#### by Isaac Goldbring

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:

- his work with Alex Wilkie, giving a nonstandard proof of Gromov’s theorem on polynomial growth [2];
- his paper with Karsten Schmidt on bounds in polynomial rings [1];
- 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).*