# Celebratio Mathematica

## Lou van den Dries

### Interview with Lou van den Dries

#### by Rob Kirby

The fol­low­ing is a tran­script of a re­cor­ded in­ter­view that took place on 8 Decem­ber, 2019. Tom Scan­lon, Rob Kirby and Cel­eb­ra­tio Math­em­at­ica’s Man­aging Ed­it­or, Sheila New­bery, spoke with Lou van den Dries at the Shat­tuck Hotel in Berke­ley, Cali­for­nia on the eve of the Ju­lia Robin­son Centen­ni­al Sym­posi­um held at MSRI. Van den Dries was as an in­vited speak­er at the Sym­posi­um, which was or­gan­ized by Hélène Bar­celo, Tom Scan­lon and Car­ol Wood.

Lou: Well, my first name is really Lauren­ti­us. Nor­mally it’s Lou, writ­ten as L-o-u, which is a bit strange.

Rob: Then you have some more names…

Lou: Yeah, OK. That’s true. Lauren­ti­us Pet­rus Dignus van den Dries.

Rob: So, is there a story be­hind that name?

Lou: Not really. Van den Dries is ori­gin­ally a Flem­ish name.

Rob: It sounds very Lat­in. Very Ro­man.

Lou: By fam­ily tra­di­tion and as the second of six I was named after my grand­fath­er on my moth­er’s side (Laurens), my fath­er (Pieter), and my old­est uncle on my moth­er’s side (Din­genis). The Lat­in­iz­a­tion is a Ro­man Cath­ol­ic habit.

Rob: So, what’s your birth date?

Lou: May 26, 1951.

Rob: And where were you born?

Lou: I was born some­where in the Neth­er­lands, not in a city, on a farm, in a cer­tain area that is be­low sea level, which is not un­com­mon in the Neth­er­lands, but not near any town.

Rob: OK. North? South? East? West? Just roughly…

Lou: The North­east­ern quarter of the Neth­er­lands.

Rob: What sort of farm? A dairy farm or…

Lou: No, a farm where you grow products like pota­toes, wheat, sug­ar beets, flax, and peas.

Rob: Your fath­er was a farm­er. Was his fath­er a farm­er?

Lou: As far as I know, as far as could trace it back, they all have been farm­ers.

Rob: On the same farm? Does that go way back?

Lou: Well, my par­ents are from Zee­land, in the south­w­est of the Neth­er­lands, and grew up on farms there which had ex­is­ted for cen­tur­ies. The farm I grew up on was a new one, dat­ing from 1949, in an en­tirely dif­fer­ent part of the coun­try.

Lou: Also. Same kind of back­ground. That fam­ily did own a farm. It’s very much a farm­ing fam­ily from both sides.

Rob: Do you have broth­ers or sis­ters?

Lou: Yes, we were six.

Rob: Same as Len­stra.

Lou: Oh!

Rob: Len­stra, I be­lieve, has five sib­lings, also.

Lou: Oh I see. Huh.

Rob: So, which one were you?

Lou: Second. I was one year after the first born.

Rob: How many boys? Girls?

Lou: Two girls. Four boys.

Rob: And the girls were which…?

Lou: OK, the third one, that is my old­est sis­ter, and then the very young­est one, the sixth one, was also a girl.

Rob: OK, is there any­thing in­ter­est­ing about them? Did they have un­usu­al ca­reers — or something of in­terest that you want to men­tion?

Lou: Well, let’s see, my old­est broth­er was ba­sic­ally the first one in my fam­ily who had high­er edu­ca­tion. He be­came an ag­ri­cul­tur­al en­gin­eer. He worked a lot in Africa. In fact he has a de­gree in trop­ic­al en­gin­eer­ing. And he still goes there a lot. And oth­er trop­ic­al areas in the world.

My young­est broth­er, the fifth one, he took over the farm. He’s the farm­er. And that’s also where I go back, still, most years.

Rob: Al­right. Your school­ing — did you just go to a reg­u­lar…?

Lou: Yeah. So the nearest vil­lage, Ens, was about five kilo­met­ers from our farm. We biked every day to school and back. That was just a primary school — a six-year primary school. And then after that, I went to the loc­al lycée, well ly­ceum, a sort of a high school in Em­meloord. At that time, that was a school that if you fin­ished it with suc­cess, you could go to the uni­versity. So, that was about ten kilo­met­ers away. Also, I went there usu­ally by bike — and back.

Rob: And so then, where did you go to col­lege?

Lou: Utrecht. And that is in the middle of the coun­try, the middle of the Neth­er­lands.

Rob: So, did you go in 1969? When you were eight­een? Or earli­er?

Lou: Yeah, ex­actly. I went there in 1969. In high school, or ly­ceum, as we called it, I did be­come very in­ter­ested in math­em­at­ics. And in fact one of the books that got me hooked was this little book called From Zero to In­fin­ity by Con­stance Re­id [e14], which of course I only found out later was the sis­ter of Ju­lia Robin­son, when I was already here in the US.

Rob: Did you ever meet her? Con­stance Re­id?

Lou: No, but I met Ju­lia Robin­son and Raphael Robin­son. I was an as­sist­ant pro­fess­or at Stan­ford for a few years, and so at that time I did meet some people in Berke­ley, in­clud­ing Ju­lia Robin­son and Raphael Robin­son.

Rob: I knew Con­stance Re­id a little bet­ter than I knew the Robin­sons. She was a lively lady!

Lou: Any­way, I cer­tainly re­mem­ber that book; it made math­em­at­ics look very in­ter­est­ing to me.

Rob: Did you have cal­cu­lus in ly­ceum?

Lou: Oh, yeah, it cer­tainly in­cluded some dif­fer­en­ti­ation and in­teg­ra­tion, yes. It didn’t go very far. We did learn some things.

Rob: So then, as an un­der­gradu­ate at Utrecht you were a math ma­jor?

Lou: Yes. In the be­gin­ning you had to choose — you couldn’t just be a math ma­jor. You had to choose so-called minors as well, and I do re­mem­ber I had phys­ics and as­tro­nomy as minors, but I was some­how able after a year or two to fo­cus on math.

Rob: When did you turn to lo­gic?

Lou: In the first two years or so I took a course in math­em­at­ic­al lo­gic, and I thought, “This is a sub­ject where you start from scratch and you can still get some non­trivi­al things!” And the book we used was Shoen­field’s [e2], I re­mem­ber. Ac­tu­ally, that book is quite a good book. I think now it would ac­tu­ally be con­sidered a little on the hard side for un­der­gradu­ates to learn it from that.

Rob: Were there any par­tic­u­larly pro­fess­ors that were im­port­ant to you?

Lou: Well, back in high school, I par­ti­cip­ated in this math­em­at­ic­al Olympi­ad, and I was second in the Neth­er­lands. When we went to the Hag­ue to get our prize, Freudenth­al was present, and prob­ably had chosen the books that we were giv­en. And this also in­flu­enced my choice of Uni­versity, since Freudenth­al was a pro­fess­or in Utrecht and I did in­deed take a course from him in the first two years, called “Struc­turen”, which was quite nice.

In col­lege — we call it simply uni­versity — one course that really did have an in­flu­ence on me was a course on sev­er­al com­plex vari­ables that I took from van der Put. He was a coau­thor, for ex­ample, with Mi­chael Sing­er. Sing­er–van der Put is a well known book on dif­fer­en­tial al­gebra [e12]. He gave a nice course which I took in my third year as an un­der­gradu­ate and I and some oth­er stu­dent [ Tom Vorst] were ap­poin­ted as note-takers. So we really had to learn it. Al­though some of it went over my head — like Dol­beau­lt co­homo­logy, I re­mem­ber. But I don’t re­mem­ber very much of it.

Rob: I think it went over my head, too.

Lou: Čech co­homo­logy, and so on. But what I did learn are things like Wei­er­strass pre­par­a­tion and that has al­ways served me very well since then.

Rob: Did you stay at Utrecht?

Lou: Yeah. It was fairly com­mon at that time to stay at the same uni­versity and con­tin­ue there.

Lou: Van Dalen. So he was from this Dutch tra­di­tion of in­tu­ition­ism. I cer­tainly had courses in that sub­ject but I de­cided for my­self, rather quickly, that this was not the way to do math. So I simply worked on my own. It was per­haps more com­mon then than it is now. Nowadays, gradu­ate stu­dents typ­ic­ally work in the ex­pert­ise of their ad­visor, but work­ing through Shoen­field, there was this chapter on mod­el the­ory and I thought, “Hey, this is the way to use lo­gic in math­em­at­ics.” In a way that is cre­at­ive. So, I got hooked on that. But ba­sic­ally it was self-study.

Of course the to­po­lo­gist Brouwer was also the fath­er of in­tu­ition­ism.

Rob: My re­ac­tion to it was that it was like do­ing math­em­at­ics with your hand tied be­hind your back.

Lou: Yeah. This was also sort of my re­ac­tion.

Rob: Maybe this is also why nobody does it!

Lou: Well, one could say that if you prove something in an in­tu­ition­ist­ic way, you of­ten get more in­form­a­tion out of it. But of course, clas­sic­al math­em­aticians also know that if you prove something in a con­struct­ive way, of­ten that simply means that you have proved more. Right? But I didn’t see why one had to have an in­tu­ition­ist­ic philo­sophy be­hind that, where one could simply be in­ter­ested in bet­ter res­ults or more con­struct­ive res­ults any­way.

Tom: Well, does this have an in­flu­ence on the way you do math­em­at­ics. I find that your work is much more con­struct­ive than oth­er people work­ing in our field.

Lou: Well, I doubt it. I some­how doubt that be­ing ex­posed to in­tu­ition­ism had a big in­flu­ence on me. I think I am able to reas­on in­tu­ition­ist­ic­ally if I want, but mod­el the­ory is also a way to ex­tract con­struct­ive in­form­a­tion that is not overtly there, but which, by mod­el the­ory, you some­how know how to ex­tract. So I def­in­itely am in­ter­ested in ex­tra in­form­a­tion that one can get this way. But I don’t think that came from be­ing ex­posed to in­tu­ition­ism.

Rob: So you really wrote your thes­is pretty much on your own.

Lou: Right, yeah, well, of course I did have con­tacts in my years as a gradu­ate stu­dent — I did get in touch with people like An­gus Macintyre and [Gregory] Cher­lin, whom I met at meet­ings in Bel­gi­um.

Rob: I was at UCLA in the 60s, and met Ab­ra­ham Robin­son. Is he…?

Lou: Yes, of course, his way of do­ing mod­el the­ory cer­tainly had a very big in­flu­ence on me, be­cause I found out later the ex­er­cises I had been do­ing in mod­el the­ory from that book of Shoen­field’s really came from his work. And then later on I read some of his things, like this nice little book on com­plete the­or­ies — that was an eye open­er for me.

Lou: Yes.

Tom: I ac­tu­ally looked at your thes­is [1] a few years ago. My stu­dent Will John­son was try­ing to look for ex­amples of mul­ti­val­ued fields, and es­pe­cially the sec­tion about mul­ti­val­ued fields and mul­tiple or­ders looks like it could have been taken al­most dir­ectly from Robin­son’s book.

Lou: Well, of course Robin­son didn’t con­sider mul­tiple-val­ued fields, but I mean it was cer­tainly the first time that someone con­sidered val­ued fields from a mod­el the­or­et­ic per­spect­ive. Yes. So I’m pretty sure I did read that book — I did read much of it when I was a gradu­ate stu­dent be­fore fin­ish­ing my thes­is. I do re­mem­ber think­ing of pos­sible thes­is top­ics. I had to, you know, in­vent some top­ic that I could work on, and at some point I de­cided, “OK, let’s look what hap­pens if you have fields with more than just one valu­ation.” And I figured out enough to say at some point to Macintyre when I met him, “I think I figured out that the the­ory of fields with more than one valu­ation has a mod­el com­pan­ion.” And I asked, “Do you think that would be in­ter­est­ing?” And he said, “Oh, yes, that would be very in­ter­est­ing to write out.” So I de­cided this will be part of my thes­is. Right.

Rob: Is Macintyre still alive?

Lou: Sure, oth­er­wise I’d have heard about it! He’s about ten years older than I am and I’m — sixty…

Rob: Sixty-eight.

[laughter]

Lou: Yeah. Nowadays I al­ways have to think. Am I sixty-sev­en or sixty-eight or sixty-nine…? It’s one of those three.

Rob: So where was your first — where did you go when you got your PhD? Did you go to Stan­ford?

Lou: No, I was a postdoc at Yale Uni­versity with Macintyre from 1979–1981.

Rob: I thought Macintyre was in Eng­land. But he moved or something…?

Lou: Be­fore I was a gradu­ate stu­dent even, Van Dalen had ac­tu­ally con­tac­ted Robin­son who was then at Yale Uni­versity, and said, “I have a stu­dent who is very in­ter­ested in your work, and has learned about it, and could he be a gradu­ate stu­dent?” and, you know — be­cause that cer­tainly would have been bet­ter than work­ing on my own. Or more ef­fi­cient — and Robin­son ac­tu­ally replied, “Yes, that would be fine.” At that time, I think these things were done without too much bur­eau­cracy. In fact, what was ac­tu­ally planned was that I was go­ing to be there as a gradu­ate stu­dent, but then he wrote back half a year later — that was ac­tu­ally the time that he got very ill. And so then I de­cided to stay in Utrecht, and he said, “I can­not take any stu­dents any­more”. That was also the time Macintyre had gone there as an as­so­ci­ate pro­fess­or, And so I think I still should have gone there, but any­way. I ended up there any­way as a Gibbs in­struct­or.

Rob: And after two years there…?

Lou: Then I was at Stan­ford for a few years. And I was also a year at the In­sti­tute for Ad­vanced Study. And then in 1986, I was hired by the Uni­versity of Illinois.

Rob: So, I think I re­mem­ber that Berke­ley made you an of­fer some­where in there…

Lou: No, that didn’t hap­pen. I think they did con­sider me at one point for a po­s­i­tion. When I was at Stan­ford, I def­in­itely came here to give a talk that was sort of con­sidered as a kind of in­ter­view talk, and I think they did con­sider me, but I don’t think they made me — well, I would have known if….

Rob: I just re­mem­ber your name from those days.

Lou: Aha.

Rob: You prob­ably chose a dif­fer­ent field.

Lou: Well, I do know that when I came for this talk, I did not ac­tu­ally talk about my own work. I had no idea that this was sup­posed to be what you should do. I just talked about something that I found in­ter­est­ing and learned about. It might have been about Denef’s work or something like that. Of course, later on I ac­tu­ally got to work with Denef on re­lated is­sues, but at that time, I was still more or less fa­mil­i­ar­iz­ing my­self with some of his work, so I talked about that but not about my own work, which would have been prob­ably — at that time I was already con­sid­er­ing things like $$o$$-min­im­al­ity. Right, yes.

Rob: Could you go back and say the per­son whose work you talked about…the name is…?

Lou: Jan Denef. He is a Bel­gian math­em­atician, with whom I did some work later on.

Tom: And, so was the in­tro­duc­tion of Wei­er­strass di­vi­sion your idea? Did it come from the clas­sic…?

Lou: Yeah, so I had in­tro­duced this $$o$$-min­im­al­ity busi­ness — I mean I didn’t call it $$o$$-min­im­al­ity, but I had in­tro­duced some things that de­veloped in­to $$o$$-min­im­al­ity [3]. In par­tic­u­lar, I also ob­served at the time that this the­ory of sub­ana­lyt­ic sets — if you just change the defin­i­tion from sub­ana­lyt­ic to what I called glob­ally sub­ana­lyt­ic, then the reals with that is an $$o$$-min­im­al struc­ture. So then it was nat­ur­al to start think­ing about “what about the $$p$$-adics?” Right? And in fact there was this ques­tion by Serre in an art­icle that he had writ­ten [e5] that seemed — that it might be solv­able if you had a reas­on­able the­ory of $$p$$-ad­ic sub­ana­lyt­ic sets. I ac­tu­ally star­ted to think about that, and I was on leave at the Max Planck In­sti­tute in Ger­many in 1986, be­fore I took up my job in Illinois, and I had some idea that went in­to the right dir­ec­tion and I ac­tu­ally gave a talk about it in Bel­gi­um, where Denef was present. Of course I knew Denef already. In fact, we had already writ­ten something re­lated to Artin ap­prox­im­a­tion, so I knew him quite well. Then, a week later, after that talk, we ba­sic­ally both had real­ized in­de­pend­ently how to use Wei­er­strass pre­par­a­tion and di­vi­sion in such a way that you could de­vel­op a the­ory of $$p$$-ad­ic sub­ana­lyt­ic sets. This in­volved a some­what tricky use of Wei­er­strass. And then we wrote the pa­per [4]. It was a very nice col­lab­or­a­tion. It did an­swer Serre’s ques­tion on Poin­caré series of $$p$$-ad­ic ana­lyt­ic vari­et­ies.

Tom: And had you worked out the sim­il­ar ar­gu­ment with the reals be­fore­hand? Or is that after…?

Lou: No, then we de­cided, OK, now that it worked so well for the $$p$$-ad­ic, let’s go back to the reals and see if we get this quan­ti­fi­er elim­in­a­tion that we had for the $$p$$-adics. Could we also get it for the reals, be­cause the the­ory of sub­ana­lyt­ic sets de­veloped by Hironaka and Gab­ri­elov did not ac­tu­ally give you quan­ti­fi­er elim­in­a­tion. It was mod­el com­plete­ness ba­sic­ally, what you get. And then we did that for the reals in a way sim­il­ar to what we did for the $$p$$-adics. But in the $$p$$-adics, it’s a little sim­pler. So it was nice that we did it there first and then went back to the reals. But in some sense it went from the reals — ba­sic­ally, with the real­iz­a­tion that the reals with the glob­ally sub­ana­lyt­ic sets form an $$o$$-min­im­al struc­ture — to the $$p$$-adics, and then back again to the reals.

Rob: Is there a ver­sion for the com­plex num­bers?

Lou: Well, for the mod­el the­or­ist, the com­plex num­bers are sim­pler than the reals. The com­plex num­bers are in­ter­pretable in the reals, as we say, so I don’t think there is a spe­cial the­ory of sub­ana­lyt­ic sets over the com­plex num­bers.

Rob: So, the math­em­at­ics you’re talk­ing about here is mostly dur­ing your first six years, ten years…?

Lou: Well, what I was just talk­ing about — this col­lab­or­a­tion with Denef happened in 1986, when I was on leave. I had some sort of semester off. I was at the Max Planck In­sti­tute. I think I was already hired by UIUC. But, for ex­ample, these ideas about $$o$$-min­im­al­ity, I got in 1982, so that was a few years earli­er.

Tom: And it ends all the work that you did on ul­traproducts of poly­no­mi­al rings and faith­ful flat­ness.

Lou: Oh yeah, that was in my thes­is.

Tom: Why is that — OK, I don’t know who your coau­thor Schmidt is.

Lou: Oh yeah, Karsten Schmidt.

Tom: How did it come about that you were work­ing with him?

Lou: He was from Kiel (Ger­many) and got in touch about these bounds when I pub­lished something from my thes­is about it in the pro­ceed­ings of a meet­ing. We then col­lab­or­ated on get­ting fur­ther bounds of this kind, con­sid­er­ing also cases where they don’t ex­ist [2]. He went on to pub­lish some pa­pers of his own on these mat­ters. Schoutens took up these things as well, and so did Aschen­bren­ner (when he was my stu­dent) in an­oth­er dir­ec­tion, poly­no­mi­al rings over the in­tegers, where it turns out there are bounds of a dif­fer­ent nature [e13].

Any­way, in the last chapter of my thes­is, I wanted to un­der­stand these com­plic­ated al­gorithmic res­ults of Seiden­berg and…what’s her name? They al­ways men­tion two people, right? Grete Her­mann and Seiden­berg. At the time I didn’t like al­gorithms. I wanted to un­der­stand from a more con­cep­tu­al point of view. Now I think al­gorithms are ac­tu­ally — I’m get­ting bet­ter at it, per­haps, or have more ap­pre­ci­ation for it, but at that time, I wanted to un­der­stand in a more con­cep­tu­al way how these bounds came about. After all, they mean that cer­tain things are of a first-or­der nature. And of course my in­terest was also piqued by Ab­ra­ham Robin­son’s list of prob­lems. He stated some of these things as open prob­lems or par­tially open prob­lems. Then I no­ticed that faith­ful flat­ness and some oth­er well known com­mut­at­ive al­gebra facts could be used to get these bounds in a very cheap way.

Rob: Was Ab­ra­ham Robin­son’s list of prob­lems pub­lished?

Lou: Yes. He gave an ad­dress as Pres­id­ent of the ASL — an ad­dress leav­ing this po­s­i­tion. He men­tioned a num­ber of prob­lems and de­scribed them in a very at­tract­ive way, and they were pub­lished in, I think, the Journ­al of Sym­bol­ic Lo­gic.1 They’re cer­tainly in his col­lec­ted pa­pers or something.

Rob: Ac­tu­ally, let me in­ter­rupt just so I don’t for­get. Is there un­pub­lished work — your notes, something of that sort — that would be valu­able to know about…?

Lou: No. With Macintyre and [Gregory] Cher­lin, the three of us wrote a fairly long pa­per on some sort of co-mod­el the­ory for Galois groups or profin­ite groups where everything is sort of dual. Later on I real­ized it’s just a form of many-sor­ted first-or­der lo­gic. But at the time, I some­how did not real­ize that.

Tom: Ah.

Lou: So that pa­per was nev­er pub­lished be­cause we couldn’t really agree on how to fin­ish it, how to put it in­to fi­nal shape. But we dis­trib­uted it to people and people picked up on it and so we de­cided, ok, what the hell, let’s just have it in the pub­lic do­main in this way. Some­times I still get re­quests for a copy, yeah.

Rob: And you’d al­low us to in­clude that…?

Lou: Well, if Macintyre and Cher­lin are OK with it.

Rob: Sure, of course we’ll ask them.

Lou: I think I must have a copy some­where.

Tom: I have a pho­to­copy. It’s an ac­tu­al phys­ic­al pho­to­copy of a copy.

Lou: You can prob­ably see that dif­fer­ent parts are in dif­fer­ent fonts.

Rob: Yeah, it may not be per­fect, by defin­i­tion, but still it’s of his­tor­ic­al in­terest.

Lou: Right. Something that I was some time ago plan­ning to pub­lish but nev­er got around to was a prob­lem that Serre once posed about Hil­ber­tian fields. I’ve ac­tu­ally giv­en a few talks about it, but nev­er got around to writ­ing the solu­tion down, un­til about a year ago I heard someone else talk about a sim­il­ar thing, and I men­tioned this. What’s his name again…this guy in Dresden?

Tom: Arno Fehm?

Lou: Fehm, yeah! I said, “This looks very much like the kind of thing I was think­ing about — I don’t know, it must have been about twenty years ago, and I still have notes about it. I think I should send it to you to see if there’s some over­lap.” And it turns out, that he had in­deed also proved these things — but in a dif­fer­ent way and pub­lished something about it [e15]. I should have been quick­er with that.

Rob: There’s a good place in Cel­eb­ra­tio Math­em­at­ica for things like this, un­pub­lished work…be­cause it’s part of the his­tor­ic­al re­cord. It shows something you were think­ing about. It’s part of your oeuvre.

Lou: I don’t know. In fact, I do re­mem­ber you [Tom Scan­lon] were at the talk that I once gave about this in Urb­ana, and I think you asked me to send you the slides.

Tom: Huh. I must have those, too! Which the­or­em is this?

Lou: Yeah, what was the prob­lem again…? It was something about…Ah, yeah. So, Serre had defined some no­tion of Hil­ber­tian vari­ety, right — sort of a gen­er­al­iz­a­tion of Hil­ber­tian fields sat­is­fy­ing the Hil­bert ir­re­du­cib­il­ity the­or­em, which means that a field of ra­tion­al func­tions in one vari­able over it would be the…or the af­fine line would be an Hil­ber­tian vari­ety in his sense. And his ques­tion was wheth­er the product of two Hil­ber­tian vari­et­ies is Hil­ber­tian. And I saw that one could use some ar­gu­ments of Ro­quette’s, or one could ad­apt — modi­fy, gen­er­al­ize — ar­gu­ments of Ro­quette’s to in­deed an­swer this ques­tion pos­it­ively. And I wrote it down. On one or two oc­ca­sions I gave a talk about it, and then I sort of for­got about it. About a year ago or two years ago, it turned out that someone else had in the mean­time also done this.2

Tom: I wanted also to ask about some oth­er work. There are two dir­ec­tions I wanted to talk about, and they’ll each one take a long time. One would be about the work you did with Dave [Mark­er] and An­gus [Macintyre] on power series mod­els…and con­nec­tions to the quan­ti­fi­er elim­in­a­tion [5], [6] .

Lou: Oh yeah.

Tom: Was that the ori­gin of your think­ing about trans­ser­ies and the twenty-year pro­ject?

Lou: Yes. Right, so let me see. I have to go back in time now. Be­fore Alex Wilkie had proved the $$o$$-min­im­al­ity of the real field of ex­po­nen­ti­ation [e10], Dahn and Göring had writ­ten a short pa­per [e6] sug­gest­ing a con­crete, non­stand­ard mod­el of this the­ory, which con­sisted of log­ar­ithmic ex­po­nen­tial series over the reals. So, after this work of Alex’s, we star­ted to think about wheth­er per­haps we could in­deed prove that this was an ele­ment­ary ex­ten­sion of the real ex­po­nen­tial field, and then we did…Well, it’s a little com­plic­ated again be­cause — first, I did some things with Chris Miller where we ba­sic­ally ad­ap­ted tech­niques of Alex’s and we did prove that the reals with re­stric­ted ana­lyt­ic func­tions and ex­po­nen­ti­ation is mod­el com­plete, and $$o$$-min­im­al as a con­sequence.

We did get some­where — we pub­lished a book about it a few years ago [7], but since then we have ac­tu­ally got­ten some fur­ther res­ults, which I think are also very in­ter­est­ing.

In some sense one can con­nect it to some ori­gin­al things that Ju­lia Robin­son did. That’s how I get to talk about this to­mor­row.

Rob: I was just go­ing to ask you that ques­tion.

Tom: Well, so do you think there’s any chance that you could de­scribe that struc­ture as a dif­fer­ence field? It has many dif­fer­ent struc­tures as a dif­fer­ence field…

Lou: Yeah. Well, it’s cer­tainly in the back of our minds to look at that also. Someone who has ideas about that would be Jor­is. But I must say I haven’t really got­ten far enough in­to that to have an opin­ion about it. There are so many dif­fer­ence op­er­at­ors there. What is the right — well, I guess maybe just the one that sends $$x$$ to $$x + 1$$ would be the most nat­ur­al one. Or the one that sends $$x$$ to $$e^x$$. Either one would be very nat­ur­al to con­sider. Well, ac­tu­ally these auto­morph­isms do play even a role in our own work. For ex­ample, to show that cer­tain things are not defin­able in our struc­ture. But we ba­sic­ally have only been us­ing it so far in this way. We have not really in­vest­ig­ated the mod­el the­ory of this as a dif­fer­ence field. But it’s cer­tainly something that at some point, you know, people should do it.

Rob: [To Tom] You had an­oth­er top­ic…?

Tom: Well, ac­tu­ally, it ends up be­ing the same top­ic.

Lou: Right.

Tom: You know, one kind of flits…

Lou: Yeah, one thing leads to an­oth­er. Go­ing back to Ju­lia Robin­son and this book by her sis­ter, From Zero to In­fin­ity… At that time — that was back be­fore I star­ted col­lege — I really found num­ber the­ory a fas­cin­at­ing sub­ject; I should really get in­to that. And of course, as one gets older it gets harder and harder to branch out in such a dir­ec­tion where there are such bril­liant people already in­volved in con­nect­ing mod­el the­ory to num­ber the­ory.

Tom: But you did the work on the al­geb­ra­ic num­bers.

Lou: Yes, right. That was nice.

Tom: And I re­mem­ber there be­ing some small talk you gave about com­plex­ity bounds for com­put­ing greatest com­mon di­visors…

Lou: Ah yeah. Those were sort of ex­cur­sions, which were very nice. One some­times needs to take a break from big­ger pro­jects to do something that —

Rob: To feel like you ac­com­plished something!

Lou: Yeah, for ex­ample, this trans­ser­ies thing that has been go­ing on for twenty years — for the first fif­teen years we made very little pro­gress. I did feel the need some­times to do something where you could ac­tu­ally show someone that you had done something.

Rob: Yeah, it’s true that as I’ve heard you talk about things that I don’t know hardly any­thing about — but the Wei­er­strass pre­par­a­tion the­or­em and things of that sort, I haven’t heard any­thing that soun­ded like Ju­lia Robin­son’s work. But there is a con­nec­tion there. Not between Wei­er­strass, but between…

Lou: Well, the thing is that Wei­er­strass pre­par­a­tion and di­vi­sion can be used to prove things that we call mod­el com­plete­ness, which means that all defin­able sets are ex­ist­en­tially defin­able. And this no­tion of an ex­ist­en­tial defin­able set plays a big role in Ju­lia Robin­son’s work — but in an­oth­er way, so to say. I mean, mod­el com­plete­ness means that all defin­able sets are ex­ist­en­tially defin­able while in Ju­lia Robin­son’s work, the ex­ist­en­tially defin­able sets turn out to be the ones that are re­curs­ively enu­mer­able. Of course that means that for Hil­bert’s tenth prob­lem there is no de­cision pro­ced­ure for solv­ing Di­o­phant­ine equa­tions, so that goes in the op­pos­ite dir­ec­tion, so to say.

Rob: Can you say what ex­ist­en­tially defin­able is in twenty-five words or less?

Lou: OK. A set of tuples, of num­bers, is ex­ist­en­tially defin­able if it’s the set of para­met­ers for which a sys­tem of equa­tions is solv­able. So we have a sys­tem of equa­tions with para­met­ers in them — you have un­knowns and para­met­ers. And then the set of para­met­ers for which it is solv­able is ex­ist­en­tially defin­able. The set of para­met­ers for which a sys­tem is solv­able. That is what we call an ex­ist­en­tially defin­able set.

Tom: If you were to write it in terms of the lo­gic­al for­mu­las, you’re say­ing there ex­ists such and such that defines it.

Lou: So, there is this pa­per by Ju­lia Robin­son in 1952 [e1] as I will men­tion to­mor­row in my talk be­cause I think it’s a sem­in­al pa­per in this whole area that led to the neg­at­ive solu­tion of Hil­bert’s tenth prob­lem — well, in the sense that there is no al­gorithm. But it also has a pos­it­ive mean­ing, namely, that the ex­ist­en­tially defin­able sets are ex­actly the re­curs­ively enu­mer­able sets.

Rob: For my own edu­ca­tion in the field of mod­el the­ory, who are the early gi­ants?

Lou: Well, Tarski and Robin­son, I would say. And Malt­sev. Those, I think, are usu­ally men­tioned as early mod­el the­or­ists. After that there are many people who have made big con­tri­bu­tions to the sub­ject. But maybe it’s too early to…

Rob: Tarski was a mod­el the­or­ist but really he did more parts of lo­gic also…?

Lou: Who?

Tom: Tarski? I think he in­ven­ted the term.

Rob: And by “Robin­son” here you mean Ab­ra­ham Robin­son.

Lou: Right, right. I guess Ju­lia Robin­son isn’t coun­ted as a mod­el the­or­ist, but some­how her work, be­cause it in­volved defin­ab­il­ity, it’s in some sense pretty close in spir­it — or part of it. And also with Raphael Robin­son…they both took the no­tion of a defin­able set quite ser­i­ously and got some very nice res­ults.

Tom: So you de­cided not to name people who are still alive on your list of im­port­ant mod­el the­or­ists.

Rob: He’s be­ing very care­ful! [laughter]

Tom: So there’s one per­son in par­tic­u­lar who is still alive —

Lou: She­lah?

Tom. Yeah.

Lou: Oh, yeah, of course. I can safely call him a gi­ant. Even his work is gi­gant­ic.

Tom: But the ap­proach that he takes to mod­el the­ory is very dif­fer­ent from the one you take. Can you see places where what he’s done or his ap­proach to clas­si­fic­a­tion the­ory or sta­bil­ity the­ory or any­thing has in­flu­enced your work, or you’ve re­spon­ded to it in some way?

Lou: In some sense, when I grew up as a mod­el the­or­ist, I grew up sort of isol­ated from what turned out to be the main­stream of mod­el the­ory, namely, this work of She­lah. In fact, I re­mem­ber even She­lah giv­ing talks in Bel­gi­um that I at­ten­ded and I thought, “Well, is this mod­el the­ory?” [laughter] “Why would any­one be in­ter­ested in know­ing how many mod­els a the­ory has in a cer­tain car­din­al­ity?” At that time I did not ap­pre­ci­ate that at all. Of course when people talk about these things all the time around you, you are ex­posed to so much of it that at some point you real­ize, “Oh, well, hmmm.” I guess there are some people like Cher­lin, Pil­lay, and then es­pe­cially of course Hrushovski… When these people star­ted to… You real­ize, “Oh wow! This is really a fant­ast­ic sub­ject!” Or a very big de­vel­op­ment, which I com­pletely missed out on. So at some point of course you do get in­flu­enced by these things. But I ba­sic­ally nev­er — I did not grow up in this style of mod­el the­ory. The things I’m do­ing for ex­ample with Mat­thi­as and Jor­is, we do pay at­ten­tion to is­sues like NIP — wheth­er a struc­ture has NIP, right? Which is a no­tion that came out of this whole de­vel­op­ment.

Rob: NIP is…?

Lou: NIP stands for the noninde­pend­ence prop­erty, which is a very ro­bust prop­erty that struc­tures or the­or­ies can have and has many con­sequences. But it’s typ­ic­ally a no­tion that was in­ven­ted in the school of She­lah and pretty far from Robin­son’s way of think­ing. But of course we learn about these things and pay at­ten­tion to it, but I was nev­er in­clined, for ex­ample, to work on this as a pure no­tion by it­self.

Rob: So is it fair to say that the sort of mod­el the­ory that you’ve done has more con­nec­tions to the rest of math­em­at­ics than She­lah’s work?

Lou: Not any more. [laughter] No, at the time that I heard first about She­lah’s work, I thought, “This will nev­er make any dif­fer­ence to math­em­at­ics.” But I was totally wrong about it. It’s just that at the time, the way he talked about this or the way every­one talked about it at the time was, “Well, we want to know how many mod­els a the­ory can have in a cer­tain car­din­al­ity,” and I had no sense that this was an im­port­ant ques­tion. Of course now we know that some­how in­vest­ig­at­ing this you ac­tu­ally in­vest­ig­ate very ro­bust prop­er­ties of the­or­ies, but this was not how people talked about it at the time. That came later.

Sheila: Is it rel­ev­ant to ask how did you come to real­ize the im­port­ance of the ques­tion? At what point and how did you come to that real­iz­a­tion?

Lou: Yeah, just be­ing ex­posed to people talk­ing about it. Also partly it was $$o$$-min­im­al­ity, right? I in­ven­ted this no­tion which turned out to be — later people called it $$o$$-min­im­al­ity, and I asked people like Pil­lay, “Why do you call it $$o$$-min­im­al­ity?” And he said, “Oh, it’s ana­log­ous to strong min­im­al­ity.” And strong min­im­al­ity is an earli­er no­tion that also came up in this pure mod­el the­ory trend that I had nev­er really paid at­ten­tion to. But then I sud­denly un­der­stood why this was such a strong, power­ful no­tion. Be­cause I saw it is ana­log­ous to $$o$$-min­im­al; then it must be good! [laughter]

Rob: What does the let­ter $$O$$ — where does that name come from?

Lou: Or­der.

Rob: Or­der min­im­al­ity?

Lou: Yeah, or­der min­im­al. So, yeah, this is something that Pil­lay and Stein­horn and Ju­lia Knight — they wrote [e8] (see also [e7]). So after I gave some talks about this, which Anand Pil­lay and Stein­horn at­ten­ded, they made this in­to a defin­i­tion: they called it $$o$$-min­im­al. Which stands for or­der min­im­al, and thought of it as ana­log­ous to an ex­ist­ing no­tion called strongly min­im­al, which ex­is­ted already in mod­el the­ory. So that’s also when I real­ized, “Aha, there is really a whole in­dustry there that one should really pay at­ten­tion to.”

I’ve al­ways been most in­ter­ested in un­der­stand­ing rather con­crete struc­tures, where one of the main things you want is, for ex­ample, quan­ti­fi­er elim­in­a­tion or mod­el com­plete­ness. It means that you already can un­der­stand a lot about it, and these are also of­ten the main tools that you need to prove oth­er prop­er­ties like, you know, NIP or sta­bil­ity or whatever.

I wish that I had paid at­ten­tion to these things earli­er but there are lots of things to do any­way. You don’t have to know everything.

Rob: This fel­low — Sela. Is he con­nec­ted at all to — ?

Lou: Oh, the group the­ory!

Tom: Zlil Sela?

Lou: Zlil Sela. Well, he’s more a group the­or­ist but did solve a very im­port­ant prob­lem of Tarski. An older prob­lem of Tarski which goes back to the 1940s, try­ing to un­der­stand mod­el-the­or­et­ic­ally the the­ory of free groups. And that is what — how do you pro­nounce his name?

Tom: Zlil Sela.

Lou: That is what Zlil Sela man­aged to do, which is very deep work, I be­lieve.

Rob: It is, to some ex­tent, mod­el the­ory…?

Lou: Yeah, def­in­itely. Right. Al­though he’s not really a mod­el the­or­ist, but it has been picked up by mod­el the­or­ists. Def­in­itely. The very hard work he did was ba­sic­ally…he did it from scratch, so to say. I guess he was ex­posed to some lo­gic. After all, the prob­lem that Tarski stated wasa posed in terms of — as a lo­gic prob­lem.

Sheila: I want to take a step back for a minute and pin down the time when you learned about the ana­log­ous­ness of $$o$$-min­im­al­ity to strong min­im­al­ity. When was this brought to your at­ten­tion?

Lou: OK, let me see. That was the mid 80s. But the She­lah re­volu­tion already star­ted fif­teen years earli­er and a lot of mod­el the­or­ists im­me­di­ately caught on to that. Not me. [laughter]

Rob: Well, the She­lah thing was in the late six­ties, wasn’t it?

Lou: The end of the 60s, yeah.

Tom: He wrote his PhD thes­is in 1968…? 1969?

Rob: Yeah.

Tom: But the book [Clas­si­fic­a­tion the­ory and the num­ber of non­i­so­morph­ic mod­els] came out in the early 70s.3

Rob: Be­cause it seems like UCLA and Berke­ley — I went to Berke­ley in 1971 — and both places were try­ing to hire him, with no suc­cess.

Sheila: I just wanted to pin down the ques­tion about the Hil­ber­tian vari­et­ies — when the product of two Hil­ber­tian vari­et­ies are Hil­ber­tian. When did you think of that? And then I know that Arno Fehm ended up prov­ing it or pub­lish­ing his res­ult. When do you think your work was?

Lou: OK. I re­mem­ber this was fairly early when I ar­rived at UIUC in 1986, so prob­ably the late 80s or the early 90s. I re­mem­ber there was a little book — or lec­ture notes — that was pub­lished by Serre [e9] or writ­ten by a stu­dent, where he star­ted to talk about Hil­bert’s ir­re­du­cib­il­ity the­or­em, and so on, and gen­er­al­ized this no­tion of Hil­ber­tian fields, and I saw these ques­tions there. I was aware that there was this treat­ment of Hil­ber­tian fields by Ro­quette and I thought, “Hey, maybe these tech­niques can help an­swer this ques­tion.” I thought a bit about it and I real­ized in­deed that that was the case. I could an­swer these ques­tions us­ing these ideas. Then, I wrote them down but some­how nev­er got to the point of writ­ing it down as a pub­lic­a­tion. I guess there are al­ways so many things that you want to do that some­times you simply don’t get to put it in a form that you’re really happy with. Any­way, it’s not a big deal.

Rob: How do you spell Ro­quette?

Lou: R-o-q-u-e-t-t-e. This is a num­ber the­or­ist in Heidel­berg, who ac­tu­ally worked with Robin­son (Ab­ra­ham Robin­son) in the early 1970s. And who gave a non­stand­ard treat­ment of Hil­bert’s ir­re­du­cib­il­ity the­or­em [e3], which, at the time, I read care­fully and liked a lot. So I learned something from that.

Sheila: Can I ask a gen­er­al ques­tion? If you were go­ing to do something with your mind and your tal­ent and not think of your­self as be­ing “too old” to be­gin something new, what would it be? What would you plunge in­to?

Lou: Ouf! Some­times I have some ideas and every time it’s a dif­fer­ent thing. [laughter]

Sheila: That’s a fair an­swer.

Lou: I mean, I could very well ima­gine start­ing over and try­ing to be­come a num­ber the­or­ist, for ex­ample. I mean, one of the things that has al­ways frus­trated lo­gi­cians is that we have Gödel’s in­com­plete­ness the­or­em, which says that if you have a little bit of arith­met­ic in your the­ory, then you get all kinds of an­noy­ing things. There is no al­gorithm to de­cide wheth­er things are true and so on. But, on the oth­er hand, num­ber the­or­ists who work, for ex­ample, with glob­al fields have a very im­port­ant ax­iom which is called the product for­mula, right? The prob­lem for lo­gi­cians is that it’s not a first-or­der state­ment. Or not in a nat­ur­al way, at least. But now there is a new — a sort of vari­ant of first-or­der lo­gic called con­tinu­ous lo­gic by which it is ac­tu­ally pos­sible to for­mu­late the product for­mula as a single ax­iom. And so Hrushovski and Itaï Ben Yaa­cov have been ex­plor­ing this in a way that sounds very prom­ising to me. And so this would be a way to com­bine mod­el the­ory with num­ber the­ory in a new way. I think it sounds quite fas­cin­at­ing. So if I had enough time, I can well ima­gine that I would like to get in­to that.

But, you know, the pro­jects that I am already in­volved in are suf­fi­ciently in­ter­est­ing that I will prob­ably stick with them. To learn new tech­niques and so on — at least at this… Well, I think I’m pretty slow any­way, so when you get older you get even slower!

Rob: I didn’t ask wheth­er you have any hob­bies… Mu­sic?

Lou: Well, no, just simple things like read­ing — es­pe­cially his­tory, for ex­ample. I like to read his­tory.

Rob: Any par­tic­u­lar time?

Lou: Well, ba­sic­ally, any time. As long as it’s not too re­cent. [laughter]

Tom: What’s wrong with re­cent his­tory?

Lou: Well, be­cause you live it any­way, so… I mean, the things that have happened since I was born, some­how it doesn’t sound so ex­cit­ing to me.

I just went to the book­store and I got these. This is a book on Kais­er Wil­helm II. This is the book by Hil­ary Man­tel called A Place of Great­er Safety. This is about the French Re­volu­tion.

Sheila: But is this…this is, uh, fic­tion!

Lou: Yes, this is a nov­el. But it’s his­tor­ic­al fic­tion! That’s good enough. [laughter] And this is a book by Ju­li­en Benda called The Treas­on of the In­tel­lec­tu­als. I’ve al­ways heard that this is a good book to read.

Rob: I used to read those sorts of things when I was much young­er.

Lou: Some­how one has the feel­ing that he [Wil­helm II] might have been a fig­ure a bit like Trump.

Sheila: You chose this in Eng­lish!

Lou: Yes, I should have bought it in…I think I ac­tu­ally have the French ver­sion but the book was so badly or­gan­ized that when I saw it here in Eng­lish and I looked at it, I thought, “Oh, this is much more read­able.”

Rob: So how many lan­guages do you speak?

Lou: Well, ba­sic­ally, just Dutch and Eng­lish and a bit of Ger­man. I can read French, but not really speak it.

Rob: It’s much easi­er to read —

Lou: Yes, es­pe­cially math­em­at­ic­al French is quite easy in my opin­ion. And of course I learned Lat­in and Greek but nev­er got any flu­ency in that.

Rob: So, you still got that!

Lou: Yeah, well, I was in a so-called gym­nas­i­um. The ly­ceum that I went to had two parts: one dir­ec­tion where you had to take clas­sic­al lan­guages and one which did not re­quire it, and for some reas­on, the part that re­quired clas­sic­al lan­guages was sup­posed to be the more pres­ti­gi­ous part. And so you were ba­sic­ally sort of steered in that dir­ec­tion if your grades were high enough, but I nev­er caught on to Greek and Lat­in very much, ex­cept his­tory. Clas­sic­al his­tory I do like.

Sheila: Ta­cit­us?

Lou: Yes, of course. I did read some Ta­cit­us. But the books that we had to read — of course we star­ted with De Bello Gal­li­co in the second grade of the gym­nas­i­um, and then after that we had to go through Livi­us. The whole his­tory of the city of Rome from the be­gin­ning. It was al­ways too hard, You could really only do it with a dic­tion­ary. I nev­er got enough fa­cil­ity with it.

Rob: My fath­er took four years of Greek and Lat­in be­cause he ex­pec­ted to be­come a min­is­ter and that’s what you did.

Lou: Oh yeah.

Rob: I took two years of Lat­in in high school and it didn’t take at all. But my neph­ew is only 26. He took four years of Lat­in in high school and liked it, so it just de­pends on…

Lou: Yeah, I mean, sure, there were people who were good in it, but I def­in­itely was not.

Rob: My fath­er al­ways knew the de­riv­a­tion of words bet­ter than any­one else in the room from his four years of Greek and Lat­in.

Lou: Of course the Greek al­pha­bet is still quite use­ful for math­em­aticians. [laughter] In fact, some­times you need more than the two al­pha­bets.

### Works

[1] L. P. D. van den Dries: Mod­el the­ory of fields: De­cid­ab­il­ity and bounds for poly­no­mi­al ideals. Ph.D. thesis, Rijk­suni­versiteit Utrecht, 1978. phdthesis

[2] L. van den Dries and K. Schmidt: “Bounds in the the­ory of poly­no­mi­al rings over fields: A non­stand­ard ap­proach,” In­vent. Math. 76 : 1 (February 1984), pp. 77–​91. MR 739626 Zbl 0539.​13011 article

[3] L. van den Dries: “Re­marks on Tarski’s prob­lem con­cern­ing $$(\mathbb{R},+,\cdot\,,\operatorname{exp})$$,” pp. 97–​121 in Lo­gic col­loqui­um ’82 (Florence, 23–28 Au­gust 1982). Edi­ted by G. Lolli, G. Longo, and A. Mar­cja. Stud­ies in Lo­gic and the Found­a­tions of Math­em­at­ics 112. North-Hol­land (Am­s­ter­dam), 1984. MR 762106 Zbl 0585.​03006 incollection

[4] J. Denef and L. van den Dries: “$$p$$-ad­ic and real sub­ana­lyt­ic sets,” Ann. Math. (2) 128 : 1 (July 1988), pp. 79–​138. MR 951508 Zbl 0693.​14012 article

[5] L. van den Dries, A. Macintyre, and D. Mark­er: “The ele­ment­ary the­ory of re­stric­ted ana­lyt­ic fields with ex­po­nen­ti­ation,” Ann. Math. (2) 140 : 1 (July 1994), pp. 183–​205. MR 1289495 Zbl 0837.​12006 article

[6] L. van den Dries, A. Macintyre, and D. Mark­er: “Log­ar­ithmic-ex­po­nen­tial power series,” J. Lon­don Math. Soc. (2) 56 : 3 (1997), pp. 417–​434. MR 1610431 Zbl 0924.​12007 article

[7] M. Aschen­bren­ner, L. van den Dries, and J. van der Ho­even: Asymp­tot­ic dif­fer­en­tial al­gebra and mod­el the­ory of trans­ser­ies. An­nals of Math­em­at­ics Stud­ies 195. Prin­ceton Uni­versity Press, 2017. MR 3585498 Zbl 06684722 book