Celebratio Mathematica

My fam­ily al­ways lived in Aus­tin ex­cept when forced for a six-year re­treat (back) to Alabama; I was born there in 1928. This hard peri­od for my fath­er in­cluded plow­ing with a mule. But he was de­term­ined to get us, now five, back to Aus­tin and the Uni­versity of Texas (there was only one un­til dec­ades later.) My moth­er’s par­ents were Le­on Wal­ter and Lu­cre­tia Brown of Runge, Texas, and my fath­er’s were Joseph Marble and Mary Mi­chael (Fones) Wil­li­ams, of Mont­gomery Alabama. The Great De­pres­sion, World War II and ri­gid Jim Crow defined the young lives of my two broth­ers and me.

I gradu­ated from high school — neither par­ent had been to high school — in 1945 and entered the Uni­versity. All three boys ful­filled our fath­er’s wish and gradu­ated from the Uni­versity of Texas: me in 1948, BA in Math­em­at­ics; El­gin, Jr. with a BA in Eco­nom­ics, 1942, Mas­ters 1943; Joe (after a stint in the Navy), with a BS in Ar­chi­tec­ture, 1949. My broth­ers were very im­port­ant to me.

My cur­riculum at UT was mostly phys­ic­al sci­ence but in­cluded eco­nom­ics, psy­cho­logy and ad­vanced Eng­lish courses. By my sopho­more year I was grad­ing math­em­at­ics home­work and had a key to the Math­em­at­ics De­part­ment of­fice. Cal­cu­lus un­der R. L. Moore had only two stu­dents dur­ing the first semester but the num­ber in­creased to sev­en in the second (an ef­fect of World War II). Next was Moore’s (un­der­gradu­ate) ana­lys­is. By this time I was much taken by the “Moore meth­od” of teach­ing and cer­tain that I wanted to be a math­em­atician. Along with my close friends, El­don Dyer and Mary-Eliza­beth Ham­strom, I was ready to show the world the Moore meth­od (now uni­ver­sally ad­mired as in­quiry-based learn­ing). The math com­munity in­cluded — at some stage of their ca­reers — Mary El­len Es­tell (later, Rud­in), R. D. An­der­son, B. J. Ball, R H Bing, Cecil Bur­gess and Bur­ton Jones. I had a year in gradu­ate school, con­tinu­ing Moore’s se­quence of courses, on which I spent most of my time.

In the fall of 1948 I ap­plied for sup­port at the two uni­versit­ies sug­ges­ted by Moore; I ac­cep­ted a po­s­i­tion at the Uni­versity of Vir­gin­ia where I con­tin­ued my gradu­ate work un­der Gor­don Why­burn. There are vari­ous stor­ies about this move: one was that Moore hoped I would ma­ture, and re­turn for my Ph.D. An­oth­er, see ([e7], p. 232), con­cerned my mod­ern dan­cing which had be­come im­port­ant to me. Could I have been a pro­fes­sion­al mod­ern dan­cer? I had a sum­mer fel­low­ship to study dance un­der Han­ya Holm, but in­stead ac­cep­ted a sum­mer job teach­ing math — which was can­celed without warn­ing.

Vir­gin­ia was a ma­jor change and at first I res­isted the more bal­anced pro­gram un­der Why­burn. With mo­mentum from Moore, I tried to prove some fool­ish things about com­plexes, but did solve a (dy­nam­ics) prob­lem sug­ges­ted by Gust­av Hed­lund [1]: I re­in­ven­ted the dy­ad­ic solen­oid and doub­ling dif­feo­morph­ism; later E. E. Floyd showed me how in­verse lim­its were good for such ex­amples. An­oth­er schol­ar sug­ges­ted a point­wise defin­i­tion for loc­al con­trac­tions, pl con­trac­tion: I showed how poorly this worked; in par­tic­u­lar I put a to­po­lo­gic­ally equi­val­ent met­ric on the circle [5] so that an ir­ra­tion­al ro­ta­tion was a pl con­trac­tion but not a pl iso­metry. I began to have more con­tact with E. E. Floyd which con­tin­ued for many years. Also Dav­id Lowdensla­ger and Bob Plun­kett (fel­low gradu­ate stu­dents) be­came im­port­ant friends.

In 1952 I mar­ried Cath­er­ine Barnes — whose BA was a double ma­jor in paint­ing and art his­tory. She con­tin­ued to paint throughout our twenty years of mar­riage which ce­men­ted my in­terest in art and the chem­istry of oil paint­ing. My thes­is was “Re­duc­tion of open map­pings” [2], [3] and my Ph.D. was gran­ted in 1954.

Bob Plun­kett and I ac­cep­ted as­sist­ant pro­fess­or po­s­i­tions at the newly cre­ated Flor­ida State Uni­versity. I left this po­s­i­tion for a one-year ap­point­ment to the Math­em­at­ics De­part­ment of The Uni­versity of Wis­con­sin. This con­tin­ued my con­tact with Pro­fess­or RH Bing, whom I knew at Texas. I con­cen­trated on his on­go­ing work on sur­faces: that a 2-sphere to­po­lo­gic­ally em­bed­ded in a 3-sphere can be ap­prox­im­ated by one which has a three-di­men­sion­al disk on one side; Bing solved it him­self. I was at­trac­ted to and wrote a pa­per on di­men­sion the­ory [4].

In 1956 I be­came As­sist­ant Pro­fess­or at Purdue Uni­versity. I took the sum­mer of 1957 off and toured Europe with my wife Cath­er­ine. My daugh­ter El­len Lind­say was born in La­fay­ette, In­di­ana, April of 1958.

There was little to­po­logy at Purdue and I began work­ing with the group around Lam­berto Cesari and Wendell Flem­ing [6]. I also made con­tact with Mo Hirsch and Steve Smale, who were nearby at the Uni­versity of Chica­go. I got a one-year NSF fel­low­ship to at­tend the In­sti­tute for Ad­vanced Study (ex­ten­ded to a second year.)

This was a real learn­ing peri­od — much of it with Smale, Hirsch and Frank Ray­mond. I began a long peri­od work­ing with Ray­mond on the Hil­bert–Smith con­jec­ture [8], [9]. The IAS pro­fess­ors helped me get a two-year post-doc at the Uni­versity of Chica­go. I fi­nally real­ized that I needed to learn lin­ear al­gebra and got to teach it. Chica­go was a friendly place and I learned a lot. I had some suc­cess here [4] but a long pro­ject con­cern­ing the agree­ment of co­homo­logy di­men­sion with to­po­lo­gic­al di­men­sion failed be­cause of an ele­ment­ary and ser­i­ous mis­take.

While I was at the Uni­versity of Chica­go, two un­der­gradu­ates, Rich Gold­stone and Steve Der­enzo, in­tro­duced me to rock climb­ing. They had dis­covered a rock wall near the uni­versity that made for chal­len­ging tra­verses. And on week­ends, there was the fine wall at Dev­il’s Lake, in Wis­con­sin. Like many climbers in these early days, Gold­stone is a math­em­atician, as are Mike Freed­man, John Gill, Rob Kirby, John Mil­nor and George de Rham, all of whom I have had great pleas­ure climb­ing with. De Rham wrote the guide book to l’Ar­gen­tine (a moun­tain in the Vaud Alps, Switzer­land). For the next four dec­ades I was an avid climber. Later, trips in the US and to roughly ten coun­tries for math­em­at­ics meet­ings, I eas­ily found climbers, moun­tains, rock and snow for my main sport.

I moved to nearby North­west­ern Uni­versity as an as­so­ci­ate pro­fess­or without ten­ure, where I fin­ished my work on the Hil­bert–Smith con­jec­ture. I rode the Chica­go “L” down to the col­loquia at the Uni­versity of Chica­go: Smale gave one about hy­per­bol­ic sub­sets of a man­i­fold un­der a dif­feo (Ax­iom “A”). Ex­ample: the doub­ling dif­feo on the dy­ad­ic solen­oid (DS), just as in [1]. Sud­denly I was at home in dy­nam­ic­al sys­tems. The ad­di­tion­al struc­ture suited me. Strug­gling with the zeta func­tion of the dy­ad­ic solen­oid, I was de­lighted to un­der­stand the fol­low­ing: a zeta func­tion is ra­tion­al if and only if there are two matrices \( A \) and \( B \) such the num­ber of fixed points of \( f^n = |f^n|= \operatorname{tr}A^n- \operatorname{tr}B^n \).

I used a pre­lim­in­ary ver­sion of Smale’s “Dif­fer­en­tial dy­nam­ic­al sys­tems” pa­per [e2] to teach a sem­in­ar. Paul Sch­weitzer was vis­it­ing North­west­ern at this time and was en­thu­si­ast­ic; we also thought about Smale’s newly for­mu­lated “stable man­i­fold the­or­em”, later proved by Hirsch, Pugh, and Shub (even­tu­ally there were sev­er­al oth­er proofs). The cli­max of this sem­in­ar was my in­tro­duc­tion of one-di­men­sion­al “branched man­i­folds” (later giv­en the great name “train tracks” by Thur­ston) which I used to char­ac­ter­ize all one-di­men­sion­al hy­per­bol­ic at­tract­ors [10].

Smale was en­thu­si­ast­ic and in­vited me to vis­it Berke­ley. He had con­struc­ted the “\( DA \)” dif­feo­morph­ism of the tor­us; I saw its beauty and showed him how it fit in­to my char­ac­ter­iz­a­tion [10]. The \( DA \) was clearly struc­tur­ally stable and a great ad­di­tion to his “horse shoe” from sev­en years earli­er. But I had no idea how much of my work in the next years would in­volve this ex­ample. (My vis­it happened to co­in­cide with thou­sands march­ing through San Fran­cisco to protest the war in Vi­et­nam.)

This was the be­gin­ning of an ex­cit­ing era for dy­nam­ics and most of my life’s work. At this time I began to work on the Weil-type zeta func­tion in­tro­duced by Artin and (Barry) Mazur and a high­er-di­men­sion­al ver­sion of [10], which even­tu­ally ap­peared in 1971 [14].

I com­puted the peri­od­ic points in sev­er­al ex­amples [10] and saw the for­mula \[ |f^n| = [\operatorname{tr} (f_*^n) + \operatorname{tr} (\tilde f_*)^n]/2, \] where \( \tilde f \) is the lift to the double cov­er ori­ent­ing the ex­pand­ing bundle of \( df \), \( |f| \) is the num­ber of fixed points of \( f \), and \( * \) in­dic­ates the map in­duced on ho­mo­logy. This led to the pa­per [11] show­ing that all hy­per­bol­ic at­tract­ors had ra­tion­al zeta func­tions. I had a sab­bat­ic­al year at the Uni­versity of Geneva where I com­pleted that work.

In fact this led to eight­een years of work (see [e6]) on the “Wil­li­ams prob­lem.” Mean­while, at War­wick, Bill Parry found the er­ror in [13] while we were work­ing on fi­nite Markov chains [24]. In a series of fine pa­pers, John Wag­on­er, Kim and Roush showed that the strong ver­sion is in fact stronger.

The Tur­bu­lence Sem­in­ar was or­gan­ized in Berke­ley fol­low­ing the sug­ges­tion by Ruelle and Takens [e3] of the “strange at­tract­or” as a mod­el for tur­bu­lence: a con­tem­por­ary defin­i­tion of strange at­tract­or was an at­tract­or that is is neither fi­nite not a sub­man­i­fold. This in­cluded those I had worked on. It also in­cludes the Lorenz at­tract­or, but this last was es­sen­tially un­noticed un­til Guck­en­heimer’s “A strange strange at­tract­or” ap­peared as a pre­print [e4]. My pa­per [26] was next; it made use of branched man­i­folds [18] and gave ad­di­tion­al in­sight in­to the Lorenz at­tract­or. Our mod­el was re­ferred to as the geo­met­ric Lorenz at­tract­or. I in­tro­duced a zeta func­tion to char­ac­ter­ize them, at the Tur­bu­lence Sem­in­ar [23]. I later wrote about these knots with Joan Birman [28].

At this point we had no com­plete proof con­nect­ing the geo­met­ric LA of Guck­en­heimer and my­self; I had done lots of veri­fic­a­tion. For ex­ample, I would “loc­ate” a peri­od­ic of say five or six trips around the peri­od­ic points, then James Curry would find it with an ODE solv­er. I wrote a pro­gram to “find” the strong stable man­i­folds and the pic­tures were strik­ing.

Curry is a long­time mem­ber of the Math­em­at­ics De­part­ment of the Uni­versity of Col­or­ado, which I had the hon­or of vis­it­ing as hold­er of the Stan­isław Ulam po­s­i­tion. It was here that I first met Marcy Barge, whose thes­is was un­der Wes Wilson. Later Marcy and I worked to­geth­er on tiling the­ory.

At the Tur­bu­lence Sem­in­ar, I showed that the peri­od­ic or­bits cor­res­pon­ded one-to-one with the aperi­od­ic words in two let­ters, say \( \{x,y\} \), and poin­ted out that they are mostly knot­ted. All my life I have liked knots, how to tie them and where to use them, but nev­er as a math­em­atician. Now I had a chance and spent many hours ty­ing these “Lorenz” knots, us­ing the sym­bol­ic dy­nam­ics I had found. I built a “knot loom” us­ing this sym­bol­ic dy­nam­ics for ty­ing them — and the very first time I used it, the knot fell in­to a shape that I saw was im­port­ant: each Lorenz knot is a pos­it­ive braid with a full twist. Shortly after this, I met Joan Birman, whose knot work was largely on braids. Not­ably every knot can be writ­ten as a braid, and the Lorenz knots are giv­en as braids. To­geth­er we wrote [28] on Lorenz knots and [29].

Earli­er, three new dy­nam­icists from Berke­ley, Clark Robin­son, Shel­don Ne­w­house and John Franks came to North­west­ern. To­geth­er we began The Mid­west­ern Dy­nam­ic­al Sys­tems Sem­in­ar with Joel Rob­bin and Charles Con­ley (at Wis­con­sin) which met weekly for a while. This was the be­gin­ning of MWDS which re­mains im­port­ant 50 years later; I am proud of this — es­pe­cially, my listen­ing closely to the much young­er Ne­w­house, Robin­son and Franks. True to his many im­port­ant con­tri­bu­tions to dy­nam­ics, Robin­son kept the re­cords and mail­ing list. Also im­port­ant were the series of oth­er young dy­nam­icists that had a year or two at NU: Bob Devaney, Paul Blan­chard, and Lai-Sang Young. I was in­vited to present a pa­per at the In­ter­na­tion­al Con­gress in Nice, 1970.

Over the next three dec­ades, I took part in many meet­ings for dy­nam­ic­al sys­tems. I had al­ways wanted to vis­it Brazil and my four vis­its there were ful­filling. In 1971 at the In­sti­tuto de Matemática Pura e Ap­licada (IMPA) I gave my slightly sim­pli­fied ver­sion of the \( DA \), and a pa­per on the com­pos­i­tion of con­trac­tions that I had worked out on a train ride from Geneva to War­wick Uni­versity. Shel­don Ne­w­house and I drove our “fus­cas” through the jungle, to the month-long in­ter­na­tion­al meet­ing in Sal­vador, Bahia (Brazil) in 1971.

Knot the­ory had been an act­ive sec­tion of math­em­at­ics for over fifty years when Vaughan Jones found a new poly­no­mi­al in­vari­ant. Defined via \( C^* \) al­geb­ras, many knot the­or­ists figured out, at the about the same time, how to com­pute it via stand­ard “knot moves”. My col­league John Franks per­suaded the un­der­grad Henry Ce­jt­in, to write a pro­gram to com­pute this. After many com­pu­ta­tions we no­ticed a cer­tain ex­po­nent was a lower bound for the braid in­dex (BI) of a knot [33]. As every knot can be writ­ten as a braid, the smal­lest num­ber of strands for such a braid is a knot in­vari­ant. This makes BI a use­ful in­vari­ant for the first time.

That one-di­men­sion­al at­tract­ors have non­trivi­al ho­mo­logy was proved in my 1967 pa­per in To­po­logy [10]. But for ex­pand­ing at­tract­ors, a new idea is needed: Den­nis Sul­li­van and I showed this us­ing co­chains with bounded coef­fi­cients [21]. Though I had known Den­nis for many years, I had the most con­tact with him while a mem­ber of the In­sti­tut des Hautes Études Sci­en­ti­fiques. This also gave me a lot of con­tact with René Thom. His friend­ship has been im­port­ant to me.

In­spired by this the­or­em which unites two areas of math­em­at­ics, I began work­ing on tilings which led to [34], [35] and [36].

The math­em­atician, Kar­en Uh­len­beck and I got to­geth­er in 1975; the out­doors with back-pack­ing, and moun­tain-climb­ing has been con­tinu­ous and great to­geth­er; we are mar­ried. I left North­west­ern Uni­versity to ac­cept a po­s­i­tion at the Uni­versity of Texas in 1986. Dur­ing this peri­od we spent sum­mers in the West. We now have a house in Boze­man, Montana and been giv­en much hos­pit­al­ity by Montana State Uni­versity. We lived in a house in the Texas Hill Coun­try, sur­roun­ded by trees. After a vis­it to Hawaii, I built an out­door shower with cir­cu­lat­ing hot wa­ter. I re­tired in 1999 and con­tin­ued as an Ed­it­or for dy­nam­ics for Trans­ac­tions of the AMS for sev­er­al years. We moved to Prin­ceton when Kar­en re­tired in 2014. Un­til this move we have al­ways had at least two cats. We are as busy as ever with our new con­nec­tions to the In­sti­tute for Ad­vanced Study. The IAS has been very ac­com­mod­at­ing; Kar­en is a Vis­it­ing Dis­tin­guished Pro­fess­or.

As I write this we are hope­ful, along with mil­lions of oth­ers, of liv­ing through the COV­ID-19 coronavir­us pan­dem­ic.