# Celebratio Mathematica

## Alain Connes

### Interview with Alain Connes

#### by Allyn Jackson

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.