Celebratio Mathematica

The first thing that strikes you is that she’s charm­ing. Only later do you real­ize that she is prob­ably the smartest per­son you’ve ever met. Some­how, you don’t ex­pect math­em­at­ic­al geni­us and a charm­ing, genu­inely warm per­son­al­ity to co­ex­ist. Yet, here sit­ting down to lunch with us is an ex­ist­ence proof: Pro­fess­or In­grid Daubech­ies of Prin­ceton Uni­versity, Ma­cAr­thur Fel­low, mem­ber of the Na­tion­al Academy of Sci­ences, the moth­er of wave­lets. Fuzzy-haired, be­spec­tacled, quick to smile and a world-class storyteller, she’s telling us about grow­ing up in a coal-min­ing town in Bel­gi­um in the six­ties.

“I grew up in a very small town. My fath­er worked in a coal mine. He was an en­gin­eer in a coal mine, and of course while I was grow­ing up I didn’t no­tice, but in ret­ro­spect coal-min­ing towns are very spe­cial kinds of small towns. I mean, there is just one big em­ploy­er, and he con­trols the whole life, even the so­cial life, of the town. My moth­er, well, for her gen­er­a­tion in Bel­gi­um it was not com­mon to have uni­versity edu­ca­tion, but she did. She had ex­pec­ted to have a ca­reer, but after mar­ry­ing my fath­er she didn’t. Partly be­cause there was no op­por­tun­ity, but also be­cause in this very small pa­ter­nal­ist­ic town, two gen­er­a­tions be­hind the wide world, it just was not done for wives of en­gin­eers to work. There was one wife who worked, she was a nurse, and every­body knew that that was why her hus­band nev­er got a pro­mo­tion. It wasn’t said that way, of course, and even then they couldn’t write things down that way. So my moth­er didn’t work.

“I re­mem­ber when we were little she did a lot with us kids. As we grew up we be­came more in­de­pend­ent be­cause she wanted to give us our in­de­pend­ence, but she was also very bit­ter at not hav­ing any big­ger frame­work for her own life. She went back to col­lege when I went to uni­versity. We had moved by then. She went for a dif­fer­ent de­gree be­cause her first de­gree, in eco­nom­ics, had be­come ob­sol­ete — she hadn’t worked for twenty years. She got a second de­gree in crim­in­o­logy and worked for about twenty years as a so­cial work­er. She worked with troubled youths, try­ing to mon­it­or them, and to help them, and to give them de­cent lives.

“She had met my fath­er while they were stu­dents at dif­fer­ent uni­versit­ies at a meet­ing that brought to­geth­er stu­dents from uni­versit­ies in Bel­gi­um. She met him, and later they de­cided to get mar­ried. They mar­ried in 1952, and he star­ted work at the coal mine. At this coal mine they would only hire en­gin­eers who were either mar­ried or en­gaged to be mar­ried very soon. They did not want trouble with single men around.

“The re­gion where my fath­er’s par­ents were born and lived was a very poor re­gion. The reas­on my grand­fath­er didn’t work in the coal mines was that his moth­er really wanted one of her chil­dren not to go down the coal mine be­cause that was the time when you would die young if you went in­to the coal mines. They didn’t know how to pre­vent black lung dis­ease; every­body died young. The coal mines were just dis­put­ing the fact that it was any­thing to do with work in the mines, so you didn’t get any com­pens­a­tion either. My grand­fath­er had been sickly when he was little and he was the first-born, so his moth­er wanted him to work else­where. Gen­er­a­tions in my fam­ily are very long, so I’m talk­ing the end of the 19th cen­tury when my grand­fath­er was born. He left school when he was nine. I said this was a poor re­gion, and this was be­fore child-labor laws. His moth­er, my great-grand­moth­er, had ar­ranged a job for him in a big glass fact­ory; he was in pack­ing. But then, these things are in­cred­ible, by the time he was 14, he had in vari­ous ac­ci­dents lost a fin­ger and an eye in the glass fact­ory. I think of my chil­dren, and I think — how is this pos­sible? My fath­er al­ways says of his fath­er that he was really very smart. He went to even­ing school at some point. He was in pack­ing all of his work­ing life, and he be­came a fore­man. At some point, for a very com­plic­ated de­liv­ery, they had to make a case the in­side of which was the in­ter­sec­tion of two cyl­in­ders, but he had to make it in wood, of course, and fold it out of ply­wood. He had tried with el­lipses and some­how it nev­er fit, so he ac­tu­ally went back to even­ing school to study math­em­at­ics in or­der to learn how to do that. An­oth­er fore­man had ex­plained how to do it, but he wanted to know what it looked like. If you took a cyl­in­der and then un­fol­ded it, what would it look like? How do you ac­tu­ally com­pute that? So, he went back to school.”

In­grid at­ten­ded the Dutch Free Uni­versity in Brus­sels where she stud­ied phys­ics. She held a re­search po­s­i­tion in phys­ics at that same uni­versity un­til 1987 when she came to the United States. Today, though, she as­serts, “I’m a math­em­atician.”

“My fath­er was al­ways in­ter­ested in math­em­at­ics, and he was al­ways in­ter­ested in ex­plain­ing things to me, and I liked it. I would ask ques­tions. I would usu­ally get an­swers which were much longer than I hoped for, so I am try­ing with my chil­dren to yes, give an­swers, but maybe not go bey­ond so far. I re­mem­ber lik­ing math­em­at­ics when I was little, but I ac­tu­ally did ma­jor in phys­ics. I think I ma­jored in phys­ics be­cause it was my fath­er’s dream to be­come a phys­i­cist; he ex­plained to me things about phys­ics. He went to ex­tra Open Uni­versity courses whenev­er he could. Some­times they would or­gan­ize a series of phys­ics lec­tures, and he would go.

“Phys­ics just seemed to be a very noble choice be­cause of my fath­er’s in­flu­ence. It was something in­ter­me­di­ate between what I really wanted — math­em­at­ics — and what my moth­er really wanted — which was that I would have be­come an en­gin­eer. She was a bit wor­ried about all this sci­ence. She thought sci­ent­ists were like artists, they really can­not make a good liv­ing. An en­gin­eer can al­ways find a good job.

“With a free choice, I think I prob­ably would have chosen math­em­at­ics. I don’t know. I liked phys­ics very much. I es­pe­cially liked some phys­ics classes that we had. At some point I was con­sid­er­ing switch­ing between math and phys­ics, and I de­cided to stay in phys­ics be­cause of one par­tic­u­lar course which I thought was won­der­ful, in which we were go­ing bey­ond geo­met­ric op­tics. If you go to the Kirchoff–Fres­nel the­ory of op­tics, you ac­tu­ally see, and can com­pute, that a lens, in fact, com­putes a Four­i­er trans­form, which I think is won­der­ful. I think this is mind-bog­gling, that a lens would, in fact, com­pute a Four­i­er trans­form and this is used in some op­tic­al com­put­ing. This was mar­velous. In fact, this course was really a course in ap­plied math­em­at­ics. It was labeled as a phys­ics course, and it was won­der­ful, and so I stayed with phys­ics. I don’t re­gret it. As a res­ult I have learned a whole lot of things that I wouldn’t have learned in a stand­ard math cur­riculum. And the math that I wanted and needed I have learned by my­self any­way.

“I think I think like a math­em­atician; I switched from the­or­et­ic­al particle phys­ics to more math­em­at­ic­al phys­ics be­cause I felt that people who were really good at particle phys­ics had an in­tu­ition about which I had no clue. I felt like I could learn how to read those pa­pers, but it was like learn­ing a lan­guage without un­der­stand­ing the mean­ing of the words, which I didn’t like at all. It’s hard to de­scribe how I think. Even in ana­lys­is, I don’t think in for­mu­las. Al­though when I work something out, I do com­pute a lot. I have some kind of mech­an­ic­al or geo­met­ric­al way of think­ing, I don’t really know where that comes from.

“Any­way, in Bel­gi­um, un­der­gradu­ate edu­ca­tion is really dif­fer­ent from here in that you track very, very early on. When you re­gister for the uni­versity, you have to say what you’re go­ing to ma­jor in. So you get very few courses out­side your ma­jor or out­side things re­lated to your ma­jor. For phys­ics, you get a lot of math, you get some chem­istry, but you don’t get any lib­er­al arts courses. I think you could go sit as an aud­it­or in some of these courses, but really there’s no time; you don’t choose your own courses. You say ‘I will ma­jor in phys­ics,’ and then the courses are spe­cified ex­cept that in later years, you have some choices, you get to choose one of four. In the first two years everything is com­pletely chosen for you, and it’s quite a heavy sched­ule. It’s a heav­ier sched­ule than I see here, but the res­ult is that you can do much less in­de­pend­ent work. I think a sched­ule where you put to­geth­er a com­bin­a­tion your­self and where you’re en­cour­aged to do a lot of in­de­pend­ent work is ac­tu­ally bet­ter.

“I was tracked with phys­ics, so I had a lot of math courses, es­pe­cially the first two years. And when I had ma­jored, I had seen a lot of phys­ics courses that would be at gradu­ate level in the States be­cause you can­not cram four years full of phys­ics courses and not get to that level. Things are not or­gan­ized so much by semester as they are for a whole year, so many courses were a full year. In the third year, that was really the heav­iest year, we had 13 dif­fer­ent phys­ics courses, and we had 5 weeks of labs. Lots of that phys­ics I have for­got­ten.

“At the gradu­ate school level, however, in most uni­versit­ies in Bel­gi­um, you don’t get courses any more. There is a move­ment there now to get some what-they-call third-cycle courses which are gradu­ate courses, but mostly you’re left to learn ex­tra things on your own or with your ad­visor. You’re as­signed or find a re­search top­ic; right away when you ar­rive you start work­ing on pa­pers. You also have a teach­ing sched­ule, and you’re ex­pec­ted to have your PhD in 5 or 6 years. I nev­er had a teach­ing sched­ule be­cause I had a spe­cial fel­low­ship, but on a teach­ing as­sist­ant­ship, you would teach (be in the classroom) 30 hours a week. I would typ­ic­ally have 8–10 hours a week. The things you would be teach­ing would be prob­lem ses­sions. Of course, you were there for the stu­dents, and typ­ic­ally the only per­son there for the stu­dents to ask ques­tions from be­cause they would not dare to ap­proach a pro­fess­or. You don’t have con­trol over what you’re do­ing; you’d be giv­en a list of prob­lems, and you’d an­swer ques­tions about them. It was very frus­trat­ing be­cause it meant you had zero in­put in­to what you did. That per­sists even geo­phys­i­cist Jean Mor­let and math­em­at­ic­al after the PhD, for a while.”

Wave­lets are every­where these days. Wave­lets are a new meth­od for en­cod­ing and com­press­ing in­form­a­tion (see box). They are be­ing used in im­age com­pres­sion (the FBI’s files of ap­prox­im­ately 200 mil­lion fin­ger­prints are be­ing con­ver­ted to wave­let-com­pressed elec­tron­ic im­ages), also sound and video com­pres­sion, med­ic­al ima­ging, and geo­lo­gic­al ex­plor­a­tion. Ron­ald Coi­f­man of Yale Uni­versity used wave­let tech­niques to re­move the noise from a cen­tury-old re­cord­ing of Brahms play­ing one of his own com­pos­i­tions.

The most ex­cit­ing thing about wave­lets might just be the way that they are draw­ing to­geth­er people and ideas from so many dif­fer­ent fields of sci­ence: math­em­aticians, phys­i­cists, geo­lo­gists, stat­ist­i­cians, com­puter sci­ent­ists, en­gin­eers of all kinds. In fact the his­tory of the idea has roots in all of these fields and more. Yves Mey­er has iden­ti­fied pre­curs­ors to the idea in math­em­at­ics, com­puter sci­ence, im­age pro­cessing, nu­mer­ic­al ana­lys­is, sig­nal pro­cessing, stud­ies of hu­man and com­puter vis­ion, and quantum field the­ory. The short ver­sion of the his­tory has geo­phys­i­cist Jean Mor­let and math­em­at­ic­al phys­i­cist Al­ex­an­der Gross­mann in­tro­du­cing the idea in the early 1980s. Mey­er and Stéphane Mal­lat pieced to­geth­er a math­em­at­ic­al frame­work for wave­lets in the mid-eighties. In 1987 Daubech­ies made her fam­ous con­tri­bu­tion of a fam­ily of wave­lets that are smooth, or­tho­gon­al, and equal to zero out­side a fi­nite Thus, in a stroke, ac­com­plish­ing what every­one sup­posed im­possible and mak­ing wave­lets very much more ap­plic­able.

“How did I get star­ted on wave­lets? For my PhD work, I had worked on something in quantum mech­an­ics which are called co­her­ent states. This is a tool to un­der­stand the cor­res­pond­ence between quantum mech­an­ics and clas­sic­al mech­an­ics. So you try to build func­tions that are well-loc­al­ized, that live in Hil­bert space, but that cor­res­pond as closely as you can with be­ing in one po­s­i­tion, in one mo­mentum in clas­sic­al mech­an­ics. I had worked in Mar­seilles with Alex Gross­mann, and Alex was one of the people who really star­ted the whole wave­let syn­thes­is. There are roots in pure math­em­at­ics, in many dif­fer­ent fields, but the syn­thes­is really, I feel, was cre­ated by Alex Gross­mann and Jean Mor­let. Mor­let was not happy with what was called the wave­let trans­form, and he wanted something else, so he in­ven­ted an al­gorithm. But he had no math­em­at­ic­al the­ory be­hind it. Alex Gross­mann real­ized there was an ana­logy with the co­her­ent state form­al­ism in quantum mech­an­ics. He re­cast it in terms of a group rep­res­ent­a­tion and then people star­ted say­ing ‘But we’ve been do­ing that all along in a dif­fer­ent con­text.’ And they were right; they had been do­ing it all along. So that’s what made the syn­thes­is hap­pen that made the jump from math­em­aticians to en­gin­eers.

“I knew Alex Gross­mann very well from my thes­is work, and I was look­ing for something else to start work­ing on. This was a time when I had many changes in my life. Be­fore my hus­band, I had a long re­la­tion­ship, and I had just left him, and I was look­ing for something else, for changes. So I changed re­search top­ics. I star­ted work­ing on wave­lets in ’85. I star­ted work­ing on wave­lets and I met Robert [Calderb­ank, the math­em­atician to whom she is mar­ried] with­in a peri­od of six weeks in ’85.

“At this time geo­lo­gists were do­ing win­dowed Four­i­er trans­forms, so they would win­dow [look at a short time seg­ment of a sig­nal], then Four­i­er trans­form [ap­prox­im­ate that bit of sig­nal with com­bin­a­tions of sines and co­sines]. This means that for very high fre­quency things that live in very short time in­ter­vals — these are called tran­si­ents — if you’ve de­term­ined your win­dow to be this wide, then you need a whole lot of high fre­quency func­tions to cap­ture that be­ha­vi­or. So, you could of course make your win­dow very nar­row, but then you don’t cap­ture a lot of them. Mor­let didn’t like that as­pect of the win­dowed Four­i­er trans­forms, so he said ‘After all, I’m us­ing loc­a­tion and mod­u­la­tion, let’s do it dif­fer­ently. Let’s take one of these func­tions that have some os­cil­la­tions and let’s put that in dif­fer­ent places and squish it so that I have a dif­fer­ent thing,’ really wave­lets. Now he didn’t really for­mu­late this pre­cisely. Ac­tu­ally the name comes from there be­cause in geo­logy when you have dif­fer­ent win­dows, which de­term­ine the shape of these multi-layered func­tions, they call them wave­lets. So he called his trans­form ‘a trans­form us­ing wave­lets of con­stant shape’ be­cause the oth­er ones didn’t have con­stant shape. If you ad­just the win­dow, or you mod­u­late this way, then of course they look dif­fer­ent. In his case, the wave­lets look the same, they were just dilated ver­sions. He called them wave­lets of con­stant shape, but then once they left that field, there was no oth­er thing around called wave­lets, so he just dropped the ‘of con­stant shape’ to the great an­noy­ance of geo­phys­i­cists be­cause in their field it has an­oth­er mean­ing.

“Wave­lets were not really something that were a trend in geo­phys­ics, Mor­let just came up with it. In har­mon­ic ana­lys­is, people had been look­ing at, not ex­actly the same way, but something sim­il­ar for ages. It goes back to Lit­tle­wood–Pa­ley the­ory [circa 1930], and even the in­teg­ral trans­form for­mula that Gross­mann and Mor­let wrote, be­cause Mor­let had no real for­mu­las, is a trans­form that you find in Calderón’s work in the six­ties. So in some sense they had re­in­ven­ted the wheel. In an­oth­er very real sense they had looked at it com­pletely dif­fer­ently. For Calderón it was a tool to carve up space in­to dif­fer­ent pieces on which he would then use dif­fer­ent tech­niques for es­tim­ates. Gross­mann and Mor­let gave these wave­lets some kind of phys­ic­al mean­ing, in a cer­tain sense, view­ing them as ele­ment­ary build­ing blocks which was a dif­fer­ent way of look­ing at it. And Yves Mey­er later told me that when he read those first pa­pers by Gross­mann, it was very hard for him be­cause it was a dif­fer­ent style. It took a while be­fore he real­ized in what sense it was really dif­fer­ent, be­cause at first you see the for­mu­las there, and you say, ‘well, yeah, we’ve been do­ing that for 20 years,’ but then you real­ize that here was a dif­fer­ent way of look­ing at it: a new paradigm shift.”

To most math­em­aticians it ap­pears that wave­lets sprang full-grown from the fore­heads of Gross­mann, Mor­let, and Daubech­ies and then were im­me­di­ately grabbed by en­gin­eers and sci­ent­ists. It is un­usu­al for a piece of math­em­at­ics to find so many ap­plic­a­tions so quickly. Daubech­ies with her abil­ity to talk the lan­guages of phys­ics and en­gin­eer­ing and math­em­at­ics is, to a large de­gree, re­spons­ible for the build­ing of so many bridges between the groups.

“In pure math­em­at­ics the idea was de­veloped start­ing in the thirties, then in great­er de­tail in the six­ties. It was a very power­ful tool which lived in a re­l­at­ively small com­munity in math­em­at­ics, and out­side the small com­munity, I felt it spread rather slowly. For ex­ample, in quantum mech­an­ics I think some of these tech­niques would have been use­ful to math­em­at­ic­al phys­i­cists earli­er than they pen­et­rated. I think it’s be­cause through Gross­mann and Mor­let there were in­ter­me­di­ate people. I treas­ure every single elec­tric­al en­gin­eer I meet and with whom I can talk. I’m in­ter­ested in talk­ing with them, I think many math­em­aticians aren’t, but I am. Even so, I find it hard to talk with many of them be­cause we’ve been trained in com­pletely dif­fer­ent ways and the words mean dif­fer­ent things. But I have found some I can talk to and I think it’s very valu­able when they are also in­ter­ested in talk­ing with me. I think it’s easi­er for me be­cause of this phys­ics back­ground I have and be­cause I have learned at least some of their lan­guage. I think Alex Gross­mann played a very im­port­ant role that way. I have met Jean Mor­let sev­er­al times, I think he’s a very in­ter­est­ing man, but I find it very hard to talk with him. I mean, of course, not talk­ing so­cially, but to really un­der­stand his ideas. Be­cause it’s not even that I can see that here’s an idea and I know that I don’t un­der­stand the form­al math­em­at­ics, it’s that I don’t even un­der­stand the idea: that I don’t even no­tice or can’t tell if there is something there or not.

“The prob­lem is you don’t know what are ideas and what aren’t: he prob­ably knows the dif­fer­ent lay­ers of what he’s say­ing, but for me it’s im­possible. Alex Gross­mann can talk to him. It’s good to have a chain of people. I think so. And I think that’s the role that I played. In some sense you could say that I didn’t dis­cov­er any­thing that any­body didn’t know be­cause there’s this one as­pect of the math­em­at­ic­al roots, but then there’s the oth­er one which has to do with an al­gorithm for im­ple­ment­ing the whole thing. If there wasn’t an al­gorithm, then none of this would be hap­pen­ing any­way. But that al­gorithm ex­is­ted in elec­tric­al en­gin­eer­ing: it’s called sub-band fil­ter­ing. There was no con­nec­tion with any of the pure math­em­at­ics, and I don’t know that that con­nec­tion ever would have been made if it wer­en’t for this chain of people. I mean, once this con­nec­tion was made, then you had math­em­aticians in­ter­ested in hear­ing about the al­gorithms and elec­tric­al en­gin­eers in­ter­ested in hear­ing about the math­em­at­ics.

“I’m a math­em­atician. I feel like one. I feel like a math­em­atician, but I am very much mo­tiv­ated by ap­plic­a­tions. I like to go off on a math­em­at­ic­al tan­gent, but I like to get back to ap­plic­a­tions. So in that sense, I’m an ap­plied math­em­atician. At one time, and still today for some people, ap­plied math­em­at­ics meant only cer­tain types of res­ults ob­tained by solv­ing cer­tain types of par­tial dif­fer­en­tial equa­tions. I’m not that type of math­em­atician at all. So I very much feel I’m an ap­plied math­em­atician, but what I ap­ply is func­tion­al ana­lys­is, rather than PDE the­ory. Ac­tu­ally, there’s no such field as ap­plied math­em­at­ics: I think there are sub­fields with­in math­em­at­ics and that, as a math­em­atician, you al­ways really like it when dif­fer­ent sub­fields get in­to con­tact with each oth­er. I think that vir­tu­ally all of these math­em­at­ic­al sub­fields can have con­tacts with ap­plic­a­tions, so in some sense I’m an ap­plied har­mon­ic ana­lyst, not an ap­plied PDE per­son, but one can just as eas­ily be, say, an ap­plied num­ber the­or­ist.

“Ideally, I think there’s an im­port­ant place for pure math­em­at­ics and an im­port­ant place for pure math­em­aticians; I see my role as identi­fy­ing and bring­ing to more pure math­em­aticians than my­self very in­ter­est­ing prob­lems com­ing from ap­plic­a­tions. I think that’s an im­port­ant role to play and that it is good for pure math­em­at­ics. There was a while when pure math­em­at­ics wasn’t open to this, but I think that math­em­aticians are start­ing to open up more. It’s very im­port­ant to re­mem­ber that whole fields in pure math­em­at­ics have come from ap­plic­a­tions. That doesn’t mean that all the pure math that was done in that area can there­fore be de­scribed as ap­plied. No, it’s just math­em­at­ics. But it also sug­gests that it’s very well pos­sible for oth­er fields of math­em­at­ics to start to be fostered by ap­plic­a­tions. I mean ap­plied math­em­at­ics is not just learn­ing some nice math­em­at­ics and not be­ing up­set by get­ting your hands dirty on some prob­lems where things are not as neat but they will have an ap­plic­a­tion — it’s also identi­fy­ing op­por­tun­it­ies for math­em­at­ic­al think­ing, which can lead to oth­er fields of math­em­at­ics. I don’t know which oth­er fields. I can’t pre­dict. I mean it’s the ones that you can­not pre­dict that are the most in­ter­est­ing.”

Prob­lems are the lifeblood of math­em­at­ics and Daubech­ies, like most math­em­aticians, has sev­er­al go­ing at once. All the ones she tells us about come from real world ap­plic­a­tions. It’s quite clear listen­ing to her ex­plain her prob­lems that she is a phe­nom­en­al teach­er — the ex­plan­a­tions are so clear and the prob­lems sound so ex­cit­ing that we’re itch­ing to get out of the in­ter­view and get to work on them.

“Be­fore this meet­ing I was at a mo­lecu­lar bio­logy meet­ing. I got really in­ter­ested in people who can add pro­teins; for some pro­teins they know in which or­der all the atoms go, but then they don’t know what the thing will look like, and they have to ‘solve it’ to know what it looks like, and these are really im­port­ant to un­der­stand their func­tions. There are groups that try to pre­dict the form of pro­teins from en­er­get­ic com­pu­ta­tions from just the for­mula, and they have it or­gan­ized so that they have a way to ob­ject­ively com­pare how good their pre­dic­tions are with what the grand truth is by find­ing out about pro­teins that will be solved. I mean you can reas­on­ably pre­dict when things will get solved, but are not solved yet. So they use all those for­mu­las, and they all work on it: they have a dead­line. At some point it’s clear when the thing will be solved, and they say ‘now, you have to sub­mit.’ And you sub­mit at that time, and it gets com­pared with the grand truth, and in that way you can score dif­fer­ent pre­dic­tion pro­grams.

“Okay, so one guy was ex­plain­ing about his pre­dic­tion pro­gram: be­cause it’s just too big a space to ex­haust­ively search, he searched in a multi-res­ol­u­tion way. He first tried to build a coarse mod­el, then find the best coarse mod­el giv­en, then build it up from there. Now his first-level coarse mod­el, was in­deed very coarse, but he was put­ting it on a reg­u­lar lat­tice, and his meth­od was do­ing very well. But it struck me that if we have a good idea of how to com­press, how to find sub­di­vi­sion schemes for curves, we could look at all the pro­teins that they know, and try to find that sub­di­vi­sion scheme that will be ad­ap­ted to the pro­tein world. I’ve been al­ways look­ing at smooth­ness; they don’t care about smooth­ness. These pro­teins ac­tu­ally do all kinds of strange things. But one could try to find a sub­di­vi­sion scheme that would ad­apt to their goal, and that could give you a good idea of what kind of coarse things to start from. So that’s something that hasn’t star­ted, but something I’m very ex­cited about and I hope to work on this spring.

“An­oth­er thing that I’m very in­volved in is un­der­stand­ing the math­em­at­ic­al prop­er­ties of coarsely quant­ized but very over­sampled au­dio sig­nals, modeled by so-called band-lim­ited func­tions. There are really neat links to dy­nam­ic­al sys­tems. I’d like to do that for oth­er wave­let trans­forms. I think we can do it. If we can do it, I think it’s go­ing to have very use­ful ap­plic­a­tions, plus I think math­em­at­ic­ally it’s go­ing to be very in­ter­est­ing. Already for the band-lim­ited func­tions it’s much more in­ter­est­ing that I had ex­pec­ted a year ago.

“I have a gradu­ate stu­dent with whom I work on ap­plic­a­tions of wave­lets to the gen­er­a­tion and com­pres­sion of sur­faces. People rep­res­ent sur­faces with tri­an­gu­la­tions with tens and hun­dreds of thou­sands of tri­angles, so you’d like to com­press that in­form­a­tion. Well, you can do that via multi-res­ol­u­tion, and then you can won­der what kind of wave­lets are as­so­ci­ated with that. Then you can think about smooth­ness. In some ap­plic­a­tions, smooth­ness again is very im­port­ant. So that’s an­oth­er pro­ject.

“What else? I look at my stu­dents and col­lab­or­at­ors be­cause everything I work on I work on with a col­lab­or­at­or. I’m still work­ing with one of my former stu­dents on a way of us­ing frames for trans­mit­ting in­form­a­tion over mul­tiple chan­nels, but that’s very the­or­et­ic­al work, and I’m not sure how close it’s go­ing to get to ap­plic­a­tions. I have the im­pres­sion I’m for­get­ting something.

“I love to talk about math­em­at­ics and so I en­joy all the courses I teach. I teach reg­u­lar un­der­gradu­ate math courses, I de­veloped a course of math­em­at­ics for non-math ma­jors. For many stu­dents in our cal­cu­lus classes this will be their last con­tact with math­em­at­ics. I don’t think this is a very good idea. Many of them are really not turned on by cal­cu­lus, and it’s hard to get a really mean­ing­ful ap­plic­a­tion in a course if you also want to teach cal­cu­lus tools. You can try to fit in some ap­plic­a­tions, but they really feel very con­trived. The real ap­plic­a­tions of cal­cu­lus are all the phys­ics and oth­er math courses, but they won’t ever see that, and they’re not in­ter­ested in see­ing that. I wanted to get math­em­at­ic­al ideas across without teach­ing tech­nique.

“I call the course Math Alive; in two-week units we vis­it dif­fer­ent con­cepts. I do one on vot­ing and fair share, and one on er­ror cor­rec­tion and com­pres­sion, and I have one on prob­ab­il­ity and stat­ist­ics, and one on cryp­to­graphy, and there’s one (ac­tu­ally not taught by me, it’s a course we co-teach) on Why New­ton Had to In­vent Cal­cu­lus, that’s a unit stu­dents have more trouble with. Then there’s one on dy­nam­ic­al sys­tems and pop­u­la­tion ex­plo­sion, it gets in­to pop­u­la­tion mod­els. I’ve en­joyed this course a lot. I’m teach­ing it this spring for the fifth time, and I’d like to doc­u­ment it so that it can be taught by some­body else the next time, be­cause that’s the best way to prove a concept, if it can be done by some­body else. “I want these stu­dents to go away know­ing that math­em­at­ics is really im­port­ant, that it turns up in lots of things where they may be im­pressed by the tech­no­logy, but they don’t real­ize there’s very deep math­em­at­ics in there. I want them to see that math­em­at­ics is neat in that you really solve a prob­lem. You think your way out of it. And ba­sic­ally, if they re­mem­ber that, that’s fine. I’m sure that when people meet his­tor­i­ans, they don’t say, “you must know all the dates.” They know it’s something else. Well, they don’t know that it’s something oth­er than bal­an­cing a check­book in math­em­at­ics. I’d like them to know that.

“I also teach gradu­ate courses — of­ten a start­ing gradu­ate course in wave­lets, some­times a more ad­vanced course. I en­joy teach­ing un­der­gradu­ate courses more than gradu­ate courses. Not be­cause I don’t like teach­ing gradu­ate courses, but be­cause at gradu­ate study level, the start­ing gradu­ate courses I can see work well, but I think an ad­vanced gradu­ate course works bet­ter as a read­ing course than as a lec­ture course.”

Daubech­ies is mar­ried to Robert Calderb­ank, a dis­tin­guished Brit­ish math­em­atician. They have two chil­dren, Car­oline and Mi­chael. If you’re won­der­ing what it must be like to have a geni­us for a mom, well, it sounds a lot like hav­ing a mom.

“I go to my chil­dren’s school and the tables are in groups and the classroom is full of won­der­ful things. In my school days classrooms might have had some stuff, but we had little desks which were all lined up in rows, maybe that’s the way things were here as well in the six­ties. I think the new way is much more fun. It may well be dif­fer­ent in Bel­gi­um now; I haven’t vis­ited the ele­ment­ary schools. All my ele­ment­ary and sec­ond­ary edu­ca­tion was in single-sex schools. That was the way it was in Bel­gi­um at that time. I went to pub­lic schools, but all pub­lic schools at that time were gender-sep­ar­ated.

“I have mixed feel­ings about that. At the time I thought it was a bad thing; it’s bet­ter if you don’t see an­oth­er gender as a dif­fer­ent spe­cies be­cause at some point you will start look­ing for a com­pan­ion, and if you haven’t really met any people of the oth­er gender un­til you’re 18, it’s very ar­ti­fi­cial. So I didn’t like it at the time, but then after I got to uni­versity I real­ized that people in classrooms were less likely to ask me to give n an­swer than some boys that were there. At least in the be­gin­ning. After a while, when they knew I was in­ter­ested, then it was dif­fer­ent. But you al­ways felt that there was a big­ger hurdle to get over as a girl to get no­ticed than as a boy. I hadn’t thought about that in great de­tail, but then com­ing to this coun­try and hear­ing all the de­bate here about it — I can see how it might be good for some girls to have sep­ar­ate gender schools. In an ideal world there would not be this ef­fect, but giv­en it ex­ists, I can see how it might help in build­ing self-as­sur­ance in girls. My daugh­ter goes to pub­lic school and it’s mixed gender, and she’s happy, but I’m won­der­ing if at some point there might be a prob­lem. I think she’s a smart little girl, and if at some point I feel that be­cause she is in a mixed-gender school she is not get­ting as much of an op­por­tun­ity, I might con­sider a single-gender school. I haven’t, she’s only sev­en, and it’s not an is­sue at this time. It’s something that ten years ago I would not have thought I would ever con­sider, but now I would.

“I don’t know if go­ing to all-girl schools had an ef­fect on me. My par­ents al­ways made it clear I could do any­thing. It didn’t oc­cur to me un­til I went to uni­versity that people could think I was less good at something be­cause I was a girl. I think I was very for­tu­nate, be­cause at that age, you’re too old to take that pre­ju­dice ser­i­ously, and when you en­counter someone with that at­ti­tude, you think, ‘You’re a jerk.’

“As I said, my par­ents, es­pe­cially my fath­er, really in­flu­enced my edu­ca­tion. So did pop­u­lar psy­cho­logy ac­tu­ally. When I was little the pre­vail­ing the­ory was that it wasn’t good to mix lan­guages too early. You might really con­fuse chil­dren and then they wouldn’t really be able to use any of the lan­guages in great depth and it would leave marks on them the rest of their lives. My par­ents were in an ideal situ­ation to bring my broth­er and I up bi­lin­gually, be­cause they spoke French to each oth­er and we lived in the Flem­ish part of Bel­gi­um, where people speak Dutch. But since there was this myth that mix­ing lan­guages early was not good, they de­cided to bring us up in Dutch, which is my moth­er’s tongue; my fath­er’s flu­ent in it since he went to school in Dutch.

“The­or­ies hav­ing changed now; I bring my chil­dren up bi­lin­gually in Dutch and Eng­lish. I thought it would be too hard do­ing it in a lan­guage that is not my moth­er tongue. My French is flu­ent, but in the be­gin­ning es­pe­cially it took quite an ef­fort to have a bi­lin­gual house­hold be­cause my hus­band is Brit­ish and he didn’t speak Dutch. He has learned to­geth­er with the chil­dren. So I speak Eng­lish with him, but Dutch with the chil­dren.

“My hus­band since he has learned Dutch would like to have prac­tice speak­ing, but my chil­dren won’t al­low it. They roll on the floor when he tries it. They think it’s quite in­cred­ibly funny. My son went through a stage where when my hus­band would say something in Eng­lish that he knew was of in­terest to me too, he would turn to me and trans­late for me.

“Eng­lish is their first lan­guage be­cause they go to school in Eng­lish. So in Dutch they some­times have more dif­fi­culty find­ing words, but I really try to en­cour­age them to find the words rather than switch to Eng­lish, and so some­times be­fore my daugh­ter tries to tell me something, she’ll say ‘how do you say that word in Dutch?’ They had been talk­ing in school about dif­fer­ent lan­guages, and she really wanted to tell me, so she said, in Dutch ‘How do you say ‘Dutch’ in Dutch?’ But at the end of the sen­tence the name was already there.

“Our son is tal­en­ted in math, and I think our daugh­ter might be too. We, of course, like to stim­u­late that when we ask ques­tions, but we are not push­ing them hard. What we are push­ing is that they have to co­oper­ate and work at school and do the best they can, not just in math, but in everything. In fact, I’m more con­cerned about writ­ing and things like that. Try­ing to get ideas in an or­gan­ized way on pa­per I think is im­port­ant in math­em­at­ics as well as else­where.”

“I like to go to my chil­dren’s school and help out, es­pe­cially on sci­ence day. When I’m there I’m not that wo­man pro­fess­or math­em­atician, I’m Mi­chael’s mom and Car­o­lyn’s mom, and I like that. I like that.”