Celebratio Mathematica

Georgia Benkart

Interview with Georgia Benkart

by J. Sriskandarajah

JS: As an un­der­gradu­ate and gradu­ate stu­dent, were there a reas­on­able num­ber of fe­male stu­dents in your math classes?

GB: In vir­tu­ally all my math­em­at­ics classes, there were only one or two wo­men.

Dur­ing the my time as a gradu­ate stu­dent at Yale, the en­tire gradu­ate math­em­at­ics pro­gram had only four or five fe­male stu­dents, and the un­der­gradu­ate col­lege at Yale was just in the pro­cess of be­com­ing coed. Per­haps this sounds a little strange com­ing from a math­em­atician, but ac­tu­al num­bers are not so im­port­ant. The at­mo­sphere is the key, and I was for­tu­nate to be sur­roun­ded by won­der­ful fel­low stu­dents and en­cour­aging pro­fess­ors who cre­ated a co­oper­at­ive, sup­port­ive en­vir­on­ment for learn­ing math­em­at­ics.

The num­bers of wo­men study­ing math­em­at­ics have changed sig­ni­fic­antly. Nowadays over 40% of the math­em­at­ics ma­jors are wo­men, and about 28% of the Ph.D.s in math­em­at­ics are earned by wo­men com­pared to the roughly 7% when I got my doc­tor­al de­gree. The faces in math­em­at­ics classrooms are quite dif­fer­ent now, but I hope that same spir­it of co­oper­a­tion and nur­tur­ing will al­ways be present. In 2005, only 9% of the ten­ured math­em­at­ics fac­ulty at four year col­leges and uni­versit­ies were wo­men. That is something that needs to change if the math­em­at­ic­al en­ter­prise has a chance of con­tinu­ing.

JS: Let’s start with your child­hood. What im­pres­sion did grade school make on you?

GB: I al­ways have loved sci­ence and math­em­at­ics and re­mem­ber join­ing the sci­ence club as soon as I could. Are there any teach­ers who had in­flu­enced you to be­come a math­em­atician? The hon­ors math­em­at­ics pro­gram at Ohio State Uni­versity was the de­term­in­ing factor in my be­com­ing a math­em­at­ics ma­jor. We were treated to small classes and ex­cel­lent teach­ers who en­cour­aged us to take gradu­ate courses when they thought we could do well in them. Pro­fess­ors Joan and Jim Leitzel and Joe Fer­rar stim­u­lated my in­terest in ab­stract al­gebra with chal­len­ging courses, and I re­mem­ber quite fondly a gradu­ate \( p \)-ad­ic ana­lys­is course I took when I was a ju­ni­or from the fam­ous num­ber the­or­ist Kurt Mahler.

As a gradu­ate stu­dent, I took a course in Lie al­geb­ras from the group the­or­ist Wal­ter Feit. Even though there were ex­perts in Lie the­ory on the fac­ulty at Yale, he was teach­ing the course that term be­cause he wanted to study the sub­ject. So we all struggled to learn the top­ic to­geth­er, and that is how I be­came in­ter­ested in Lie al­geb­ras.

JS: What town did you grow up in and how did you end up in UW-Madis­on?

GB: I grew up in Young­stown, Ohio, and had nev­er been in the state of Wis­con­sin be­fore I ac­cep­ted a two-year postdoc­tor­al in­struct­or­ship at UW-Madis­on. Some­how two years trans­lated in­to a ca­reer at Madis­on.

JS: Did your fam­ily in­flu­ence your in­tel­lec­tu­al de­vel­op­ment in any par­tic­u­lar dir­ec­tions? Tell me about grow­ing up and be­com­ing a math­em­atician?

GB: My par­ents en­cour­aged our edu­ca­tion­al activ­it­ies but didn’t try to in­flu­ence the sub­jects we stud­ied. I star­ted out want­ing to be­come a chem­ist but soon found I was al­ler­gic to most things in the lab in­clud­ing the work. My sis­ter went in an en­tirely dif­fer­ent dir­ec­tion and earned a Ph.D. in his­tory.

JS: You have re­ceived UW’s highest prize award for dis­tin­guished teach­ing in 1987, and you were named the Polya Lec­turer of the MAA in 2000. What is the best part of be­ing a math­em­atician?

GB: The two things that I have en­joyed most are the ex­cite­ment of dis­cov­ery and the people I have en­countered along the way.

JS: What is the worst part?

GB: It is dif­fi­cult to com­mu­nic­ate to oth­ers, es­pe­cially to non-math­em­aticians, what math­em­at­ics is and what it is we do. So many people have such neg­at­ive im­pres­sions about the sub­ject.

JS: I want to talk about how you do math­em­at­ics and how you did it. Has it changed over the years? Did you do it dif­fer­ently at 30 than you did at 40, 50?

GB: Math­em­at­ics is a rap­idly chan­ging field, and learn­ing is really a life­time pro­cess. The most suc­cess­ful math­em­aticians seem to be those who are will­ing and able to con­tin­ue to learn. Prob­ably the thought-pro­cesses re­main pretty much the same throughout a ca­reer, but the ac­tu­al prob­lems one con­siders might be con­stantly evolving.

JS: Tech­no­logy plays a crit­ic­al role in math­em­at­ics and what is your im­pres­sion on that? Does it hurt the stu­dent in any­way?

GB: Tech­no­logy has ex­pan­ded our cap­ab­il­it­ies im­meas­ur­ably. With sev­er­al clicks we can ac­cess in­form­a­tion that used to take weeks to loc­ate. An older col­league once told me about the days be­fore Xer­ox ma­chines when he would copy by hand an art­icle that he needed in his re­search.

Tech­no­logy has in­flu­enced how we ap­proach some prob­lems and has altered which prob­lems we tackle. It has en­abled us to col­lab­or­ate with people we haven’t even met. But we need to con­vey to stu­dents that they still need to think and cre­ate and not just push but­tons.

JS: Where do you think math­em­at­ics is go­ing, and then closely al­lied to that, where do you think it should go?

GB: The num­bers of ma­jors in what might be re­garded as ivory tower sub­jects (such as philo­sophy, clas­sics, and per­haps even math­em­at­ics) are de­clin­ing while those in the more ca­reer-ori­ented ma­jors of the health and bio­lo­gic­al sci­ences, busi­ness, and en­gin­eer­ing are in­creas­ing. Math­em­at­ics plays an im­port­ant role in these and oth­er sub­jects (that’s a mes­sage that should be com­mu­nic­ated widely), and there will be a need to de­vel­op the rel­ev­ant math­em­at­ics for these areas.

JS: What do you think makes a math­em­atician suc­cess­ful?

GB: Per­sever­ance and per­sever­ance and a healthy dose of en­thu­si­asm for math­em­at­ics.

JS: What of your math­em­at­ic­al work do you like best?

GB: I have en­joyed work­ing on the clas­si­fic­a­tion of simple Lie al­geb­ras of prime char­ac­ter­ist­ic, on the clas­si­fic­a­tion of Lie al­geb­ras graded by fi­nite root sys­tems, and on com­bin­at­or­i­al prob­lems that arise in rep­res­ent­a­tion the­ory from con­sid­er­ing com­mut­ing ac­tions.

JS: What have been some mo­ments that have stood out for you in your ca­reer so far?

GB: The times I have giv­en in­vited ad­dresses at the joint math meet­ings stand out but mostly for the sheer pan­ic of speak­ing be­fore such large audi­ences. Also ment­or­ing gradu­ate stu­dents and see­ing them suc­cess­fully com­plete their doc­tor­al de­grees has been one of the best as­pects of the job.

JS: I un­der­stand you have in­ven­ted a new branch of Al­gebra called “down-up al­geb­ras.” Can you elab­or­ate on this in lay­man’s lan­guage?

GB: If a set of ob­jects has a par­tial or­der where one ob­ject may be lar­ger or smal­ler than an­oth­er, or per­haps they are not re­lated at all, then one can define an up op­er­at­or that takes an ob­ject to the sum of the lar­ger ones that lie dir­ectly over it and a down op­er­at­or that takes an ob­ject to the sum of the smal­ler ones that lie im­me­di­ately be­neath it. Down-up al­geb­ras gen­er­al­ize the al­geb­ras gen­er­ated by such op­er­at­ors. Of­ten there is a beau­ti­ful com­bin­at­or­ics un­der­ly­ing the situ­ation, and these op­er­at­ors can re­veal much in­form­a­tion about the com­bin­at­or­ics of the ob­jects.

JS: You have served as an ed­it­or of the Journal of Algebra since 1991. Are you con­tinu­ing to serve in this ca­pa­city?

GB: After al­most 15 years as an ed­it­or of Journ­al of Al­gebra, I de­cided to resign from the board. About a year ago I joined the ed­it­or­i­al board of a new journ­al, Al­gebra and Num­ber The­ory.

JS: Who else in your fam­ily is good in math­em­at­ics?

GB: My fath­er was an en­gin­eer and was good in math­em­at­ics. My moth­er was a suc­cess­ful teach­er at all levels from kinder­garten through to col­lege. She ma­jored in Eng­lish, bio­logy, and edu­ca­tion. My own ca­reer as a math­em­at­ics pro­fess­or re­flects their in­terests.

JS: What is your ad­vice to col­lege stu­dents and new teach­ers?

GB: Find something you en­joy and de­vote your en­er­gies to it, but don’t be afraid to try new things. Chal­lenge your­self.

Jegan­ath­an Sriskanda­ra­jah, known to his stu­dents as “Sri,” has been teach­ing math­em­at­ics at Madis­on Area Tech­nic­al Col­lege (MATC) since 2000. Sriskanda­ra­jah foun­ded and ad­vises the award-win­ning MATC math club, and chairs the Wis­con­sin chapter of the Math­em­at­ic­al As­so­ci­ation of Amer­ica.