Celebratio Mathematica

A com­mon pre­ju­dice holds that wo­men can’t match the strength of men in math­em­at­ics. But for more than three dec­ades, Bel­gian phys­i­cist/math­em­atician In­grid Daubech­ies has been prov­ing the pre­ju­dice wrong — and us­ing maths to make a bet­ter world.

As a little girl, In­grid Daubech­ies felt she was al­ways a bit dif­fer­ent from her friends. Be­fore the age of 6, she was already fa­mil­i­ar with com­plex math­em­at­ic­al con­cepts, and when she couldn’t sleep she did not count num­bers as oth­ers do, but in­stead men­tally com­puted powers of 2.

As a young and prom­ising sci­ent­ist, she entered the Vrije Uni­versiteit Brus­sel at 17. After com­plet­ing her un­der­gradu­ate stud­ies in phys­ics she worked at both the CNRS Cen­ter for The­or­et­ic­al Phys­ics in Mar­seille and her own alma ma­ter, to de­vel­op the ma­ter­i­al for her thes­is in quantum mech­an­ics (1980). Dur­ing the first years of her ca­reer, and un­til 1987, she re­mained at Vrije Uni­versiteit, but then headed to the US to be­come a Mem­ber of Tech­nic­al Staff at AT&T Bell Labor­at­or­ies.

Dur­ing a vis­it in 1986 to the Cour­ant In­sti­tute of Math­em­at­ic­al Sci­ences, an in­de­pend­ent di­vi­sion of New York Uni­versity, Daubech­ies made her most im­port­ant dis­cov­ery, in the field of wave­lets (wave­lets are math­em­at­ic­al func­tions use­ful in di­git­al is the sig­nal pro­cessing and im­age com­pres­sion as well as in many oth­er branches of ap­plied and pure math­em­at­ics). After her move to the US, she con­tin­ued in this dir­ec­tion, which even­tu­ally led her to en­gage in more in­ter­dis­cip­lin­ary work. From 1994 to 2010, she was act­ive in the Pro­gram in Ap­plied and Com­pu­ta­tion­al Math­em­at­ics at Prin­ceton. In 2011, she moved to Duke Uni­versity, where today she serves as James B. Duke Pro­fess­or of Math­em­at­ics, nur­tur­ing in­terests that cov­er math­em­at­ics as well as arts and oth­er fields.

She is a glob­al au­thor­ity on tech­niques to de­com­pose and ana­lyze im­ages on a math­em­at­ic­al basis. She is the re­cip­i­ent of nu­mer­ous awards, and be­came the first fe­male full pro­fess­or of Math­em­at­ics at Prin­ceton. Today, she serves as the first wo­man pres­id­ent of the In­ter­na­tion­al Math­em­at­ic­al Uni­on (IMU). She vis­ited Trieste dur­ing the 2 to 21 June joint ICTP-TWAS School on Co­her­ent State Trans­forms, Time-Fre­quency and Time-Scale Ana­lys­is, Ap­plic­a­tions, for which she also served as co-dir­ect­or.

Talk­ing with TWAS staff writer Cristina Serra, In­grid Daubech­ies ex­plained why more wo­men should en­gage in math­em­at­ics.

There’s a com­mon as­sump­tion that wo­men are less good than men at math­em­at­ics. What could be the reas­on for this, as­sum­ing it is true?

I dis­agree with this view — com­pletely. There is a highly vari­able per­cent­age of wo­men in aca­demia and in de­part­ments of math­em­at­ics across Europe. Dif­fer­ences are so enorm­ous that it be­comes ob­vi­ous that it has something to do with cul­tur­al habits, which dif­fer from one na­tion to an­oth­er, and not with in­tel­li­gence. I have a very cyn­ic­al col­league who says that the num­ber of wo­men math­em­aticians who are in the aca­demia is in­versely pro­por­tion­al to some av­er­age of the amount of money and prestige that this job can grant: If there is little money and no prestige, there you’ll find more wo­men. I agree: These as­pects seem to play a much lar­ger role than be­ing smart.

You are the first fe­male pres­id­ent of the In­ter­na­tion­al Math­em­at­ic­al Uni­on (IMU). This proves that the glass ceil­ing can be broken: Have you been elec­ted for your cur­riculum or for your sem­in­al dis­cov­ery?

My elec­tion as IMU pres­id­ent had little to do with my spe­cif­ic sci­entif­ic achieve­ments. Of course, there is a gen­er­al level of dis­tinc­tion that the IMU Nom­in­at­ing Com­mit­tee looks for. Bey­ond that, it has to do with the need to find a per­son who really cares for the math­em­at­ic­al com­munity, who wants to cre­ate net­works. Many math­em­aticians be­lieve math­em­at­ic­al tal­ent is dis­trib­uted more or less uni­formly around the globe, and the IMU cares about edu­ca­tion in de­vel­op­ing coun­tries. This is not just about spot­ting ex­tremely rare top geni­uses, but also about fos­ter­ing the growth of strong, healthy maths com­munit­ies that in­ter­act pro­duct­ively with the whole math­em­at­ics world. Rais­ing aware­ness about and try­ing to re­medi­ate the scarce num­ber of wo­men in math­em­at­ics is, to me, part of that whole pack­age.

As the IMU pres­id­ent you must be in a power­ful po­s­i­tion. How does it feel?

I do not to see this po­s­i­tion in terms of power, but in terms of ser­vice. I like to work with people rather then steer them. I think it’s im­port­ant to find out what people like to do: Some­times I feel I must push harder in a situ­ation, but that is still just to change it for the bet­ter, if oth­ers in­deed agree. Most of the time my po­s­i­tion is just about build­ing con­sensus, talk­ing with people, com­ing up with new ideas and see wheth­er they can work.

Did you de­vel­op this pas­sion for math­em­at­ics early in your life?

I al­ways wanted to un­der­stand things and I nev­er had prob­lems with math­em­at­ics, be­cause it has al­ways been a tool that I used to do phys­ics. Later in my life, as I be­came in­ter­ested in not only phys­ics, but also in en­gin­eer­ing and oth­er sub­jects, I used the math­em­at­ics that I knew to do the rest. If I have al­ways wanted to learn more it was be­cause dif­fer­ent types of prob­lems re­quire dif­fer­ent types of math­em­at­ic­al tools. I use and de­vel­op maths to ap­proach prob­lems. Some­times that in­volves de­vel­op­ing new math­em­at­ic­al tools.

You have joined a num­ber of European and Amer­ic­an uni­versit­ies, like Vrije Uni­versiteit in Bel­gi­um, Rut­gers, Prin­ceton, and now Duke. Did you ex­per­i­ence ma­jor dif­fer­ences between the two aca­dem­ic sys­tems?

When I left Europe, al­most 30 years ago, life in US academies was much more flu­id than here. Young sci­ent­ists in Europe were not giv­en re­spons­ib­il­ity for their re­search, there was al­ways a boss to con­trol everything. Even for their teach­ing activ­it­ies young sci­ent­ists couldn’t ap­ply for fund­ing. Now the situ­ation is much more flu­id, but it wasn’t then, in Europe. I’m very happy at Duke: It’s a place where sci­ent­ists can eas­ily en­gage in in­ter­dis­cip­lin­ary work. I find this at­ti­tude very help­ful work now I work on ap­plic­a­tions of math­em­at­ics to fields like art his­tory and art con­ser­va­tion. At Duke they like this a lot and I find great en­thu­si­asm around.

As a math­em­atician your name is linked to the field of “wave­lets in im­age com­pres­sion”. What is it about?

Wave­lets are used in sig­nal pro­cessing. A high-res­ol­u­tion di­git­al im­age is com­posed of a huge num­ber of tiny pixels (a pixel is the smal­lest con­trol­lable ele­ment of a pic­ture rep­res­en­ted on a screen), each of them a small square of one con­stant col­or or gray level (if the im­age is not in col­or). Num­bers can quanti­fy the ex­act col­or or gray level of the pixels, so that the im­age cor­res­ponds to an enorm­ous ar­ray of num­bers. Yet most pixels in an im­age are very sim­il­ar to their neigh­bors; al­though of course there are also many places in the im­age (wherever something in­ter­est­ing hap­pens) where a pixel dif­fers ap­pre­ciably from at least some if the neigh­bor­ing pixels, these loc­a­tions are still in the minor­ity. A wave­let de­com­pos­i­tion of an im­age ex­ploits that as­pect. Wave­lets de­com­pose the im­age in­to build­ing blocks of dif­fer­ent scale that, to­geth­er, de­scribe what’s go­ing on in the im­age. Sim­pli­fy­ing a lot, this ap­proach tells you where you need to put lots of de­tail (be­cause pixels dif­fer a lot from their neigh­bors) and where not, in im­age ana­lys­is.

What ex­actly do you do in the ap­plic­a­tion of im­age ana­lys­is to art con­ser­va­tion?

We ana­lyze old paint­ings with ima­ging tech­niques, to provide artists and people who re­store old paint­ings with tools to un­der­stand the nature of the mas­ter­piece from yet an­oth­er point of view than what they are already us­ing. The first step is talk­ing to people to find out what kind of ques­tions they have. The in­ter­est­ing thing is that we of­ten do not have ready-made solu­tions for them, so their ques­tions also pose in­ter­est­ing chal­lenges for us.

Can you make an ex­ample?

We re­cently ap­plied di­git­al im­age-pro­cessing to the fam­ous Ghent Al­tarpiece, a polyp­tych loc­ated in the Saint Bavo Cathed­ral in Ghent, Bel­gi­um, which is com­posed of 12 pan­els, eight of which are hinged shut­ters. Dat­ing back to the 14th cen­tury, it is con­sidered not only a mas­ter­piece, but also a key ele­ment in un­der­stand­ing the art his­tory of that peri­od in West­ern Europe. The Hubert and Jan van Eyck’s Ad­or­a­tion of the Mys­tic Lamb, also Like most paint­ings of its age and ma­ter­i­als, it known as the Ghent Al­tarpiece has many small cracks or breaks in the paint lay­er, caused by dif­fer­ences in the way the paint lay­ers and the un­der­ly­ing wood sup­port re­act to tem­per­at­ure and hu­mid­ity changes. We carry out vir­tu­al res­tor­a­tions of high-res­ol­u­tion di­git­al ver­sions of paint­ings by auto­mat­ic­ally de­tect­ing and re­mov­ing cracks, which turns out to be re­mark­able chal­len­ging.

In the case of the Ghent Al­tarpiece we com­bined three de­tec­tion meth­ods to make a map of all the cracks. Then we vir­tu­ally in­painted these cracks, thus re­con­struct­ing a sharp­er view of, in this par­tic­u­lar case, the let­ters in a me­di­ev­al book de­pic­ted in the back­ground on one of the pan­els. This made it pos­sible for pa­leo­graph­ers to de­cipher many more words than the 2 they could read be­fore; as a res­ult they can now un­am­bigu­ously identi­fy not only that the paint­er re­ferred to a par­tic­u­lar text by Thomas of Aqui­nas about the An­nun­ci­ation, which turns out to be es­pe­cially ap­pro­pri­ate for the pan­el in ques­tion.

Where does maths come in, in this case?

Ba­sic­ally every­where. To ana­lyze the Al­tarpiece, di­git­ized at an in­cred­ibly high res­ol­u­tion, we de­signed math­em­at­ic­al al­gorithms to de­tect the cracks, to fill then in, and to de­tect let­ters in the “re­stored” writ­ing on the book. In oth­er pro­jects con­nect­ing im­age ana­lys­is and art res­tor­a­tion, we de­vel­op tech­niques that will re­lieve art con­ser­vat­ors from some of their more te­di­ous tasks; after we de­vel­op the al­gorithms, we passed them on to oth­er soft­ware ex­perts, to turn them in­to user-friendly pack­ages for pro­fes­sion­al art ren­ov­at­ors. This has led to new in­ter­dis­cip­lin­ary col­lab­or­a­tions that are giv­ing in­ter­est­ing res­ults.

Is maths an im­port­ant sub­ject for sci­ent­ists from de­vel­op­ing coun­tries? What is your ex­per­i­ence on this?

Math­em­at­ics is very pop­u­lar in de­vel­op­ing coun­tries — it has a great ap­peal be­cause it is so neat — you lit­er­ally solve prob­lems and build ap­proaches by just the power of thought. What it is some­times sur­pris­ing to me is that people prefer the­or­et­ic­al to ap­plied math­em­at­ics — I think it is best to al­ways de­vel­op both: ap­plied math­em­at­ics helps in build­ing strong STEM edu­ca­tion, and will be­ne­fit oth­er sci­ences and en­gin­eer­ing; the­or­et­ic­al math­em­at­ics is the es­sen­tial found­a­tion, on which everything is built and that makes it pos­sible to ex­tend ideas use­ful in one dir­ec­tion to ap­par­ently com­pletely dif­fer­ent frame­works.

The good news is that math­em­at­ics is spread­ing thanks to the In­ter­net and to the at­ti­tude that math­em­aticians have to make their work avail­able (about a quarter of all pa­pers are now avail­able on In­ter­net even be­fore they are ac­cep­ted for pub­lic­a­tion). I think this is great be­cause it makes lit­er­at­ure ac­cess­ible to all. Also, maths is cheap­er to do than oth­er sci­ences, as it needs less in­vest­ment in terms of labs and equip­ment.

As far as my ex­per­i­ence with de­vel­op­ing coun­tries, I lived in Mad­a­gas­car for three months, which is not very long, but nev­er­the­less was an eye-open­ing ex­per­i­ence. I still have links with people there, and I’m try­ing to help the loc­al de­part­ment of math­em­at­ics to fur­ther de­vel­op.

What is your im­pres­sion of TWAS and its role in the de­vel­op­ing world?

I think TWAS is won­der­ful in what it does for de­vel­op­ing coun­tries, es­pe­cially the fel­low­ships pro­gram.