I felt the IAS was a rather strange place. There was a huge cafet­er­ia, where people would sit at dif­fer­ent tables. I didn’t find it very con­geni­al, ex­cept for meet­ing my col­lab­or­at­or Henri.

Jack­son: Really? Many math­em­aticians talk about the great at­mo­sphere in the IAS din­ing hall and how they love to sit at the math table.

Connes: Some­how, there was a huge con­trast with IHES. At IHES the cafet­er­ia is small, and people are forced to be to­geth­er, where­as in Prin­ceton you could eas­ily sit alone at a table and be ig­nored.

For­tu­nately I met Henri Mo­scov­ici at this time, and we did for many years a lot of work to­geth­er. Oth­er­wise I would have been quite isol­ated, I think. He had many ideas that I was miss­ing and knew things I did not know. This also happened in 1980 when I met Paul Baum. It was an en­counter with some­body who didn’t have the same way of think­ing as I had. It was com­ple­ment­ary. I met Paul Baum at a con­fer­ence in King­ston, at the time when I had dis­covered non­com­mut­at­ive geo­metry. Be­fore I went to King­ston I wrote a Comptes Ren­dus note9 about an idea that came from fo­li­ations. The point was that fo­li­ations not only have meas­ure the­ory, which I found gives rise to factors that are exot­ic, but they also have dif­fer­en­tial geo­metry. I real­ized this dif­fer­en­tial geo­metry could be brought to bear in the non­com­mut­at­ive frame­work. In the Comptes Ren­dus note I just men­tioned, I had done the full non­com­mut­at­ive dif­fer­en­tial geo­metry for non­com­mut­at­ive tori.

Then I met Paul Baum, and he had ex­actly what I was miss­ing. I had con­struc­ted, us­ing geo­met­ric trans­vers­als, mod­ules on the al­geb­ras of fo­li­ations, which meant that I had con­struc­ted ele­ments of K-the­ory. But I didn’t know how to con­struct them in gen­er­al. Baum had ex­actly the idea, in a com­pletely dif­fer­ent top­ic, that would lead to the con­struc­tion of gen­er­al ele­ments of K-the­ory. We met, and poof, there was a spark.

Jack­son: What was the oth­er top­ic that he was look­ing at?

Connes: He was work­ing on a geo­met­ric real­iz­a­tion of what are called the K-ho­mo­logy cycles. K-ho­mo­logy was de­veloped first in the Hil­bert space lan­guage by Atiyah, and then by [Aleksandr] Mis­chen­ko, [Gen­nadi] Kas­parov, and many oth­ers. The point of Paul Baum was to make it geo­met­ric. He had defined a geo­met­ric ob­ject that he was us­ing only for Rieman­ni­an man­i­folds. What I saw im­me­di­ately was that this worked for fo­li­ations as well.

Noncommutative geometry and physics

Jack­son: What is the main idea of non­com­mut­at­ive geo­metry?

Connes: There are spaces, like the space of leaves of fo­li­ations or the space of Pen­rose tilings, that, when you try to view them as or­din­ary spaces, are in­tract­able. They be­come tract­able provided you gen­er­al­ize the idea of Descartes of us­ing co­ordin­ates, to situ­ations where the co­ordin­ates no longer com­mute. Once you ac­cept the use of non­com­mut­at­ive al­geb­ras as al­geb­ras of co­ordin­ates, then you dis­cov­er that you can treat spaces that, with the or­din­ary tools, would be totally in­tract­able.

At the end of the 1970s and in the 1980s, I began to de­vel­op geo­metry — full geo­metry, in­clud­ing dif­fer­en­tial geo­metry and de Rham the­ory, which gave cyc­lic co­homo­logy — so that all the tools that we nor­mally have would be avail­able in this gen­er­al­ized, non­com­mut­at­ive setup. The beauty here comes from the fact that you are not just gen­er­al­iz­ing something; these new spaces have totally new fea­tures. One of them is this God-giv­en time evol­u­tion. The or­din­ary spaces are stat­ic, while these new spaces have the great prop­erty that they are dy­nam­ic­al and have this time evol­u­tion.

When you dis­cov­er something truly ori­gin­al, you can be sure that people will be against you and will try to dis­miss it. This is a fact of life. If you do something in the or­din­ary way, every­body will be happy and can un­der­stand it. But as soon as you do something that people can­not un­der­stand be­cause they are not in the right frame­work, then you can be sure there will be a lot of op­pos­i­tion.

Jack­son: What is be­hind the op­pos­i­tion? Is it just a tech­nic­al bar­ri­er?

Connes: No, it’s not a tech­nic­al bar­ri­er. Math­em­at­ics is evolved and com­plic­ated as it is. You do not want to in­tro­duce something new. This is the re­ac­tion of people.

It’s nor­mal of course to be con­fron­ted with skep­ti­cism. For in­stance, with Henri Mo­scov­ici we solved the Novikov con­jec­ture for hy­per­bol­ic groups us­ing our work.10 This was a prob­lem that was known in­de­pend­ently of the new tech­nique. The new tech­nique has to make its mer­its on prob­lems that were posed be­fore. Oth­er­wise people will not ac­cept it and will say, “We knew this be­fore”, or “Why are you in­ter­ested in this?”

In the middle of the 1980s I dis­covered something I found very sur­pris­ing. When you have these new spaces avail­able, then you can re­think or­din­ary space­time. What I found is that space­time has a fine struc­ture, which is not the or­din­ary con­tinuum and which is just a little bit more com­plic­ated; it is non­com­mut­at­ive. When you take this fine struc­ture in­to ac­count, you find that pure grav­ity will give you the Stand­ard Mod­el coupled to grav­ity. At the time I wrote just one pa­per about this.11 The idea came to full fruition in the 1990s, in my work with Ali Chamsed­dine.

What is mys­ter­i­ous and strange in the Stand­ard Mod­el is what is called the Higgs sec­tor, though it is ac­tu­ally due to three people, [Robert] Brout, [François] En­glert, and [Peter] Higgs. This sec­tor was called the “toi­let” of the Stand­ard Mod­el: it is something you really need in your house but you would not show it off to your guests. This sec­tor is very strange. It gives masses to all particles, but it is due to a scal­ar field, so it is a field with spin zero. This comes out of the blue.

Now, from the point of view of non­com­mut­at­ive geo­metry, the men­tal pic­ture is in­cred­ibly neat. If you think of space­time as like a sheet of pa­per, it is two-sided. When you dif­fer­en­ti­ate a func­tion on this space, you can dif­fer­en­ti­ate it on its re­stric­tion to the up­per side of the sheet, or you can dif­fer­en­ti­ate it on its re­stric­tion to the bot­tom side. But you can also dif­fer­en­ti­ate it by tak­ing the fi­nite dif­fer­ence across the two sides — the dif­fer­ences of the val­ues of the func­tion on the two sides of the pa­per. That gives you a field of spin zero, the Higgs field. This tells you that, provided you re­fine the geo­metry of space­time, you will un­der­stand why the Stand­ard Mod­el looks so com­plic­ated, even though it is just pure grav­ity.

Jack­son: So you have the piece of pa­per with the two sides — where is the non­com­mut­at­ive as­pect?

Connes: The non­com­mut­at­ive as­pect comes from the fact that when you look at that fi­nite dif­fer­ence, it be­comes non­com­mut­at­ive dif­fer­en­tial geo­metry. There is also a slight amount of non­com­mut­ativ­ity in the al­gebra of func­tions, and it is this amount of non­com­mut­ativ­ity that ac­tu­ally gen­er­ates the gauge fields of the strong force and the elec­troweak force. The de­vel­op­ment of these ideas came to a cul­min­a­tion in 2014, in a pa­per12 with Chamsed­dine and [Vi­atcheslav] Mukhan­ov, where we really un­der­stood the non­com­mut­ativ­ity that one had to in­clude in or­der to get the full story.

The start­ing point for that was in the 1980s, when de­vel­op­ing non­com­mut­at­ive geo­metry oc­cu­pied a lot of my time. I had some very ex­pli­cit ex­amples, like space­time, which were mo­tiv­at­ing the gen­er­al the­ory, and of course fo­li­ations. An­oth­er ex­ample is Pen­rose tilings. In the late 1980s I went to a con­fer­ence held in a castle near Mu­nich, Schloss Ring­berg. [Ro­ger] Pen­rose gave an in­triguing talk about Pen­rose tilings. These are tilings of the plane that are not peri­od­ic. They were dis­covered by lo­gi­cians. The ini­tial tilings used many dif­fer­ent tiles, but Pen­rose sim­pli­fied them to only two tiles, which is quite re­mark­able. You can tile the plane in many dif­fer­ent ways with these two tiles.

Pen­rose showed an amaz­ing prop­erty of these tilings. If you have two tilings that are not the same, then you can take a por­tion of one of the tilings, and you can find that por­tion in­fin­itely many times with­in the oth­er tiling. He put up two trans­par­en­cies show­ing this, and he said, “There is something quantum be­hind that.” When I came back from the con­fer­ence, I real­ized im­me­di­ately that the space of Pen­rose tilings was a non­com­mut­at­ive space.

Jack­son: What does that mean, that “there is something quantum” about this space?

Connes: What it means is that the al­gebra of func­tions will be non­com­mut­at­ive, so it will be a Hil­bert space story. But here is what it means at the level of Can­tor and of set the­ory. If you view the col­lec­tion of Pen­rose tilings as a set, it has the car­din­al­ity of the con­tinuum. But the claim that I make — and this is a char­ac­ter­ist­ic prop­erty of non­com­mut­at­ive spaces — is that you can­not put it ef­fect­ively in bijec­tion with the real num­bers. In fact, you can­not in­ject it ef­fect­ively in the real num­bers. If I have two dif­fer­ent real num­bers and look at their decim­al ex­pan­sions, they will be dif­fer­ent at some point. But this is not the case for Pen­rose tilings, be­cause if I look at them loc­ally, I can­not dis­tin­guish between two of them.

When I wrote my book Non­com­mut­at­ive Geo­metry13 in 1994, I put Pen­rose tilings at the be­gin­ning, be­cause it is an ex­ample that is very strik­ing. Pen­rose had the right in­tu­ition. The space has a to­po­logy, a non­com­mut­at­ive to­po­logy. And the Golden Ra­tio comes out by a mir­acle from the al­gebra.

Jack­son: How does it come out?

Connes: When you have a non­com­mut­at­ive al­gebra, you have its K-the­ory, which was first in­ven­ted by Grothen­dieck and was ad­ap­ted by Atiyah to the to­po­lo­gic­al frame­work and which makes sense in the non­com­mut­at­ive case. You can com­pute the K-the­ory and also map it to the real num­bers by the trace, if there is a trace on the al­gebra. For the case of the Pen­rose tilings, when you map its K-the­ory to the real num­bers, you get the Golden Ra­tio, just by a mir­acle.

In the new non­com­mut­at­ive world, there are things that were easy to ad­apt, like K-the­ory, which was al­most built for non­com­mut­at­ive situ­ations. Then there are things that were much harder to ad­apt to the non­com­mut­at­ive world, and this is what I did with cyc­lic co­homo­logy at the be­gin­ning of the 1980s.

Connecting to the Riemann Hypothesis

Jack­son: Your work made con­nec­tions to the Riemann Hy­po­thes­is. How did that come about?

Connes: In the 1990s I col­lab­or­ated with Jean-Benoît Bost on a sys­tem of quantum stat­ist­ic­al mech­an­ics that had a very strik­ing prop­erty called spon­tan­eous sym­metry break­ing. This can be ex­plained quite simply. Ima­gine you are sit­ting at a round table with sev­er­al people. On each side of each per­son there is a bread plate. As soon as one of the people de­cides to take the plate on the left, it is clear that all oth­ers will have to take the one on the left. But that first per­son could have picked the plate on the right, and then every­body else would have to do the same. That’s called spon­tan­eous sym­metry break­ing.

With Jean-Benoît Bost, we found a sys­tem with spon­tan­eous sym­metry break­ing.14 Its par­ti­tion func­tion was the Riemann zeta func­tion, which was bizarre. It came out of the blue. Be­cause of that pa­per with Bost, I got in­vited in 1996 to a con­fer­ence in Seattle in hon­or of Atle Sel­berg, who had made a lot of dis­cov­er­ies about the Riemann zeta func­tion. I went to the con­fer­ence, and there was quite an in­ter­est­ing crowd of people, in­clud­ing Paul Co­hen for in­stance and sev­er­al phys­i­cists. I gave a talk about the work with Bost. Af­ter­ward Sel­berg came to me and said, “You know, it’s un­clear that your work will truly be re­lated to the zer­os of the Riemann zeta func­tion.”

Jack­son: Why did Sel­berg think that?

Connes: In my talk, the func­tion was just ap­pear­ing as a func­tion; the zer­os did not have any mean­ing for my talk. So of course for him, it was not clear at all that there would be any re­la­tion.

I came back from Seattle, and in­stead of try­ing to ad­apt to the loc­al time — as you know there is nine hours of jet lag — I kept liv­ing on Seattle time, more or less. I could do this thanks to the un­der­stand­ing of my wife! I was not work­ing. I was read­ing The Right Stuff, which re­counts the story of Apollo 13. After one week, I sud­denly real­ized that there is a space that pops out ex­tremely nat­ur­ally from the sys­tem we had with Jean-Benoît Bost, which was a non­com­mut­at­ive space and from which the zer­os of zeta were ap­pear­ing com­pletely nat­ur­ally. The terms of the Riemann–Weil ex­pli­cit for­mula also came in ex­tremely nat­ur­ally.

A prob­lem much stressed by some phys­i­cists in Seattle is that, when you try to real­ize the zer­os of zeta as a spec­trum — which every­one was try­ing to do — it is prob­lem­at­ic be­cause of a per­sist­ent minus sign in some of the terms. This minus sign pre­vents the na­ive ex­pres­sion of the zer­os as a spec­trum. When I found the terms of the Riemann–Weil ex­pli­cit for­mula were ap­pear­ing nat­ur­ally, I un­der­stood that one shouldn’t look for an emis­sion spec­trum but an ab­sorp­tion spec­trum. Let me ex­plain the dif­fer­ence between the two, be­cause it is cru­cial in my work.

When you pass sun­light through a prism, the light de­com­poses in­to vari­ous wavelengths, or fre­quen­cies, and this gives you a beau­ti­ful im­age of a rain­bow. After New­ton, a Ger­man op­ti­cian named [Joseph von] Fraunhofer, who lived at the be­gin­ning of the 19th cen­tury, stud­ied this. He de­com­posed sun­light through a prism and then looked at it al­most at the level of a mi­cro­scope. What he found is that, in the beau­ti­ful rain­bow, there are some dark lines. In fact, one dark line had been dis­covered be­fore Fraunhofer, the line of so­di­um. But he dis­covered that there are in fact more than 500 dark lines. They are called ab­sorp­tion lines.

When light goes through the at­mo­sphere of the sun, the chem­ic­als in the at­mo­sphere make trans­itions by ab­sorb­ing photons at cer­tain wavelengths. The cor­res­pond­ing wavelengths of light do not reach the earth, and thus you have the dark ab­sorp­tion lines. Around 1860, [Gust­av] Kirch­hoff and [Robert] Bun­sen dis­covered that the dark lines of ab­sorp­tion spec­tra co­in­cide with bright lines of emis­sion spec­tra, which ap­pear when ele­ments are heated.

In math­em­at­ics, it is very easy to de­scribe an emis­sion spec­trum and much harder to de­scribe an ab­sorp­tion spec­trum. My idea was that the spec­trum giv­ing the zer­os of the Riemann zeta func­tion was an ab­sorp­tion spec­trum, which ex­plained the minus sign: The ab­sorp­tion is like tak­ing the neg­at­ive of a pic­ture, and it is this neg­at­ive that gives you the minus sign. So I did a cal­cu­la­tion at that time, in 1996, and it gave me the right spec­trum. I was quite ex­cited and wrote a Comptes Ren­dus note15 ex­plain­ing this.

Caterina Consani and Alain Connes at Connes’ country house.
Photo by Danye Chéreau.

At that time I had the hope that this would give some in­sight on the zer­os of zeta. But it was an ab­sorp­tion spec­trum, so it is much more dif­fi­cult to handle than an emis­sion spec­trum. That star­ted a long story, which has con­tin­ued up to now, in work with my col­lab­or­at­or Ca­ter­ina Con­sani. The ideas re­main very power­ful. In a very re­cent pa­per16 that we put on the arX­iv, these ideas al­lowed us to make more pro­gress. The out­come is that the space of primes could be seen com­ing out of a non­com­mut­at­ive space that was ex­tremely nat­ur­al.

Jack­son: Why are the ab­sorp­tion lines so much more dif­fi­cult to deal with than the emis­sion lines?

Connes: When you have an ab­sorp­tion line, it’s a single line, so nor­mally it would not be seen. Un­less the line had some thick­ness, some width, you would not see it. Math­em­at­ic­ally speak­ing, the zer­os of zeta don’t have a thick­ness. You have to ar­ti­fi­cially make them a little bit thick in or­der to be able to see them. And that’s very hard. It is ex­actly this tech­nic­al point that is now be­ing treated much bet­ter but that took enorm­ous time to un­der­stand.

This is a de­vel­op­ment on which I have been work­ing, par­al­lel to the phys­ics de­vel­op­ment of the Stand­ard Mod­el, since the 1990s. These two de­vel­op­ments for me are cru­cial. If non­com­mut­at­ive geo­metry were just deal­ing with very strange spaces, I don’t think it would be very con­vin­cing to people. But for two fun­da­ment­al spaces — for space­time, the space where we live, and for the space of primes — it can bring something new and em­body the in­tu­ition be­hind them.

Understanding renormalization

Connes: An­oth­er de­vel­op­ment also played a key role. Since the 1970s, I have been fas­cin­ated by a tech­nique in phys­ics called renor­mal­iz­a­tion.

In the 1930s, when Dir­ac cre­ated quantum field the­ory, he quant­ized the elec­tro­mag­net­ic field in a way that is truly mind-blow­ing. Ein­stein had this in­cred­ible in­tu­ition that some­how fre­quen­cies had to be quant­ized, so their en­ergy was not ar­bit­rary but had to be in­teg­ral mul­tiples of \( h\nu \). This was like an An­satz, a pre­scrip­tion. Dir­ac was able to make this a math­em­at­ic­al fact. His idea was to use ex­actly the non­com­mut­ativ­ity to force a cer­tain quant­ity to be an in­teger.

Once Dir­ac had done that, he tried to ap­ply the same tech­nique in or­der to deal with quantum fields of more com­plic­ated sys­tems. But then noth­ing worked. The ex­pres­sions he wrote down didn’t make any sense. In the late 1940s, more pre­cise meas­ure­ments of what is known as the Lamb shift were ob­tained. Ef­forts to ex­plain the meas­ure­ments from the phys­ics failed be­cause the quant­it­ies that should have ex­plained it were mean­ing­less be­cause they were giv­en by di­ver­gent in­teg­rals.

It was a ter­rible time for phys­ics. Then brave people like [Sin-Itiro] Tomon­aga and [Ju­li­an] Schwing­er and [Richard] Feyn­man came to the fore. They ma­nip­u­lated the in­fin­it­ies in or­der to ex­tract fi­nite quant­it­ies and com­pared those quant­it­ies to meas­ure­ments. For what is called the an­om­al­ous mo­ment of the elec­tron, the pre­ci­sion of the agree­ment was the pre­ci­sion of the width of a hair in pro­por­tion to the dis­tance from Par­is to New York. Nobody could deny that they had stumbled onto something great. On the oth­er hand, if you put a math­em­atician to look at what they were do­ing, you would hear the math­em­atician scream­ing!

Jack­son: Be­cause of the way the phys­i­cists were ma­nip­u­lat­ing the in­fin­it­ies?

Connes: Right. They were ma­nip­u­lat­ing the in­fin­it­ies in a way that was totally un-un­der­stand­able. What they were do­ing is called renor­mal­iz­a­tion. Start­ing in the 1970s, I be­came fas­cin­ated with this.

Alain Connes (left), Henri Moscovici (center) and Dirk Kreimer (right) during a talk at the IHES.
Photo by Jean-François Dars.

At the end of the 1990s, I was work­ing with my col­lab­or­at­or Henri Mo­scov­ici. We were work­ing on the cyc­lic co­homo­logy ad­ap­ted to a cer­tain Hopf al­gebra. There was a vis­it­or in IHES, Dirk Kreimer, who is really a hard-core phys­i­cist. He had a won­der­ful new idea that, when one is ma­nip­u­lat­ing Feyn­man graphs, there is be­hind the scenes something like a Hopf al­gebra. We star­ted work­ing to­geth­er with Dirk; that was in 1998. We put to­geth­er all the math­em­at­ics that was re­quired be­hind this Hopf al­gebra.

Then there was a mo­ment of rev­el­a­tion that came to me in Septem­ber of 2000. Nor­mally if you make a dis­cov­ery, you are up for one hour, and then you come back to earth. But then I was up for a week. The dis­cov­ery was that this in­cred­ibly com­plic­ated tech­nique that phys­i­cists are us­ing for renor­mal­iz­a­tion is in fact a well known tech­nique in math­em­at­ics called the Birk­hoff de­com­pos­i­tion. It was pi­on­eered by [G. D.] Birk­hoff, and Grothen­dieck also used it to prove a very im­port­ant the­or­em about vec­tor bundles on the sphere.

After that, when I col­lab­or­ated with Mat­ilde Mar­colli, we found that be­hind renor­mal­iz­a­tion was not only the Birk­hoff de­com­pos­i­tion, but also an even more fun­da­ment­al prob­lem, the Riemann–Hil­bert prob­lem. We worked for sev­er­al years on that. But in a sense, I had stopped think­ing about renor­mal­iz­a­tion, be­cause in my mind it’s re­solved.

Jack­son: The math­em­aticians are no longer scream­ing?

Connes: Right, now they un­der­stand. But it takes a lot of time be­cause math­em­aticians don’t know what renor­mal­iz­a­tion is, and phys­i­cists don’t know what the Riemann–Hil­bert prob­lem is!

Overcoming prejudice against algebraic geometry

Connes: I would like to con­tin­ue the story of the num­ber the­ory side, be­cause the phys­ics and the num­ber the­ory were con­stantly in­ter­twined in my mind.

Be­fore I met Ka­tia Con­sani and star­ted a very long col­lab­or­a­tion with her, I had a lot of pre­ju­dice against al­geb­ra­ic geo­metry. I was stu­pid, be­cause when you have pre­ju­dice against something, very of­ten it’s just ig­nor­ance. I was ig­nor­ant. When Ka­tia Con­sani and I star­ted work­ing to­geth­er, I learned about con­cepts in­ven­ted by Grothen­dieck, like the concept of scheme.

I talked be­fore about the non­com­mut­at­ive space I dis­covered that em­bod­ies the prime num­bers. If you looked at it as a non­com­mut­at­ive space, you could feel that fun­da­ment­al un­der­stand­ing of this space was miss­ing, in par­tic­u­lar its re­la­tion to al­geb­ra­ic geo­metry and to oth­er fun­da­ment­al points of view in math­em­at­ics.

In 2014, we dis­covered with Ka­tia Con­sani that there is a topos — in the sense of Grothen­dieck — that is the topos of arith­met­ic. This topos im­me­di­ately gave rise to the same space as the non­com­mut­at­ive space that I had found in 1996. So this meant that this space, rather than be­ing ar­bit­rary or be­ing con­struc­ted for the pur­pose of do­ing something, was in fact an ab­so­lutely fun­da­ment­al space. We wrote a Comptes Ren­dus note17 about this.

Let me ex­plain the no­tion of topos, be­cause it is equally as im­port­ant as non­com­mut­at­ive geo­metry and very much con­nec­ted with it. Around 1958, Grothen­dieck dis­covered a new no­tion of geo­met­ric space, which he called topos. Nor­mally when you do geo­metry, you put onto the stage the space you are study­ing. The main act­or is the space, and you talk about the points of the space, you talk about its to­po­logy, its dif­fer­en­tial geo­metry, and so on. The idea of Grothen­dieck is that there is an­oth­er way to com­pre­hend the space. This oth­er way is not to put it onto the stage but to hide it be­hind the stage, in what we call in French la cou­lisse. I am not sure of the term in Eng­lish.

Jack­son: In the con­text of the theat­er, that would be “back­stage.”

Connes: Yes, like in a theat­er, “back­stage.” The space in ques­tion will be in the back­stage. You will nev­er see the space. Its role will be that, while you are do­ing math­em­at­ics with the or­din­ary char­ac­ters — the in­tegers, the real num­bers, the spaces you are used to work­ing with — all of these things will ac­tu­ally de­pend on a para­met­er that is in the back­stage. What is in the back­stage will gov­ern a ran­dom­ness that will be in­her­ited by the usu­al char­ac­ters of math­em­at­ics that you are work­ing with.

This is a fant­ast­ic idea. Tech­nic­ally speak­ing, what does it mean? In­stead of look­ing at the space, you look at what are called the sheaves of sets over the space. Then when you do set the­ory, you can do any math­em­at­ics you want, you can look at sheaves of groups or sheaves of to­po­lo­gic­al spaces. You can re­cov­er the space and its to­po­logy, which is in the back­stage, just by look­ing at set the­ory “with para­met­er”. What is a point of the space? A point of the space is a way to sup­press the ran­dom­ness in the events that are oc­cur­ring on the stage. Then when you use a point to look at what is go­ing on on the stage, it is as if it is no longer ran­dom.

What I find in­cred­ibly re­veal­ing, is that when you com­pute the points of a topos, even of a very simple topos, you get in gen­er­al a non­com­mut­at­ive space! This is what makes the con­nec­tion between the point of view of Grothen­dieck, of topos, and the point of view of non­com­mut­at­ive geo­metry. They are deeply in­ter­con­nec­ted.

Jack­son: You nev­er got to dis­cuss this with Grothen­dieck.

Connes: No, un­for­tu­nately. By the early 1990s he had dis­ap­peared to a place in the Pyrénées. Even his fam­ily didn’t know where he was. He stayed there un­til his death in 2014. It would have been very dif­fi­cult to dis­cuss with him. He be­came a mys­tic over the years. Dur­ing the time when he was se­cluded in the Pyrénées, he wrote an enorm­ous amount, tens of thou­sands of pages. The main top­ic was the prob­lem of evil. I have read many of his writ­ings, in­clud­ing an un­pub­lished text called La Clef des Songes [The Key to Dreams], in which he tells the story of his fath­er.

I nev­er met Grothen­dieck, but I think that I know him so well, from his writ­ings. In sev­er­al of them, he com­plains that people don’t un­der­stand what a topos is. It shows how com­mon it is that math­em­aticians say “This is not math­em­at­ics”, or “This is not ser­i­ous”, just be­cause they don’t un­der­stand. He suffered a lot from that. The concept of topos is an amaz­ing dis­cov­ery that gives a com­pletely new way of think­ing about math­em­at­ics. But un­less you do an ep­si­lon of pro­gress in some def­in­ite top­ic that already ex­ists and that is well paved, people don’t pay at­ten­tion.

I am not ex­empt of pre­ju­dices my­self, so I un­der­stand per­fectly why people would have these pre­ju­dices.

Jack­son: You had a pre­ju­dice about the idea of topos, and about al­geb­ra­ic geo­metry.

Connes: Yes. Be­fore I un­der­stood what a topos was, I would say, “This is bull­shit!” You really un­der­stand something only when you use it on some oth­er pur­pose, and that de­pends on the oc­ca­sion you get. In the work with Ka­tia Con­sani, we fi­nally un­der­stood that there is a topos, and it is ex­tremely nat­ur­al. This opened up a point of view that is totally dif­fer­ent.

Sir Michael Atiyah and Alain Connes at the Michael and Lily Atiyah Portrait Gallery of Mathematicians at the University of Edinburgh in 2015.
Photo courtesy of Alain Connes.

Jack­son: You made a link to new things, and you just jumped in­to them. You have done that sev­er­al times in your life. What al­lows you to do that? Is that con­fid­ence?

Connes: No, no, no. I am not really mo­tiv­ated by con­fid­ence. Nor by curi­os­ity. What I would say is, it’s more anxi­ety. I spend much more time be­ing anxious than be­ing con­fid­ent or be­ing curi­ous. My mind sort of con­stantly wor­ries. It’s not con­fid­ence — okay, I have of course some self-con­fid­ence, but it’s not a kind of over­reach­ing con­fid­ence, by no means. I knew only one per­son who had over­reach­ing con­fid­ence, that was Mi­chael Atiyah. I really liked him a lot. He could jump to oth­er top­ics. But I am not like him. I am much more mo­tiv­ated by the fact that when I do not un­der­stand something, it makes me suf­fer. It puts me in­to a state of misery. I am feel­ing bad un­til I un­der­stand. That’s ex­actly the mo­tiv­at­ing force.

This is also why I like very much to col­lab­or­ate be­cause then you share this un­eas­i­ness. You are not alone! And I love to col­lab­or­ate with people who are more con­fid­ent than I am, ex­actly be­cause of my prob­lem.

Grothen­dieck wrote something in Récoltes et Se­mailles that I like to quote. He said that to fear the er­ror is the same as to fear the truth. But if one is ready to con­front the er­ror, then this fear be­comes a bless­ing. One crosses this dif­fi­cult time and comes out with much more.

A grand unified theory from gravity

Jack­son: You talked be­fore about look­ing at space­time from the non­com­mut­at­ive point of view, which gives the Stand­ard Mod­el coupled with grav­ity. That sounds like a “grand uni­fied the­ory” of the fun­da­ment­al forces. Is this what you are say­ing?

Connes: Let me ex­plain that in some de­tail. In the mid-1980s, I had real­ized that you could get the Higgs sec­tor of the Stand­ard Mod­el from the geo­met­ric pic­ture. But I did not have something that would uni­fy grav­ity with the oth­er forces; that only came in 1996, when I began to work with Ali Chamsed­dine. We real­ized that if one takes a spec­tral point of view of geo­metry, then there is a nat­ur­al man­ner of de­fin­ing what is called in phys­ics an “ac­tion”, for that geo­metry. This will meas­ure how suit­able the geo­metry is. This ac­tion turned out to be spec­tral and de­pends only upon the line ele­ment, upon its spec­trum.

To ex­plain this I have to make a di­gres­sion, but first, to an­swer your ques­tion: It is a uni­fic­a­tion. You con­sider pure grav­ity on a geo­met­ric space, and when you com­pute what you get from pure grav­ity, you not only get the or­din­ary grav­it­a­tion field, but you also get the bo­son­ic fields of the Stand­ard Mod­el and the fer­mi­on­ic fields. So you do get the full pic­ture out of pure grav­ity. Some people have tried for in­stance to ob­tain grav­ity from gauge fields, but what I am say­ing is quite dif­fer­ent. What I am say­ing is that once you in­tro­duce some fine struc­ture in the geo­metry of space­time, then pure grav­ity will give you not only the or­din­ary grav­it­a­tion­al force, but also the oth­er forces of nature, which are the elec­troweak and the strong force. So it’s not a uni­fic­a­tion from gauge fields, but it’s a uni­fic­a­tion from grav­ity.

Now let me enter the di­gres­sion. Why is it nat­ur­al to view a space spec­trally and to define the ac­tion from a spec­tral in­vari­ant? This goes back to the story of the meas­ure­ment of length. It starts some­time be­fore the French Re­volu­tion. At that time France had no uni­fic­a­tion of the meter of length. If you were for in­stance in the trade of lin­en, you would need a unit of length to meas­ure pieces of lin­en to sell. So every town or vil­lage dis­played at its en­trance a unit of length. But the units were of­ten dif­fer­ent, which made things ex­tremely com­plic­ated. People star­ted to say they needed a way to uni­fy the meas­ure­ment of length. Sci­ent­ists in France, and also in Eng­land, thought a lot about this. They de­cided that the best idea would be to take the largest avail­able ob­ject, which is the earth, and then define the unit of length as some pro­por­tion of the cir­cum­fer­ence of the earth. They de­cided one meter would be one-forty mil­lionth of the cir­cum­fer­ence of the earth.

To meas­ure the cir­cum­fer­ence, you meas­ure an angle that can be defined by look­ing at stars and then con­cretely meas­ure the dis­tance between the two points that lim­it the angle. The angle chosen was between Dunkerque in the north of France and Bar­celona in the north of Spain, which are more or less on the same lon­git­ude. In 1792 two French as­tro­nomers were put in charge of meas­ur­ing this dis­tance. They did tri­an­gu­la­tions, which means that they would put a tele­scope on top of a hill and make some meas­ure­ments. Of course when they did that in Spain, which was in a war with France, it was quite dif­fi­cult to ex­plain that they were not spies! There are a lot of in­ter­est­ing stor­ies about what happened to them.

They even­tu­ally ob­tained a reas­on­ably pre­cise meas­ure­ment. The meas­ure­ment was used to cast a plat­in­um bar to rep­res­ent one meter, and the bar was de­pos­ited in the Pa­vil­lon de Breteuil, which is near Par­is. This was con­sidered the uni­ver­sal unit of length in the met­ric sys­tem. Of course, this was not very prac­tic­al, be­cause if you are in a for­eign coun­try and you want to meas­ure your bed, you have to travel to the Pa­vil­lon de Breteuil to know how long a meter is! So rep­licas were made and dis­trib­uted.

That was fine un­til the be­gin­ning of the 20th cen­tury. By 1925, people had bet­ter ways to meas­ure length us­ing spec­tro­scopy, by com­par­ing a giv­en length with a wavelength of a known atom­ic trans­ition. They then real­ized that the fam­ous unit of length de­pos­ited near Par­is didn’t have a con­stant length.

Jack­son: Be­cause it was made of met­al?

Connes: Ex­actly, be­cause met­al con­tracts and ex­pands. They de­cided that the ap­par­at­us that al­lowed people to see that the length of the plat­in­um bar was chan­ging was a bet­ter device to define the unit of length than the bar it­self. For some time they used krypton, which was not very sat­is­fact­ory, be­cause krypton is very rare. Even­tu­ally they switched to cesi­um. Today one can buy in a store a cheap in­stru­ment based on the cesi­um trans­ition that will give you meas­ure­ments of length with a pre­ci­sion of 10 decim­als.

When I defined non­com­mut­at­ive geo­metry, I defined it as a spec­tral triple. The shift from the clas­sic­al stand­point, which was the Riemann stand­point on geo­metry, to the new stand­point that I had defined, which is spec­tral, is ex­actly par­al­lel to the shift that oc­curred in phys­ics between the defin­i­tion of the unit of length by means of the plat­in­um bar, and the defin­i­tion by means of com­par­is­on with wavelengths of a fixed chem­ic­al. It’s very strik­ing.

The ac­tion prin­ciple, which we defined with Chamsed­dine, then al­lows you not only to re­cov­er grav­ity but to find grav­ity coupled with mat­ter. This ac­tion is simply meas­ur­ing the spec­trum of the line ele­ment. The for­mula for dis­tance in non­com­mut­at­ive geo­metry will use the fact that the line ele­ment does not com­mute with the co­ordin­ates in the space.

On the con­cep­tu­al level, what this means is that a paradigm of non­com­mut­at­ive geo­metry, that of a spec­tral triple, in fact is very closely re­lated to phys­ics. It has a big ad­vant­age in the quest to uni­fy grav­ity with oth­er forces, which is that it is both quantum and geo­met­ric. After the quantum was dis­covered by Heis­en­berg, von Neu­mann un­der­stood that the cor­rect stage on which to de­vel­op quantum mech­an­ics was Hil­bert space. So the geo­metry I am talk­ing about is the geo­metry that is on that stage, in Hil­bert space. Hence it’s con­geni­al to quantum from the start.

The influence of quantum entanglement

Jack­son: Some com­plex­ity the­or­ists re­cently proved a res­ult18 in­volving quantum com­put­ing and en­tan­gle­ment, and they thereby re­solved something called Tsirelson’s prob­lem. This in turn re­solved the Connes Em­bed­ding Con­jec­ture. Can you tell me what this con­jec­ture is, and how you see this new work?

Connes: First of all, it’s not a con­jec­ture, it’s a prob­lem. When I was work­ing in King­ston in 1975, at some point I stumbled on a cer­tain prop­erty of a factor. I saw im­me­di­ately that this prop­erty was less re­strict­ive than be­ing hy­per­fin­ite. I was prov­ing something about hy­per­fin­ite factors and had found ex­amples of factors that were not hy­per­fin­ite, but which did ful­fill this prop­erty. The prop­erty is that the factor is not hy­per­fin­ite but re­sembles a hy­per­fin­ite factor as much as pos­sible. Tech­nic­ally, it means that the factor can be em­bed­ded in an ul­traproduct of hy­per­fin­ite factors. The ques­tion is wheth­er every factor of what is called Type II\( _1 \) has this prop­erty. What I had no­ticed at the time was that all the books and pa­pers I knew about were veri­fy­ing this prop­erty for the as­so­ci­ated factor. So I wrote about this in three lines in my pa­per. And — this is the hon­est truth! — I nev­er thought more about it.

Then these few lines that I had writ­ten were picked up by a num­ber of dif­fer­ent people. [Eber­hard] Kirch­berg proved that this prob­lem is equi­val­ent to something he was work­ing on. It was used by [Dan Vir­gil] Voicules­cu in de­fin­ing his new no­tion of en­tropy and by oth­er people in quantum the­ory like [Bor­is S.] Tsirelson.

I don’t know how much the com­plex­ity the­ory pa­per has been checked. Ap­par­ently it’s quite long. What it is say­ing, roughly, is that there are things that can­not be ap­prox­im­ated at all by something that is fi­nite-di­men­sion­al. It’s something quite weird. It would prob­ably be quite sig­ni­fic­ant if they have really found an ex­ample of this. Wheth­er the ex­ample would have rel­ev­ance for phys­ics, I have no idea. I al­ways had the feel­ing, or the be­lief, that nature is really fi­nite-di­men­sion­al in a sense; even though we are ap­prox­im­at­ing it by something con­tinu­ous, everything is es­sen­tially fi­nite-di­men­sion­al.

As I said, I nev­er thought my­self about the prob­lem. I am really the worst per­son to ask!

Jack­son: The com­plex­ity the­ory res­ult has to do with quantum en­tan­gle­ment. Does this in­terest you?

Connes: En­tan­gle­ment is something I find ex­tremely in­ter­est­ing and im­port­ant, but for a dif­fer­ent reas­on. The book we wrote with my wife and Dixmi­er con­tains a pro­voc­at­ive phrase. In French, it’s “l’aléa de quantique est le tic-tac de l’hor­loge di­vine.” In Eng­lish this could be “the vag­ary of the quantum is the tick-tock of the di­vine clock.”

The reas­on time is passing, and passing in a way that we don’t con­trol at all, is pre­cisely the lack of re­pro­du­cib­il­ity of the quantum. When you send a photon to a tar­get, you will nev­er be able to re­pro­duce the res­ult. It’s something totally ran­dom and un­con­trol­lable. I was able, to some ex­tent, to de­vel­op a the­ory that would have time spring­ing from this quantum ran­dom­ness, as I ex­plained earli­er. But if you have a time at point A and a time at point B, and quantum ran­dom­ness at A and quantum ran­dom­ness at B, there would be no link at all. Right? No, that’s not right, be­cause if at A and B you are meas­ur­ing the quantum ran­dom­ness from en­tangled states, then you will get res­ults that are cor­rel­ated. The quantum ran­dom­ness is cor­rel­ated by quantum en­tan­gle­ment.

One would need a mind like Ein­stein’s to in­vent a no­tion of time that would spring from the quantum and that would make us at peace with en­tan­gle­ment and tell us that en­tan­gle­ment is just the har­mony or the cor­rel­a­tion between the vari­ous ran­dom­ness at vari­ous points. I be­lieve the per­son who might be closest to that is [Ant­on] Zeilinger, a Swiss phys­i­cist who has done ex­per­i­ments on en­tan­gle­ment for very dis­tant points, points that are more than 100 kilo­met­ers apart.19 I heard a talk by him in which he said that they are look­ing for things that can­not be en­tangled in their ex­per­i­ments. The math­em­at­ics of the time evol­u­tion, plus un­der­stand­ing of en­tan­gle­ment — there is enough stuff there to cre­ate a com­pletely new point of view on time.

What is ex­tremely troub­ling about en­tan­gle­ment is that if you have two en­tangled states and you make an ob­ser­va­tion on one, it im­me­di­ately acts on the oth­er and gives you a cor­rel­ated res­ult. Ein­stein was up­set about this and called it “spooky ac­tion at a dis­tance.” Spooky is the right word. Alain As­pect has made meas­ure­ments show­ing that the ef­fect is much faster than the speed of light. This seems to con­tra­dict the prin­ciple of re­lativ­ity. But in fact it does not, be­cause you can­not trans­mit the in­form­a­tion. Sup­pose you have two cor­rel­ated states, so that if you find a plus in one, the oth­er will be minus, or if you find minus, then oth­er will be plus. The an­swer, plus or minus, is not my own choice; it is a res­ult of the ex­per­i­ment. The oth­er guy had minus or plus, but he will get no in­form­a­tion from me. It is not a way to trans­mit in­form­a­tion, so it doesn’t con­tra­dict the prin­ciple of re­lativ­ity.

Still, it’s trouble­some. If I am in a dif­fer­ent frame, it will not be that something was first done at point A and then point B re­acted; it will be that it was done first at point B and then A re­acted. When you have space-like events seen from dif­fer­ent frames of ref­er­ence, one can be be­fore and the oth­er can be after, or vice versa. It’s your own choice. This means that the no­tion of caus­al­ity, or the no­tion of time, is totally up­set by the phe­nomen­on of en­tan­gle­ment. I in­ter­pret it as mean­ing that there is something more prim­it­ive than the passing of time, which is quantum ran­dom­ness.

The gift of the Riemann Hypothesis

Jack­son: How do you see the pro­spects for prov­ing the Riemann Hy­po­thes­is?

Connes: I worked a lot on that and made some pro­gress re­cently with Ka­tia Con­sani. But un­til you are done, you can­not say any­thing.

But there is a men­tal pic­ture of the prob­lem that is com­fort­ing. The prob­lem is like an in­fin­ite pole, and you want to show the pole is ver­tic­al. This is the men­tal pic­ture. With Ka­tia, it is as if we are build­ing found­a­tions that are tight­er and tight­er and tight­er. There is an in­fin­ite set of stairs, but each step you take on the stairs is ac­tu­ally block­ing the pole, to make it more and more ver­tic­al. What we have done re­cently is to cross the first step of the stair.

The stairs are in­fin­ite. But the beauty is that, be­cause of an idea of An­dré Weil, you only have to con­sider fi­nitely many primes at a time to solve the prob­lem. If you stick to that idea, then you are sure not to fall in­to the black hole of prov­ing res­ults that are equi­val­ent to the Riemann Hy­po­thes­is. What I found in 1996 was something that is dif­fi­cult when you ap­ply it to the in­finitude of primes, but that, when you ap­ply it to only fi­nitely many primes, gives you ex­actly the Hil­bert space and the quant­ized cal­cu­lus frame­work that is ap­par­ently very suit­able to at­tack the prob­lem.

One can be hope­ful, but un­til you are done, for­get it — you can­not say any­thing. I like to work on this prob­lem be­cause it is a test of my­self that I can­not es­cape from. It’s not like you build a new the­ory, and then you can think you are the greatest. In math­em­at­ics, there is no bet­ter way to pro­gress than to be con­fron­ted with a prob­lem that you can­not solve. If you work on a prob­lem that you can solve, it means that it’s not the right prob­lem. Fight­ing with a very hard prob­lem is a much bet­ter way to build a men­tal pic­ture than when you are work­ing on an easy prob­lem. When the mind is blocked, it has much more power to build and to con­ceive. I see a prob­lem like that as a gift.

With my wife and my teach­er, Jacques Dixmi­er, we wrote a second nov­el called Le Spectre d’Atacama.20 It re­counts the story of a math­em­atician who is con­fron­ted with the Riemann Hy­po­thes­is. He real­izes that be­ings from out­er space are send­ing spec­tra to the earth and these spec­tra con­tain the zer­os of the Riemann zeta func­tion. I re­cently pub­lished in the Journ­al of Math­em­at­ics and Mu­sic a pa­per21 that came from an is­sue we were ex­posed to while writ­ing the book.

The math­em­atician and the oth­er sci­ent­ists had to be sure that the be­ing com­mu­nic­at­ing with them from out­er space was an in­tel­li­gent be­ing and not a ma­chine. So they had to de­vise a Tur­ing Test that would make them com­pletely sure. Such a test is pos­sible, and it is re­lated to An­dré Weil’s work on the Riemann Hy­po­thes­is. He dis­covered that when you work with func­tion fields, all zer­os of the ana­logue of zeta are on the crit­ic­al line, and they are peri­od­ic — they re­peat peri­od­ic­ally.

What we real­ized when we were writ­ing the book is the link with what the com­poser [Olivi­er] Mes­si­aen had in­ven­ted with his rythme non-ret­ro­grad­able. The rhythmic pat­terns of Mes­saien have ex­actly the same prop­erty as the peri­od­ic pat­terns that you find from the zer­os of Weil’s ana­logue of the Riemann zeta func­tion.

Weil’s rhythmic pat­terns are as­so­ci­ated to each prime. What the char­ac­ters in the book did was to send in­to out­er space the pat­terns as­so­ci­ated to the primes, but they would omit one prime. If the be­ings re­ceiv­ing the mes­sage were really in­tel­li­gent, they would an­swer by send­ing the pat­tern for the miss­ing prime.

Jack­son: Is your wife a writer?

Connes: My wife is re­tired now, but she was a teach­er of Lat­in and Greek in high school. She is a lit­er­ary per­son. We are very com­ple­ment­ary. She knows so much that I don’t know.

Jack­son: Dixmi­er wrote sci­ence fic­tion when he was young­er, right?

Connes: Yes, he wrote sci­ence fic­tion and also po­lice stor­ies. The way our first nov­el star­ted is that in the sum­mer of 2012 Dixmi­er sent us a post­card. The post­card said, “I have the title of the book. You write it, and I will proofread it!” Of course we laughed.

Then my wife and I made a trip to Venice and were vis­it­ing a small mu­seum with an ex­hib­it of some very strik­ing mod­ern art, in­clud­ing a sculp­ture by [Maur­iz­io] Cat­telan. The sculp­ture showed nine dead bod­ies, full size, aligned next to each oth­er. I saw the sculp­ture, and just then a prize had been giv­en to nine people!22 This im­me­di­ately kicked something in my mind, and I star­ted writ­ing a draft of the book that Dixmi­er had sug­ges­ted.

When my wife saw the draft she said, “No, this is aw­ful!” Then she took over. She man­aged to sal­vage the idea, though what she wrote was totally dif­fer­ent. She star­ted writ­ing about a young wo­man phys­i­cist who is asked to be­come the head of CERN. So that was the start­ing point of the book. Amaz­ingly, a few years after the book ap­peared, an Itali­an wo­man, Fa­biola Gi­an­otti, was nom­in­ated as the head of CERN!

I al­ways like these mo­tiv­a­tions that are very odd, very strange, like a kick, to start something.

Jack­son: You seem to have a really fun life.

Connes: Sure. I try at least! But as I said, it’s not so much fun be­cause of the per­sist­ent anxi­ety.

Jack­son: But as you said in con­nec­tion with Grothen­dieck, the anxi­ety pushes you to­wards the truth.

Connes: Ex­actly.