Celebratio Mathematica

Tom was a gi­ant in the field of prob­ab­il­ity, and was ap­poin­ted to the Na­tion­al Academy of Sci­ence, one of the very highest hon­ors in sci­ence. There were sev­er­al ma­jor ob­it­u­ar­ies writ­ten after Tom’s passing, not­ably here and here. For math­em­at­ic­al con­tent, see es­pe­cially this write-up. Here I will not du­plic­ate those ex­cel­lent ac­counts but will in­stead write some more per­son­al ma­ter­i­al, and as noted, use it as a basis for of­fer­ing some tips to fu­ture PhD stu­dents and their ad­visers.

My fel­low stu­dent Di­ane Schwartz and I were Tom’s first PhD stu­dents. He was an As­sist­ant Pro­fess­or at the time, and thus only a few years older than we were.

In his ob­it­u­ary of Tom, Prof. Rick Dur­rett, also a renowned prob­ab­il­ist, makes this ob­ser­va­tion: “The psy­cho­logy of how stu­dents choose their ad­visers is mys­ter­i­ous.” It wasn’t mys­ter­i­ous at all in my case. Even though my ul­ti­mate ca­reer goal was stat­ist­ics rather than prob­ab­il­ity, I was in awe of Tom’s bril­liance. I had taken two year-long gradu­ate courses from him, one on real ana­lys­is and the second on meas­ure-the­or­et­ic prob­ab­il­ity. To me, there was no ques­tion who my first choice for ad­viser would be.

Al­though the real ana­lys­is course was fairly straight­for­ward, taught out of Rud­in’s Real and Com­plex Ana­lys­is, it was eas­ily my fa­vor­ite among the three courses I took that first year at UCLA. I had at­ten­ded my loc­al state col­lege, Cal Poly Pomona, 10 minutes from home, as an un­der­gradu­ate. It was a non­re­search in­sti­tu­tion, and I was struck by Tom’s per­son­al be­ha­vi­or as a math­em­atician — his bear­ing, tone of voice and so on. I was even fas­cin­ated by Tom’s use of “Math­em­at­ics-ese,” a “lan­guage” spoken by math­em­aticians, who use words and phrases such as “res­ult”, “due to”, and “ar­gu­ment” in ways dif­fer­ent from nor­mal Eng­lish, e.g., “This res­ult, due to Jones, uses a com­pact­ness ar­gu­ment.”

That prob­ab­il­ity course was an ab­so­lute gem. While Tom did nom­in­ally fol­low the stand­ard text by Chung, he had his own al­tern­ate proofs for everything, usu­ally us­ing the tools of func­tion­al ana­lys­is, and al­ways el­eg­ant and in­sight­ful. At Cal Poly every course was taught straight out of the text­book, so I was ab­so­lutely blown away by that prob­ab­il­ity course. I still am. Tom truly epi­tom­ized the no­tion that good re­search and good teach­ing should go hand-in-hand.

This ideal has def­in­itely af­fected my own teach­ing and writ­ing. I’ve been for­tu­nate to win a couple of teach­ing awards at UCD, and in the awards ce­re­mony for the cam­pus-wide one, I made a point of cit­ing Tom in my speech as hav­ing deeply in­flu­enced my teach­ing. I’ve also been for­tu­nate that my book, Stat­ist­ic­al Re­gres­sion and Clas­si­fic­a­tion: from Lin­ear Mod­els to Ma­chine Learn­ing, was the re­cip­i­ent of the 2017 Ziegel Award. Though the book is not deeply math­em­at­ic­al at all, I could not res­ist giv­ing my own al­tern­ate proof of the Tower Prop­erty, us­ing Hil­bert space geo­metry in­stead of meas­ure the­ory — very Lig­gett-esque! I be­lieve many things in that book can be traced back at least in part to Tom’s in­flu­ence.

In pre­par­ing this es­say, I came across won­der­ful news (though again, be­lated) of the Lig­gett Teach­ing Awards, es­tab­lished ini­tially by a very gen­er­ous dona­tion by Tom’s wife Christina. She had been the de­part­ment ad­min­is­trat­ive as­sist­ant for gradu­ate mat­ters back when I star­ted grad school, the first per­son I talked to when I ar­rived. (She mar­ried Tom some years later.) So I felt like “the circle had been com­pleted.”

I was quite na­ive about re­search when I began work with Tom, na­iv­ete that was ex­posed even on my first week at UCLA. The head of the math grad pro­gram held some in­form­al meet­ings with new grad stu­dents, three or four at a time. He men­tioned that, in his role as head of the pro­gram, oc­ca­sion­ally a grad stu­dent would come to him in great dis­tress. He would try to re­solve the mat­ter, an emo­tion­ally drain­ing ex­per­i­ence for him. Re­spond­ing to that re­mark, one of my fel­low new grad stu­dents asked, “How does that af­fect your work?” That of course was a ref­er­ence to re­search work, but I didn’t get it. I thought, “But deal­ing with stu­dent prob­lems IS your work.” The en­su­ing con­ver­sa­tion then fi­nally made it clear to me that they were all talk­ing about re­search.

This then was the my first real­iz­a­tion that I had now entered a com­pletely dif­fer­ent world than the non­re­search en­vir­on­ment I’d had at the state col­lege.

I then began course­work, treat­ing it as “more of the same” fol­low­ing un­der­grad work, and to a large ex­tent, it was so. I’d had sev­er­al pro­fess­ors as an un­der­grad who as­signed very chal­len­ging proofs for home­work, and this turned out to be ex­cel­lent pre­par­a­tion for my grad courses at UCLA. I loved the work.

And though some grad stu­dents will be hor­ri­fied to read this, I loved pre­par­ing for the PhD qual­i­fy­ing ex­ams. Lots of chal­len­ging proofs, of course, which I pre­pared for by work­ing as many prob­lems as I could get my hands on — in books, old UCLA ex­ams and even old ex­ams at oth­er schools that I wrote away for. The sum­mers of prep­ping for the ex­ams (four writ­ten tests, taken two at a time) were the most en­joy­able of my in­tel­lec­tu­al life to date.

After I ap­proached Tom to ex­press an in­terest in work­ing with him, he had the tra­di­tion­al math re­sponse: He gave me a small re­search pro­ject as kind of a tri­al run, in which the po­ten­tial fac­ulty ad­viser thinks, “Let’s see what this stu­dent can do.” I solved the prob­lem and proudly showed my proof to Tom — only to find he was not com­pletely happy with it. My proof was cor­rect, he agreed, but he had had an­oth­er path in mind that he thought was bet­ter. View­ing the in­cid­ent today as a com­puter sci­ence pro­fess­or, it is a bit iron­ic, as my solu­tion was re­curs­ive, a tech­nique com­mon in com­puter sci­ence. But at any rate, this first in­tro­duc­tion to do­ing re­search un­der Tom’s high stand­ards was an eye open­er.

Tom was in­ter­ested in a re­search area known as in­fin­ite particle sys­tems, an ex­tremely ab­stract and tech­nic­ally dif­fi­cult stochast­ic pro­cess mod­el in­spired by, though much more gen­er­al than, stat­ist­ic­al phys­ics. Even prov­ing the ex­ist­ence of the pro­cess in­volves some very ad­vanced ma­chinery (de­veloped by Tom and UCLA col­league Mi­chael Cran­dall)!

The field it­self had ori­gin­ated from the le­gendary Frank Spitzer. Among oth­er things, Spitzer had de­vised a clev­er way to ana­lyze in­fin­ite particle sys­tems, via an aux­il­i­ary Markov chain. This trans­formed an in­tract­able un­count­ably in­fin­ite prob­lem to a tract­able count­ably in­fin­ite one. The read­er of this note need not un­der­stand those math­em­at­ic­al terms, but just keep the point in mind that Spitzer’s device is the key to ana­lyz­ing in­fin­ite particle sys­tems. And it will be key in the thoughts I share be­low on PhD edu­ca­tion.

At the time, Tom had just pub­lished his first ma­jor work in the in­fin­ite particle sys­tems area, on what he called the voter mod­el. In­stead of think­ing of the particles in phys­ics terms, say each one hav­ing a spin in one of two dir­ec­tions, Tom used a “voter” meta­phor. Each voter changes his/her polit­ic­al stance at ran­dom times, but with some sub­stan­tial like­li­hood of fol­low­ing the ex­ample of the voter’s neigh­bors.

Ever the rebel, I pro­posed to mod­el a set­ting in which each voter tends to vote the op­pos­ite of his/her neigh­bors. Tom liked the idea (he even men­tioned that his non­mathem­atician wife liked the idea), so I star­ted pur­su­ing it.

However, I was totally ig­nor­ant of a fun­da­ment­al prin­ciple of re­search: One builds on the pre­vi­ous work of oth­ers. Some­how I’d nev­er real­ized this ba­sic ten­et. I thus wrongly ap­proached re­search as just an­oth­er course home­work prob­lem, to be solved largely from first prin­ciples rather than on lever­aging pre­vi­ous lit­er­at­ure. I had star­ted read­ing Tom’s pa­pers, find­ing them tough go­ing, es­pe­cially after I en­countered an er­ror (which Tom con­firmed). So I then did a curs­ory read­ing, if that, and none at all of Frank Spitzer’s work.

What I ur­gently needed was to use Frank’s device of the aux­il­i­ary Markov chain. Without it, I spent sev­er­al un­pro­duct­ive months on wild goose chases, try­ing to de­vel­op my own ad hoc tools. Mean­while, Tom had a Sloan Fel­low­ship and was con­stantly trav­el­ing. No e-mail in those days, so he and I com­mu­nic­ated only oc­ca­sion­ally by “snail mail.” Even­tu­ally he asked, for­tu­nately very tact­fully, “Why aren’t you us­ing the aux­il­i­ary Markov chain?”, and I then fin­ished most of the dis­ser­ta­tion in a few months. Duh.

I pub­lished two pa­pers on my dis­ser­ta­tion work, view­able here and here. I then moved on to stat­ist­ics, e.g., this, and even­tu­ally com­puter sci­ence, e.g., this.

In ad­vising stu­dents who do re­search with me, wheth­er they be at the PhD, Mas­ter’s or even un­der­gradu­ate level, I as­sume they have tal­ent and in­terest, but oth­er­wise I make no as­sump­tions, al­ways aware that some stu­dents may be as clue­less as I was.

In most cases in which an un­der­grad does re­search with me, they are think­ing of grad school. I high­light this “dif­fer­ent world” point, which most are not aware of. Un­like the un­der­grad level, where many stu­dents choose a ma­jor for largely prac­tic­al reas­ons, most stu­dents in a good grad pro­gram have a keen in­tel­lec­tu­al curi­os­ity and in­terest in the sub­ject mat­ter. That already makes it a dif­fer­ent world! Sim­il­arly, I ex­plain that in con­trast to un­der­grad work, in which the pro­fess­or provides lots of struc­ture, in PhD re­search the stu­dent to a large ex­tent for­mu­lates his/her own prob­lems.

Giv­en my own dis­ser­ta­tion his­tory of ig­nor­ing the re­search lit­er­at­ure, I re­mind them of the phrase, “Stand­ing on the shoulders of gi­ants,” em­phas­iz­ing the im­port­ance of a thor­ough lit­er­at­ure search. For­tu­nately, Google makes that easi­er these days. I also warn them that many re­search pa­pers are poorly writ­ten, and give them tips for pick­ing out the ma­jor points. I stress that al­though the more in­de­pend­ence the stu­dent has in his/her work, the bet­ter, it is es­sen­tial to keep con­stant con­tact with the ad­viser.

In ad­di­tion, I talk about the prac­tic­al as­pects of pub­lish­ing, e.g., how to make one’s writ­ing en­ti­cing, how to judge the qual­ity of a journ­al or con­fer­ence and so on. Ac­tu­ally, as I have worked in sev­er­al fields, I feel I my­self am still learn­ing these things! I also am very frank with them in ex­plain­ing that the qual­ity of re­search re­view­ers var­ies greatly, and that one must learn to deal with re­jec­tion.

Though quite caring, Tom was a rather re­served man. In ad­di­tion, his bril­liance could be a bit in­tim­id­at­ing. He and I were nev­er close, and I be­lieve that may have been true for some of his oth­er stu­dents, at least for us early ones. We had worked with him for quite some time be­fore he shared his amaz­ing back­ground — he had grown up in South Amer­ica in a mis­sion­ary fam­ily, was flu­ent in Span­ish and so on. Who knew? I’m told that he did be­come closer with later stu­dents.

I do try to con­nect with my stu­dents on a per­son­al level. I con­sider my re­search stu­dents to be my col­leagues and friends, in spite of our hav­ing dif­fer­ent aca­dem­ic statuses, be­ing in dif­fer­ent life stages and so on. I try to put my­self in­to their shoes, and hope they have some un­der­stand­ing of be­ing in mine. Fi­nally, I value that I learn as much from them as they learn from me.

Tom and I lost touch over the years, but in 2009 I had a de­light­fully sur­pris­ing en­counter with a re­cent pa­per of his. At that time, a num­ber of us at UCD, from sev­er­al quite var­ied de­part­ments, had formed an in­form­al group on the sub­ject of ran­dom net­works. This is the “six de­grees of sep­ar­a­tion” field, of­ten used to mod­el so­cial net­works. One day we held a video con­fer­ence with re­search­ers at sev­er­al UC cam­puses, and someone from the UCLA So­ci­ology Dept. was speak­ing. Sud­denly, out of the blue, the speak­er men­tioned a the­or­em Tom had proven in sup­port of the speak­er’s so­ci­olo­gic­al mod­el! This was a pleas­ant sur­prise, but ac­tu­ally to be ex­pec­ted as the nat­ur­al evol­u­tion of Tom’s voter mod­els. It turned out that Tom had been work­ing in the field since at least as far back as 2004 (see T. Lig­gett and S. W. W. Rolles, “An in­fin­ite stochast­ic mod­el of so­cial net­work form­a­tion”, Stoch. Pro­cess. Ap­pl. 113:1 (2004), 65–80).

I guess that lack of con­tact with Tom is re­flec­ted in the fact that I did not learn of his passing un­til two full years later. I deeply re­gret not ever telling him of his pro­found in­flu­ence on me, which con­tin­ues to this day.

Tom once men­tioned a re­search­er who had claimed some im­press­ive re­search res­ult that had been ques­tioned by, among oth­ers, Paul Lévy, one of the pi­on­eers of mod­ern prob­ab­il­ity the­ory. Tom told me, “Well, Lévy was someone to be reckoned with, so this re­search­er hast­ily rechecked his res­ult, etc.” That phras­ing stuck with me, and I must say that Tom was def­in­itely someone to be reckoned with.

Norm Matloff is a pro­fess­or of com­puter sci­ence at the Uni­versity of Cali­for­nia, Dav­is. He was one of the founders of the De­part­ment of Com­puter Sci­ence, and also the De­part­ment of Stat­ist­ics. His cur­rent re­search in­terests are data pri­vacy and fair­ness in ma­chine learn­ing.