Celebratio Mathematica

Lou van den Dries

Interview with Lou van den Dries

by Rob Kirby

Lou van den Dries.
Photo courtesy of the University of Illinois Department of Mathematics.

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.

Rob: Let’s start with your full name.

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.

Rob: Did your moth­er…?

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.

Rob: Who was your ad­visor?

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.

Tom: Did you read that be­fore you wrote your thes­is?

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.


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…?

Tom: What about your work on co-lo­gic. Was it ever pub­lished?

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.

I think that only gave mod­el com­plete­ness, and then right after that — yeah, what is the story ex­actly? The three of us — Dav­id, An­gus and I — were ba­sic­ally think­ing about this in­de­pend­ently. We had already been talk­ing and I think we de­cided to col­lab­or­ate, and then we found a bet­ter way of prov­ing this res­ult which in ad­di­tion gave a quan­ti­fi­er elim­in­a­tion and the fact that the field of log­ar­ithmic-ex­po­nen­tial series is an ele­ment­ary ex­ten­sion of the real ex­po­nen­tial field. And then, of course, I got also very in­ter­ested in this struc­ture as a dif­fer­en­tial field, which is a far more com­plic­ated ob­ject. So then I star­ted to work on that around the same time as Mat­thi­as Aschen­bren­ner ar­rived as a gradu­ate stu­dent in Urb­ana, and to­geth­er we did some things, and then a few years later we found out that Jor­is van der Ho­even had ac­tu­ally writ­ten a thes­is about this dif­fer­en­tial field [e11]. We got in touch with him and the three of us (Mat­thi­as, Jor­is and I) de­cided to take it on as a sort of pro­ject to really un­der­stand the ele­ment­ary the­ory of this dif­fer­en­tial field of trans­ser­ies. So that has been my main oc­cu­pa­tion for twenty-five years now. In fact, I will talk about it to­mor­row.

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.


[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