Celebratio Mathematica

This bio­graphy is based on an in­ter­view with Krystyna Ku­per­berg on Feb­ru­ary 19, 2020. Quoted words are Krystyna Ku­per­berg’s own un­less oth­er­wise noted.

Krystyna was born on 17 Ju­ly, 1944 in Tarnów in south­ern Po­land. Her par­ents were Jan Try­bulec and Bar­bara Kur­lus. They were both phar­macists and owned a phar­macy in Szczu­cin, a small town near Tarnów. Both had mas­ter’s de­grees in phar­ma­ceut­ic­al sci­ence. They met as res­id­ents in a phar­macy in the city of Bytom in Up­per Silesia, which among oth­er things provided emer­gency sup­plies dur­ing coal min­ing ac­ci­dents in the many mines in the area.

Krystyna’s pa­ternal grand­fath­er had a shoe re­pair shop and her pa­ternal grand­moth­er had a dairy farm. “My moth­er’s fath­er was Walenty Kur­lus. They lived near Pozn­ań and had a coun­try store, like the stores in old West­ern movies, selling grain and food and some ma­chinery.”

Krystyna had a broth­er, the late An­drzej Try­bulec, who was three and a half years older. He was ad­mit­ted to the Med­ic­al Uni­versity in Gdańsk, but his med­ic­al ca­reer ended early after he went on a school-sponsored trip to a con­fer­ence in Warsaw on cy­ber­net­ics. There he met Stan­isław Lem, the sci­ence fic­tion writer, and he be­came in­ter­ested in ar­ti­fi­cial in­tel­li­gence. An­drzej ma­jored in philo­sophy be­cause that de­part­ment al­lowed him to study AI. He took a math­em­at­ics minor, which he liked even more, so he con­tin­ued to gradu­ate school in math­em­at­ics at the Uni­versity of Warsaw. An­drzej ob­tained a Ph.D. in to­po­logy un­der Ka­rol Bor­suk. After ob­tain­ing his Ph.D., An­drzej ob­tained fac­ulty po­s­i­tions in Po­land and did re­search in com­pu­ter­ized form­al­iz­a­tion of math­em­at­ics; he in­ven­ted a proof as­sist­ant sys­tem called Miz­ar.

Krystyna her­self was drawn to math­em­at­ics from a young age. “I was in­ter­ested in math as far back as I can re­mem­ber. When I was around sev­en or eight, I would some­times help my moth­er with the phar­macy book­keep­ing. My job was to tally the daily re­ceipts, adding columns of about 30 entries by hand. Much later, I learned that this was just to keep me busy; but to my moth­er’s sur­prise, she could use my work as a backup, to check her own cal­cu­la­tions.”

“A few times, a teach­er would take me from my second-grade classroom to an up­per grade only to solve a prob­lem on the board, to em­bar­rass the oth­er stu­dents: ‘You could not solve a prob­lem that this little kid can!’ ”

“We had grades one through sev­en, and then four years of high school. We did not have cal­cu­lus in high school, but we did have lo­gic and truth tables in tenth grade, and three-di­men­sion­al geo­metry (which I ab­so­lutely en­joyed) in eighth grade. Re­lated to this was a draw­ing class in fifth grade, teach­ing per­spect­ive draw­ing. I look at this as my first ser­i­ous geo­metry ex­per­i­ence.”

Krystyna went to a board­ing school in Tarnów for eighth grade and then the fam­ily moved north and she fin­ished high school in Gdańsk in 1962. She then ap­plied to the Uni­versity of Warsaw. There were en­trance ex­ams in math­em­at­ics and phys­ics, and an un­ser­i­ous Rus­si­an lan­guage ex­am that every­one passed. In­stead of a bach­el­or’s de­gree as in the United States, the Uni­versity of Warsaw offered a five-year mas­ter’s de­gree, which Krystyna fin­ished in four years.

In her second semester at Warsaw, 1962–63, Krystyna at­ten­ded an ad­vanced class taught by Bor­suk, where she met Włodz­i­mierz (Włodek) Ku­per­berg, who was three years fur­ther along. “Bor­suk was de­fin­ing sim­pli­cial ho­mo­logy, care­fully de­fin­ing everything pre­cisely without go­ing too fast. He was very calm and care­ful with the de­tails. Un­der­gradu­ate stu­dents in­ter­ested in to­po­logy were wel­come to Bor­suk’s sem­in­ar at Warsaw Uni­versity. At the be­gin­ning of the sem­in­ar, he as­signed stu­dents to give present­a­tions later, of­ten present­a­tions on pa­pers. If any­one solved an open prob­lem, they had pri­or­ity. He would pro­pose twenty or thirty prob­lems in shape the­ory or re­tracts, which were top­ics on his mind at the time, and some were very easy and some very dif­fi­cult. If no one solved a prob­lem, then we would go through pa­pers. He gave me a pa­per of Steph­en Smale [e3] on a Vi­et­or­is the­or­em for ho­mo­topy (not Čech ho­mo­topy which I stud­ied later), so that was the first re­search pa­per that I read.”

“I mar­ried Włodek in 1964, ob­tained my MS de­gree in 1966, and star­ted my Ph.D. with Bor­suk.” From 1966 to 1969, Krystyna taught courses in a po­s­i­tion sim­il­ar to an Amer­ic­an TA­ship.

Krystyna and Włodek’s son Greg was born in 1967. Their daugh­ter Anna was born in Septem­ber 1969, just after the fam­ily emig­rated to Sweden in Au­gust. “We were kicked out!” In 1968, the gov­ern­ment in Po­land began a cam­paign to pres­sure Jews to leave Po­land on the pre­text that their real loy­alty was to Is­rael. If they emig­rated, as most of the re­main­ing Pol­ish Jews did, their Pol­ish cit­izen­ship was also re­voked. Włodek is of Jew­ish des­cent, al­though his par­ents and sib­lings were all sec­u­lar and lived as as­sim­il­ated Poles. His fam­ily had sur­vived World War II by flee­ing east to the So­viet Uni­on, where he was born in 1941; they re­turned to Po­land soon after the war. Dur­ing the later anti-Jew­ish cam­paign, Włodek’s par­ents and all of their chil­dren and grand­chil­dren left Po­land per­man­ently. Krystyna and Włodek changed their life plans fairly quickly, to­ward the end of the emig­ra­tion win­dow. They con­sidered that they might later face oth­ers forms of per­se­cu­tion and be un­able to leave Po­land. They ar­ranged to go to Sweden, which was one of the coun­tries that ac­cep­ted Pol­ish Jews in their situ­ation. Włodek had his Ph.D. by then, and began teach­ing at Stock­holm Uni­versity in Janu­ary 1970. Krystyna re-en­rolled in gradu­ate school at Stock­holm Uni­versity with her own teach­ing du­ties, again sim­il­ar to those of a TA­ship.

Three years later in 1972, Włodek was in­vited by An­drew Lelek (who had been a stu­dent of Bron­isław Knas­ter at Wro­claw) to the spring to­po­logy con­fer­ence in Hou­s­ton. This led to a vis­it­ing po­s­i­tion for a year for Włodek at the Uni­versity of Hou­s­ton. Mean­while, Krystyna had mostly fin­ished a Ph.D. thes­is, but the fi­nal de­gree re­quire­ments were com­plic­ated. She was in­ter­ested in geo­met­ric to­po­logy and the Uni­versity of Hou­s­ton was not the best fit. She vis­ited Rice Uni­versity, where she met with Mor­ton Curtis, who was in­ter­ested in her work on Vi­et­or­is-type the­or­ems in shape the­ory. Curtis said to talk to John Hempel, who was in charge of gradu­ate stu­dents. Since Krystyna had let­ters from Bor­suk and Stan­isław Mazur, the dir­ect­or of the Math In­sti­tute at Warsaw Uni­versity, and two pub­lished pa­pers, she was ad­mit­ted on the spot with a tu­ition waiver.

Wil­li­am Jaco be­came Krystyna’s thes­is ad­viser at Rice, partly due to her bud­ding in­terest in 3-man­i­folds, and she ob­tained her Ph.D. from Rice after two years. Krystyna and Włodek then sought uni­versity po­s­i­tions in math­em­at­ics, but they faced the bad job aca­dem­ic mar­ket in the United States of the 1970s. Au­burn Uni­versity in Alabama offered a solu­tion to their two-body prob­lem; they ar­rived in Au­burn as new fac­ulty mem­bers in 1974. At first, their teach­ing load was quite heavy: three courses per quarter, in­clud­ing courses that met five hours a week. But with­in sev­er­al years, the teach­ing load at Au­burn lessened.

When Krystyna and Włodek ar­rived at Au­burn in 1974, the math­em­at­ics de­part­ment had a close-knit so­cial circle with many house parties and oth­er so­cial events. Many of the chil­dren of the math fac­ulty also be­came close friends. A mem­or­able party was held to cel­eb­rate their nat­ur­al­iz­a­tion as Amer­ic­an cit­izens in 1979.

While Krystyna and Włodek both star­ted out in gen­er­al to­po­logy in the Bor­suk school, Włodek turned his at­ten­tion to con­vex geo­metry in his later ca­reer at Au­burn. Many of his later pa­pers were with Krystyna and Włodek’s friend and Au­burn col­league, An­dras Bezdek.

Krystyna and Włodek stayed at Au­burn Uni­versity un­til their re­tire­ment in 2020 after 46 years of ser­vice. They de­cided to re­tire in Decem­ber 2019; the pan­dem­ic that came the next year made their de­cision for­tu­it­ous.

Many (but not all) of Krystyna’s main res­ults in math­em­at­ics are counter­examples. She has found counter­examples in sev­er­al areas of math­em­at­ics, all with­in geo­metry and to­po­logy taken broadly.

In her Ph.D. work, Krystyna es­tab­lished Vi­et­or­is-type map­ping the­or­ems for Vi­et­or­is-Čech ho­mo­logy in the con­text of Bor­suk shape the­ory [2], [15]. She also found an in­ter­est­ing counter­example, that the Hurewicz iso­morph­ism the­or­em is not stable un­der in­verse lim­its; as a co­rol­lary, the Hurewicz map is not an iso­morph­ism in shape the­ory [3]. She had some earli­er pos­it­ive res­ults along the same lines [1].

In 1980, Krystyna with Coke Reed con­struc­ted a fixed-point-free smooth flow in \( \mathbb{R}^3 \) with uni­formly bounded tra­ject­or­ies [4]. This counter­example answered a ques­tion of Ulam (prob­lem 110) in The Scot­tish Book [e14]. As prom­ised in the state­ment of the ques­tion, Ulam re­war­ded Krystyna and Coke with a bottle of wine. In this re­search pro­ject, they re­dis­covered what is now called the “Wilson plug”.

Krystyna and Coke later re­fined their solu­tion us­ing the “Sch­weitzer plug”, from Sch­weitzer’s \( C^1 \) counter­example to the Seifert con­jec­ture [5], [e4]. The concept of a plug is use­ful in both Ulam’s prob­lem and in the Seifert con­jec­ture. Krystyna thus be­came fa­mil­i­ar with the Seifert con­jec­ture while work­ing on Ulam’s prob­lem, al­though she only star­ted to work on the Seifert con­jec­ture it­self rather later, in 1993.

Krystyna answered a ques­tion of Knas­ter by con­struct­ing a Peano con­tinuum (a com­pact, con­nec­ted, loc­ally con­nec­ted, met­riz­able space) which is to­po­lo­gic­ally ho­mo­gen­eous (the group of homeo­morph­isms is trans­it­ive), but not bi­ho­mo­gen­eous (there are pairs of points that can­not be swapped by a homeo­morph­ism) [7]. The ex­ample is loc­ally the Cartesian product of a man­i­fold and the square of a Menger sponge \( M \), and makes use of the to­po­lo­gic­al ri­gid­ity of the fac­tor­iz­a­tion of \( M^2 \) as \( M \times M \).

Krystyna found a pack­ing of in­fin­ite round cyl­in­ders in \( \mathbb{R}^3 \) with pos­it­ive dens­ity such that no two cyl­in­ders are par­al­lel [6]. It is easy to see that the cross-sec­tion­al shape of the cyl­in­der does not mat­ter; it can be any com­pact planar do­main. The ex­ample can be called “a dense pile of skew pen­cils”.

An­swer­ing a ques­tion of Mike Freed­man, Krystyna con­struc­ted a fi­nite set of points on the sur­face of a round ball in \( \mathbb{R}^3 \) such that the shortest tree con­nect­ing them is knot­ted [13]. It is easy to find a counter­example in a suf­fi­ciently flat el­lips­oid, but in a round ball there is less avail­able room. Krystyna’s idea to cre­ate ex­tra room is to an­chor the ends of the knot on the bound­ary of the ball with split teth­er lines.

Krystyna’s most fam­ous counter­example is a smooth counter­example to the Seifert con­jec­ture. In 1950, Her­bert Seifert [e1] asked wheth­er every con­tinu­ous vec­tor field on \( S^3 \) has a closed (or peri­od­ic) tra­ject­ory. Al­though the ques­tion was posed neut­rally, the as­ser­tion that it is al­ways true came to be known as the Seifert con­jec­ture. Seifert him­self es­tab­lished his name­sake con­jec­ture in a neigh­bor­hood of the Hopf flow. Much later, Seifert’s res­ult was gen­er­al­ized to the Wein­stein con­jec­ture for con­tact flows [e7], which was then proven by Hofer in the case of \( S^3 \) [e10] and later for all con­tact 3-man­i­folds by Taubes [e13]. At a more ele­ment­ary level, the Brouwer fixed point the­or­em for \( D^2 \) also es­tab­lishes the Seifert con­jec­ture in many cases; it is a ba­sic ob­struc­tion to pos­sible counter­examples.

Without a con­tact struc­ture as­sump­tion, the Seifert con­jec­ture ad­mits counter­examples. Two dec­ades after Seifert’s pa­per, Paul Sch­weitzer found the first counter­example [e4], with the pro­viso that his flow is only \( C^1 \) dif­fer­en­ti­able. This led to an ef­fort to modi­fy his con­struc­tion to im­prove its dif­fer­en­ti­ab­il­ity; even­tu­ally, Jenny Har­ris­on showed that a more com­plic­ated ver­sion of Sch­weitzer’s con­struc­tion can be \( C^{2+\delta} \) [e8], [e9]. Then in 1993, Krystyna found a \( C^\infty \) counter­example us­ing a com­pletely dif­fer­ent con­struc­tion [10]. Her con­struc­tion can be made real ana­lyt­ic; in an­oth­er vari­ation, it can be real­ized by a flow which is piece­wise con­stant re­l­at­ive to a tri­an­gu­la­tion of \( S^3 \) [11].

The key con­struc­tion be­hind all known counter­examples to any ver­sion of either the Seifert con­jec­ture or Ulam’s prob­lem is a spe­cial flow in a cyl­in­der \( D^2 \times I \) called a plug \( P \). Such a flow is ver­tic­al on the bound­ary of the cyl­in­der; it also be­haves as the ver­tic­al flow in the sense that any tra­ject­ory that enters the bot­tom also leaves the top in the same po­s­i­tion, if it leaves at all. \( P \) is also re­quired to trap some of the tra­ject­or­ies that enter the bot­tom for in­fin­ite pos­it­ive time, while oth­er tra­ject­or­ies that were trapped for in­fin­ite neg­at­ive time exit the top. Cop­ies of a plug \( P \) can be in­ser­ted in­to an­oth­er flow \( F \) on a 3-man­i­fold \( M \), so that every tra­ject­ory is either trapped in one copy of \( P \) or trans­itions between two of them. In this way, one can over­throw the glob­al dy­nam­ic of the am­bi­ent flow \( F \), and render the to­po­logy of \( M \) ir­rel­ev­ant. All that is left to ana­lyze is the struc­ture of the flow in­side \( P \).

Any plug must have at least one com­pact min­im­al set of or­bits that are trapped for all times. In the ori­gin­al Wilson plug, the min­im­al sets con­sist of two circles. Sch­weitzer re­placed these two circles with two Den­joy lam­in­a­tions, which is what lim­its the dif­fer­en­ti­ab­il­ity of his plug. Krystyna in­stead in­ser­ted a Wilson plug partly in­to it­self so that it breaks its own or­bits. She showed that if this is done in the right way, then no new peri­od­ic or­bits are cre­ated. In­stead, the self-in­ser­tion cre­ates a new type of min­im­al set whose to­po­logy and dy­nam­ics has since been stud­ied [e15], [11].

I first heard of Krystyna’s dra­mat­ic counter­example when I got an email from her son Greg which simply said: “My mom has a \( C^{\infty} \) counter­example to the Seifert con­jec­ture”. Has a son ever been able to write such an email about his moth­er (in math­em­at­ics)?

Soon after, Bill Thur­ston, who was then dir­ect­or of MSRI, gave a lec­ture to a fas­cin­ated audi­ence about this strik­ing res­ult, where he ob­served that it should also be real ana­lyt­ic. Like the Wilson plug be­fore it, all of the ex­pli­cit data in Krystyna’s plug is real ana­lyt­ic. However, it is sur­pris­ing that one can in­sert any non­trivi­al plug in­to an­oth­er flow in the real ana­lyt­ic cat­egory; this is­sue also arises in Krystyna’s self-in­ser­tion. Thur­ston’s point was that these cut-and-paste con­struc­tions can be made real ana­lyt­ic us­ing the Mor­rey–Grauert the­or­em. The real ana­lyt­ic counter­example is in her joint pa­per with her son, Greg Ku­per­berg [11]. Be­sides the real ana­lyt­ic form, this pa­per also de­scribes a piece­wise-lin­ear form of her plug and counter­example on \( S^3 \), giv­en by a piece­wise-con­stant vec­tor field re­l­at­ive to some tri­an­gu­la­tion.

Krystyna’s res­ult later led to oth­er hon­ors such as a Bourbaki sem­in­ar [e11] and an ICM lec­ture [12].

Not all of Krystyna’s ma­jor works are counter­examples. One the­or­em in par­tic­u­lar is re­lated to the Cartwright–Lit­tle­wood the­or­em. The Cartwright–Lit­tle­wood the­or­em says that if \( f \) is an ori­ent­a­tion-pre­serving homeo­morph­ism of the plane \( \mathbb{R}^2 \) with an in­vari­ant non­sep­ar­at­ing con­tinuum \( X \) (a con­tinuum be­ing a com­pact, con­nec­ted, met­riz­able space), then \( f \) has a fixed point in \( X \) [e5], [e2]. This res­ult was gen­er­al­ized to the ori­ent­a­tion-re­vers­ing case by Bell [e6], where the proof is much harder. In the ori­ent­a­tion-pre­serving case, the res­ult is false as stated when \( X \) sep­ar­ates the plane; for in­stance if \( X \) is a circle, then \( f \) has a fixed point, but not ne­ces­sar­ily in \( X \) it­self. No such simple counter­example is evid­ent in the ori­ent­a­tion-re­vers­ing case. Re­fin­ing Bell’s res­ult, Krystyna showed that if \( f \) is ori­ent­a­tion-re­vers­ing, then any in­vari­ant con­tinuum \( X \) must have a fixed point, in­deed at least two fixed points when the do­main en­closed by \( X \) is con­nec­ted [8].

Sev­er­al of Krystyna’s pa­pers, in both to­po­logy and in Eu­c­lidean geo­metry, are with her hus­band Włodek. She also has a joint pa­per with her son Greg, and all three have a joint pa­per on the top­ic of com­pletely sat­ur­ated pack­ings. Fi­nally, in an­oth­er kind of math­em­at­ic­al leg­acy, Krystyna’s grand­daugh­ter Vivi­an is also a re­search math­em­atician, work­ing mainly in ana­lyt­ic num­ber the­ory.