Celebratio Mathematica

Lou van den Dries is a world-class math­em­atician who com­bines in­genu­ity and tech­nic­al mas­tery to ob­tain phe­nom­en­al res­ults. Read­ing one of his pa­pers is al­ways a de­light: he writes with an un­rivaled clar­ity that res­ults in ex­pos­i­tions as il­lu­min­at­ing as they are me­tic­u­lous. Ask any mod­el the­or­ist to name their top three fa­vor­ite Lou pa­pers and you will al­ways re­ceive an im­me­di­ate and en­thu­si­ast­ic re­sponse (al­though they might per­haps grumble at be­ing re­stric­ted to nam­ing only three). For the re­cord, my top three list goes like this:

1. his work with Alex Wilkie, giv­ing a non­stand­ard proof of Gro­mov’s the­or­em on poly­no­mi­al growth [2];
2. his pa­per with Karsten Schmidt on bounds in poly­no­mi­al rings [1];
3. his pa­per with An­gus Macintyre on the lo­gic of Rumely’s loc­al glob­al prin­cip­al [3].

While I could write an en­tire trib­ute to Lou fo­cused ex­clus­ively on his math­em­at­ics, I in­stead would like to em­phas­ize an equally im­port­ant as­pect of Lou’s math­em­at­ic­al leg­acy: his pro­found im­pact on the gradu­ate stu­dents he has ad­vised. Lou is a de­voted ad­visor, and in­vests count­less hours in guid­ing his stu­dents to­wards find­ing prob­lems, aid­ing them in seek­ing solu­tions, and (per­haps his fa­vor­ite part) care­fully edit­ing their writ­ten works un­til they meet his high ex­pos­it­ory stand­ards. The stu­dents who emerge from his tu­tel­age do so as well-roun­ded math­em­aticians, pos­sessed of all his many tricks at suc­ceed­ing in re­search in ad­di­tion to the abil­ity to com­mu­nic­ate those ideas in writ­ten form. Above all of this, Lou cre­ates per­son­al con­nec­tions with his stu­dents — and his friend­ship may even be the most valu­able as­set a Lou stu­dent can re­ceive in their time with him.

My re­la­tion­ship with Lou began in 2004, when I ar­rived in Cham­paign–Urb­ana for the first time. Lou was in­deed the reas­on I chose to go to UIUC for gradu­ate school; vari­ous people had re­com­men­ded him as a great ad­visor. Hav­ing spent my en­tire life in Los Angeles and hav­ing at­ten­ded UCLA as an un­der­gradu­ate, I re­ceived the shock of a life­time when I stepped off the air­plane in Illinois to be met by fields of corn ex­tend­ing to the ho­ri­zon in every dir­ec­tion. With the real­iz­a­tion that I would be spend­ing, at a min­im­um, the next five years of my life in this re­mote place, a wave of anxi­ety came over me such as I’d nev­er felt be­fore. This trep­id­a­tion stuck with me through my first few days in Cham­paign–Urb­ana, and I figured that I might al­le­vi­ate these feel­ings by head­ing to cam­pus and in­tro­du­cing my­self to Lou. When I stepped in­to Lou’s of­fice for the first time, I was greeted by a friendly, if not a some­what tim­id, man who im­me­di­ately stopped what he was do­ing to talk with me about my plans. I am fairly sure that at this ini­tial meet­ing it was de­cided that Lou would be my thes­is ad­visor, for I nev­er form­ally asked him to be my ad­visor and he nev­er seemed to mind that I kept stick­ing around! He agreed to do a read­ing course with me on cer­tain parts of mod­el the­ory that I did not get around to study­ing dur­ing my earli­er self-guided read­ing in the sub­ject, and I left his of­fice feel­ing much bet­ter about the years to come.

Fast for­ward a week or so to the day of the Lo­gic Qual­i­fy­ing Ex­am. Hav­ing taken the gradu­ate course in lo­gic at UCLA, I figured I would try to pass the UIUC lo­gic ex­am be­fore the term star­ted. I vividly re­call sit­ting in the hall­way in front of the ex­am room in Alt­geld Hall, study­ing pre­vi­ous ex­ams and on the verge of open­ing my sack lunch, when a group of lo­gi­cians, led by Lou him­self, walked out of the mail­room, every­one ap­pear­ing to be in quite a jovi­al mood. Lou re­cog­nized me im­me­di­ately and asked if I would like to join the group for the weekly “Lo­gic lunch,” which, as I would learn, was an event that pre­ceded the weekly Lo­gic sem­in­ar. I po­litely de­clined the in­vit­a­tion, cit­ing my ur­gent need to study for the lo­gic ex­am I was about to take. Lou (who I be­lieve ac­tu­ally cowrote that ex­am) told me that I “would be just fine” and mildly in­sisted that I join the group. Not want­ing to make a bad first im­pres­sion with the rest of the lo­gi­cians, I ac­qui­esced and joined in the group head­ing out to Green Street for what turned out to be a de­li­cious and en­joy­able meal. Lou’s in­sist­ence was for­tu­it­ous, as was my idea to fol­low his sug­ges­tion: dur­ing that lunch, I met a group of lo­gic fac­ulty and stu­dents that was so friendly and en­thu­si­ast­ic about both lo­gic and life in Cham­paign–Urb­ana that I in­stantly felt ex­cited to be join­ing them. Lou knew the im­port­ance of the per­son­al as­pect of my gradu­ate study, and I will al­ways look fondly on this par­tic­u­lar event. (In­cid­ent­ally, he was also right and I was in­deed “just fine” in re­gards to the lo­gic qual­i­fy­ing ex­am.)

When I ar­rived in Urb­ana, Lou already had a large num­ber of stu­dents (around five or six) un­der his wing and so I spent the first couple of years mainly read­ing things he asked me to read in or­der to get com­fort­able with the lit­er­at­ure and to see what as­pect of mod­el the­ory I might find in­ter­est­ing. I did not mind “wait­ing” for his at­ten­tion as it was clear to me that he in­ves­ted so much of his time in each of his ad­vanced stu­dents that it was not feas­ible for him to of­fer the same amount of time to his ju­ni­or stu­dents.

Once my turn came around, we had a stroke of luck. At that time, Lou was asked to read a thes­is by someone at an­oth­er uni­versity. The top­ic of the thes­is was an at­tempt to use non­stand­ard ana­lys­is to solve a loc­al ver­sion of Hil­bert’s fifth prob­lem. Hil­bert’s fifth roughly asks wheth­er every to­po­lo­gic­al group that loc­ally looks like eu­c­lidean space in­deed has the struc­ture of a Lie group. Shortly after Hil­bert’s fifth prob­lem was re­solved by Gleason, Mont­gomery, and Zip­pin [e1], [e2] in the 1950s, Jac­oby [e3] claimed to have solved the loc­al ver­sion of the prob­lem, which is based on “loc­al groups” rather than full-fledged groups, and was later used in oth­er ap­plic­a­tions. However, sev­er­al dec­ades later, a fatal er­ror was found in the Jac­oby proof (which was no small task: Jac­oby’s proof was writ­ten in Quine’s New Found­a­tions!) and the thes­is sent to Lou claimed to have presen­ted a cor­rect solu­tion us­ing non­stand­ard ana­lys­is. Since nobody at that in­sti­tu­tion had the re­quired ex­pert­ise in both non­stand­ard ana­lys­is and Lie the­ory to read the thes­is, they sought ex­tern­al as­sist­ance. (Is there a more per­fect situ­ation that sums up Lou’s abil­it­ies? — he is one of a hand­ful of math­em­aticians who pos­sessed the re­quired pro­fi­ciency in these seem­ingly dis­par­ate fields.)

While Lou was of course cap­able of look­ing at this thes­is him­self, he knew that I was look­ing for a thes­is prob­lem, and wisely asked me to check the valid­ity of the ar­gu­ment, hop­ing that read­ing it would in­spire me to con­sider re­lated top­ics for my own thes­is. Read­ing it was very tough go­ing for me, but I even­tu­ally found a mis­take that seemed ir­re­par­able. When I presen­ted this con­cern to Lou, he con­firmed my ana­lys­is and we re­layed our con­vic­tions back to the people that asked for Lou’s ad­vice.

Not long after this epis­ode, Lou showed me a pa­per by Joram Hirsh­feld [e4], which presen­ted a nice non­stand­ard solu­tion to Hil­bert’s ori­gin­al fifth prob­lem, and sug­ges­ted that I try modi­fy­ing these ideas to the loc­al set­ting. This was a bril­liant sug­ges­tion on his part, and led to the main res­ult of my thes­is, which ended up be­ing pub­lished in the An­nals of Math­em­at­ics [e5] (where the ori­gin­al flawed solu­tion to the loc­al H5 had also ap­peared, some dec­ades earli­er). Modi­fy­ing Hirsh­feld’s tech­nique to the loc­al set­ting was a non­trivi­al en­deavor, and Lou gave me un­stint­ing guid­ance along the way. For ex­ample, he fre­quently shared with me typed notes in which he tried to re­cre­ate vari­ous ideas he had de­veloped about this prob­lem dec­ades pri­or, when he was act­ively in­ter­ested in this top­ic. Such in­volve­ment is surely above and bey­ond the typ­ic­al amount of work put in by a thes­is ad­visor, and ex­em­pli­fies the level of at­ten­tion Lou gives his stu­dents. I will be forever grate­ful for his act­ive in­volve­ment, for it clearly aided me to­wards prov­ing this fant­ast­ic res­ult.

When it came time to writ­ing up the res­ults of my thes­is, Lou was even more act­ive in the pro­cess. As any­one who has ever read a pa­per of Lou’s — or who has ever writ­ten a pa­per with Lou — can at­test, he has very strong opin­ions about math­em­at­ic­al ex­pos­i­tion. While I thought I was a fairly ad­ept writer (both math­em­at­ic­ally and oth­er­wise), Lou brought an en­tirely new de­gree of ex­pos­it­ory in­tens­ity. Al­most every draft I sent to him was re­turned with a seem­ingly end­less list of com­ments, some of them sub­stan­tial and some minute. At first I took the sea of red marks sprayed on the pages of my work quite per­son­ally. However, after con­sulta­tion with oth­er Lou stu­dents, I soon learned that this was not quite a re­flec­tion on any ser­i­ous de­fi­cien­cies on my part, but rather an ex­treme de­sire for math­em­at­ic­al pre­ci­sion on his. Once I em­braced this ec­cent­ri­city, I real­ized that there were many ways I could im­prove. Any com­pli­ments I re­ceive now on my math­em­at­ic­al writ­ing are en­tirely due to Lou’s di­li­gence in edit­ing my earli­er ef­forts, and I owe him yet an­oth­er debt of grat­it­ude for this ex­tra as­sist­ance.

Since de­fend­ing my thes­is, I have al­ways been able to de­pend on Lou, wheth­er for writ­ing let­ters of re­com­mend­a­tion or for guid­ance in im­port­ant ca­reer de­cisions. I view Lou not only as a ment­or and a bril­liant math­em­atician but, much more im­port­antly, as a friend who is al­ways will­ing to help in any way I need. Whenev­er my wife and I find ourselves geo­graph­ic­ally near to Lou, we make every ef­fort to share a meal or a cof­fee with him. He is a warm, genu­ine, kind, and thought­ful per­son whose im­pact on those who come in con­tact with him is im­me­di­ate and all-en­com­passing. The mod­el the­ory com­munity owes him a huge debt of grat­it­ude not only for his leg­acy of top-tier math­em­at­ic­al res­ults, but for the pos­it­ive in­flu­ence he has had on all of his stu­dents.

Isaac Gold­bring is an As­so­ci­ate Pro­fess­or of Math­em­at­ics at the Uni­versity of Cali­for­nia Irvine. His re­search is in ap­plic­a­tions of mod­el the­ory to a num­ber of dif­fer­ent areas of math­em­at­ics, most re­cently in­clud­ing op­er­at­or al­geb­ras and com­bin­at­or­i­al num­ber the­ory. He has been mar­ried to his wife Karina for twelve years and has two daugh­ters, Kaylee Si­ena (age 5) and Dani­ella Par­is (age 3).