Celebratio Mathematica

Ruth J. Wil­li­ams is one of the world’s lead­ing prob­ab­il­ists. Her work com­bines a pen­chant for ap­plied prob­lems with a love of rig­or­ous the­or­et­ic­al de­vel­op­ment. Much of her re­search has trained her for­mid­able tech­nic­al prowess on math­em­at­ic­al ques­tions arising in the dy­nam­ics of stochast­ic mod­els of com­plex net­works. Her work helped to build and ex­pand the the­or­et­ic­al found­a­tions for re­flect­ing dif­fu­sion pro­cesses, par­tic­u­larly re­flect­ing Browni­an mo­tions. She has de­veloped flu­id and dif­fu­sion ap­prox­im­a­tions for the ana­lys­is and con­trol of gen­er­al stochast­ic net­works and has brought sys­tem­at­ic ap­proaches to prov­ing heavy traffic lim­it the­or­ems. Her most cur­rent work, in col­lab­or­a­tion with sys­tems bio­lo­gists, cen­ters on stochast­ic mod­els for epi­gen­et­ic cell memory.

Born in Aus­tralia, Wil­li­ams com­pleted a Bach­el­or of Sci­ence (Hon­ors) de­gree in math­em­at­ics in 1976 at the Uni­versity of Mel­bourne. She also earned a Mas­ter of Sci­ence de­gree there in 1978 be­fore mov­ing to the United States to at­tend Stan­ford Uni­versity, where she earned her PhD in 1983 un­der the dir­ec­tion of Kai Lai Chung.

After a postdoc­tor­al po­s­i­tion at the Cour­ant In­sti­tute of Math­em­at­ic­al Sci­ences at New York Uni­versity, she joined the fac­ulty of the Uni­versity of Cali­for­nia at San Diego (UC­SD), where she is now Dis­tin­guished Pro­fess­or of the Gradu­ate Di­vi­sion, Dis­tin­guished Pro­fess­or of Math­em­at­ics Emer­ita, and former hold­er of the Charles Lee Pow­ell Chair in Math­em­at­ics I.

Wil­li­ams re­ceived a Na­tion­al Sci­ence Found­a­tion Pres­id­en­tial Young In­vest­ig­at­or Award in 1987 and an Al­fred P. Sloan Fel­low­ship the fol­low­ing year. The In­sti­tute for Op­er­a­tions Re­search and the Man­age­ment Sci­ences (IN­FORMS) awar­ded her both the 2016 John von Neu­mann The­ory Prize (jointly with her col­lab­or­at­or Mar­tin Re­iman) and the 2017 Award for the Ad­vance­ment of Wo­men in Op­er­a­tions Re­search and the Man­age­ment Sci­ences. Wil­li­ams was elec­ted to the Amer­ic­an Academy of Arts and Sci­ences in 2009, to the US Na­tion­al Academy of Sci­ences in 2012, and as a Cor­res­pond­ing Mem­ber of the Aus­trali­an Academy of Sci­ence in 2018.

What fol­lows is an ed­ited ver­sion of an in­ter­view with Ruth Wil­li­ams that took place over two ses­sions in June 2024.

To start at the be­gin­ning — can you tell me about your child­hood grow­ing up in Aus­tralia?

I was born in Mel­bourne, which is the cap­it­al city of Vic­tor­ia, one of the south­ern states in Aus­tralia. When I was in grade three, we moved to an in­land city called Bendigo, about 100 miles north of Mel­bourne. That’s where I spent most of my child­hood. We moved be­cause of my fath­er’s job. He ori­gin­ally worked for the fed­er­al gov­ern­ment, the Aus­trali­an Pub­lic Ser­vice, in Mel­bourne. When we moved to Bendigo, he be­came head of busi­ness stud­ies at the loc­al tech­nic­al col­lege, the Bendigo In­sti­tute of Tech­no­logy.

What was his back­ground?

He grew up in the de­pres­sion years and ori­gin­ally worked for the post of­fice in ac­count­ing, I think. He was also study­ing com­merce part-time at the Uni­versity of Mel­bourne. Then World War II came along, and he was an of­ficer in the navy. After the war, he worked as an audit in­spect­or for the navy and the fed­er­al gov­ern­ment pub­lic ser­vice. He also went to night school to com­plete his Bach­el­or of Com­merce de­gree at the Uni­versity of Mel­bourne.

He stud­ied ac­count­ing, so he must have been good at math.

I be­lieve so. He was very lo­gic­al and math­em­at­ic­ally in­clined. He was also vis­ion­ary about com­puters in edu­ca­tion. When he moved to the tech­nic­al col­lege in Bendigo, it was the early 1960s, and there were very few com­puters in edu­ca­tion in Aus­tralia. He was very ef­fect­ive at get­ting main­frame com­puters for the col­lege. That was a big deal. He also served on an Aus­tralia-wide com­mit­tee, called a Com­mon­wealth Com­mit­tee, ad­vising about com­puters in edu­ca­tion. He made a couple of over­seas trips for that com­mit­tee, in 1968 and 1972, to look at how com­puters were used in edu­ca­tion around the world. I re­mem­ber those trips. He was away for — it must have been three or four months at a time. Travel in those days took longer and was very ex­pens­ive. Once one was over­seas, one made the most of it.

Where did he travel to?

He went largely to the US, Canada, and the UK. He wrote ex­tens­ive re­ports; I still have one of them. He helped bring com­puters in­to edu­ca­tion in Aus­tralia, es­pe­cially in Bendigo, and also made strong con­nec­tions to in­dustry as well as to the loc­al high schools. He star­ted a com­puter club for high school stu­dents with the slo­gan, “Your Com­puter for 20 Cents.” It was 20 cents a year to join the com­puter club, and you could sub­mit as many com­puter jobs as you wanted to the main­frame com­puter — but the max­im­um num­ber of runs you could get per day was one! That was a really great thing. I got to see early on how com­puters could be a very use­ful tool.

There is a pic­ture of three girls, in­clud­ing you, who were mem­bers of the Bendigo Com­puter Club. That was the club your fath­er put to­geth­er?

Yes. There were many more mem­bers of the club; only three are in the photo. Ini­tially there were pro­gram­ming classes on Sat­urday morn­ing where any­body from the Bendigo com­munity could go. Quite a few teach­ers from loc­al high schools went and learned a little bit of pro­gram­ming, and high school stu­dents too. Even­tu­ally I taught those classes as a mem­ber of the com­puter club. It was really a great thing.

What lan­guage did you pro­gram in?

Ini­tially I wrote in FOR­TRAN, but I also wrote a few pro­grams in AL­GOL and CO­BOL. I wrote about 100 pro­grams while I was in high school. They were batch jobs, so every time you made a mis­take you had to wait a day to send in a cor­rec­ted pro­gram. Ac­cess was via key­punch cards, which made it a long pro­cess. There were no ter­min­als at that time, oth­er than the one that was at­tached to the com­puter it­self.

What were the pro­grams you wrote? What prob­lems were you work­ing on?

I re­mem­ber try­ing to write a pro­gram to clas­si­fy plants, one to make a de­cision tree, one for air­line re­ser­va­tions. There were oth­er, sim­pler things like com­put­ing us­ing some math­em­at­ic­al for­mu­las. This was be­fore there were hand-held cal­cu­lat­ors and the like.

So you were do­ing this in high school.

In my spare time!

And pri­or to that, did you al­ways like math­em­at­ics? Did you have in­terests in oth­er areas?

I liked math­em­at­ics and sci­ence equally well. I liked solv­ing prob­lems — un­der­stand­ing why things are true, work­ing things out, as op­posed to just mem­or­iz­ing them.

In math­em­at­ics you can un­der­stand things to the very bot­tom.

Yes. I really like to dig down and un­der­stand at a fun­da­ment­al level. That’s one at­trac­tion of math­em­at­ics for me.

Did you have good math teach­ers?

Yes, I had a teach­er in fifth and sixth grade — it was the same teach­er for both — Mr. Dav­ies, who really en­cour­aged me in math­em­at­ics. I liked to al­ways get things ex­actly right, and he en­cour­aged that. Then in high school I had a num­ber of math and sci­ence teach­ers who en­cour­aged me, in chem­istry, phys­ics, and math. Some went out of their way to help; I had a chem­istry teach­er, Mr. Lake, who had me read ahead to the next year’s ma­ter­i­al.

Did you en­counter the at­ti­tude that girls can’t do math and sci­ence?

I don’t think so. I had a strong sup­port struc­ture. I had par­ents and teach­ers who were very sup­port­ive of me.

Is there any­thing dif­fer­ent about Aus­tralia that would make it more ac­cept­able for girls to be in­ter­ested in these sub­jects?

I’m not sure. I was good at math­em­at­ics, and people who were good at something were en­cour­aged. I was al­ways in­ter­ested in hav­ing some kind of sci­entif­ic ca­reer, which in­cludes math­em­at­ics. I al­ways felt like that was pos­sible.

You didn’t feel there were obstacles, or that this was something strange for a girl to do. That’s great.

Well, there wer­en’t that many wo­men I could name who had sci­entif­ic ca­reers! But I just felt that it was a nat­ur­al thing that I was good at and en­joyed.

And your moth­er, did she have a pro­fes­sion also?

Yes, she was trained as a nurse. When she had chil­dren she stopped work­ing, but as we got older, she went back to nurs­ing. She be­came a nurse at a home for the aged blind in Bendigo, and even­tu­ally be­came dir­ect­or of nurs­ing there. My moth­er was great. She was a very caring per­son and also was al­ways very en­cour­aging. I had great par­ents; I was very lucky.

And sib­lings?

I have a broth­er and a sis­ter, both young­er.

What did they end up do­ing?

My broth­er is an elec­tric­al en­gin­eer in Sydney. My sis­ter lives in Bendigo, and she was a nurse — an op­er­at­ing theat­er nurse ac­tu­ally, which is an in­tense pro­fes­sion.

You have to have strong nerves for that! Did your broth­er get a PhD?

No, he earned a Bach­el­or of Sci­ence (Hon­ors) de­gree, spe­cial­iz­ing in elec­tric­al en­gin­eer­ing from the Uni­versity of Mel­bourne. My sis­ter trained as a nurse at the Roy­al Mel­bourne Hos­pit­al School of Nurs­ing. Nurs­ing was not a uni­versity de­gree at that time. Later she earned her Bach­el­or of Health Sci­ence in nurs­ing from La Trobe Uni­versity and took vari­ous post­gradu­ate spe­cial­ized train­ing courses, for ex­ample, re­lated to sur­gery.

It sounds like your par­ents had a huge pos­it­ive in­flu­ence on all three of you. After high school, you went to the Uni­versity of Mel­bourne for your un­der­gradu­ate stud­ies. Was that the closest uni­versity to Bendigo?

The way it worked was that when you ap­plied to go to uni­versity in Aus­tralia, you usu­ally ap­plied to the uni­versit­ies in your own state. It was dif­fi­cult to go to a uni­versity in an­oth­er state, oth­er than maybe to the Aus­trali­an Na­tion­al Uni­versity, which is in Can­berra, the cap­it­al of Aus­tralia. It would ac­cept stu­dents from any state.

Mel­bourne was the best uni­versity in the state of Vic­tor­ia. There were com­pet­it­ive statewide ex­ams at the end of high school, ex­ams in dif­fer­ent sub­jects. That de­term­ined what uni­versit­ies you got ad­mit­ted to. You could give a pref­er­ence, but the ex­ams gave you a rank­ing. Mel­bourne was the hard­est one to get in­to.

Did you know what you wanted to study?

I knew I wanted to study sci­ence, but I wasn’t sure wheth­er I wanted to do chem­istry, phys­ics, or math­em­at­ics. In the first year I stud­ied all three — ac­tu­ally I stud­ied two kinds of math­em­at­ics, pure and ap­plied. At the end of the first year I needed to de­cide, and I de­cided to do math­em­at­ics. Again, it goes back to be­ing able to drill down and un­der­stand why things were true. I felt it was a bit harder to do that with phys­ics and chem­istry. In chem­istry, the mod­el of the atom seemed to change every year, which was a little dis­turb­ing! Phys­ics was closer to math­em­at­ics. I was good at all three, but I thought math­em­at­ics en­abled me to un­der­stand things at a fun­da­ment­al level and that you could do a lot of things with it.

I also in­form­ally minored in what would now be called com­puter sci­ence and was then called in­form­a­tion sci­ence. So I took com­put­ing courses as well. That was easy to do.

You were already a hack­er, right? So to speak!

I don’t think I was a hack­er! In the first year, there was a little com­pon­ent of com­puter pro­gram­ming in an ap­plied math course, and I found that pretty easy be­cause I could already pro­gram, be­cause of my ex­per­i­ence with the Bendigo Com­puter Club. So then I was help­ing oth­er stu­dents. But it got more in­ter­est­ing in later years. There were things that I hadn’t stud­ied be­fore, such as al­gorithms and data struc­tures, and also dif­fer­ent lan­guages. I learned SNO­BOL and Pas­cal at some point.

It’s of­ten said today that girls go in­to com­puter courses lack­ing the kind of pro­gram­ming ex­per­i­ence that boys are like­li­er to have. But at that time you were un­usu­al even to have any pro­gram­ming ex­per­i­ence at all.

That’s true. That was one of the lucky things about be­ing in Bendigo, where there was the com­puter club that gave you ac­cess to a com­puter. At the time it was very un­usu­al, es­pe­cially in a re­gion­al city like Bendigo. Maybe it was a bit more com­mon in Mel­bourne. There were a few stu­dents I ran in­to at uni­versity who had a bit of pro­gram­ming ex­per­i­ence, but it was very un­usu­al at that time to have that kind of ac­cess.

Those two girls in the photo from the Bendigo Com­puter Club — those were friends of yours?

Yes. We were all mem­bers of the com­puter club. I knew them pretty well.

Did they go on to do something in math or sci­ence?

I don’t know. They didn’t go to the Uni­versity of Mel­bourne, and I lost track of them. There were some oth­er people at the high school I went to who did go to the Uni­versity of Mel­bourne. Since I was from the “coun­try”, I couldn’t live at home to go to uni­versity, so I lived in a res­id­en­tial col­lege, St. Hilda’s Col­lege. There were quite a few girls from my high school who also lived in St. Hilda’s Col­lege, so that was nice. And all of them were study­ing sci­ence. One of them, Sue Win­zar, had been a mem­ber of the Bendigo Com­puter Club. Sue had a very suc­cess­ful ca­reer in IT, even­tu­ally head­ing up IT for Esso in Aus­tralia.

Did you have oth­er in­terests as a young per­son, for ex­ample, mu­sic? Did you play a mu­sic­al in­stru­ment?

I learned the re­cord­er in primary school! I’m not really good at mu­sic­al in­stru­ments. I like out­door things, that’s my re­lease.

Does that come from liv­ing in Aus­tralia?

I think so. I think that’s a part of the Aus­trali­an spir­it, to like to be out­doors. I played some ten­nis and did some run­ning for a while. Also, my fam­ily would of­ten spend week­ends and hol­i­days work­ing at a wheat and sheep farm that we owned. This was an­oth­er as­pect of my love of the out­doors.

A wheat and sheep farm! Can you say more about that?

It was ori­gin­ally the farm that my moth­er had grown up on. Her sis­ter and broth­er-in-law had owned it for a while, but then they moved in­to town, and we bought it from them when I was around five years old. It’s one of the reas­ons we moved from Mel­bourne to Bendigo, be­cause it was closer to the farm. We would go there on the week­ends. Farms al­ways need something to be done! Wheat and sheep are good be­cause they tend to look after them­selves fairly well, but there were al­ways things to be done, like mend­ing fences and mak­ing sure wa­ter didn’t flow in the wrong places. Shear­ing time was a busy time, and we would help with round­ing up the sheep and gath­er­ing wool from the shear­ing. I re­mem­ber jump­ing on the wool bales to press them down. Feed­ing the shear­ers was a non­stop task be­cause they con­sume a lot of en­ergy.

Pro­gram­ming com­puters dur­ing the week and then go­ing on the week­end in­to the coun­try — that sounds like a great bal­ance.

Yes, I think it was a great ex­per­i­ence for me as a child. I have a lot of good memor­ies from that.

Did you ever par­ti­cip­ate in math com­pet­i­tions when you were grow­ing up in Aus­tralia?

No. I didn’t even know they ex­is­ted — and I am not sure they did ex­ist in Aus­tralia when I was a kid. But I did have one really nice ex­per­i­ence when I was in the last year of high school. There was a con­fer­ence in Mel­bourne cel­eb­rat­ing Edis­on’s birth­day, and they se­lec­ted one stu­dent from each high school in the state of Vic­tor­ia to go. And I was chosen to go from my high school. That was a really in­ter­est­ing ex­per­i­ence. It was a few days in Mel­bourne, and people came to speak about the fu­ture of sci­ence. I got to meet oth­er stu­dents from around the state of Vic­tor­ia. Edis­on, of course, had been from the United States, so the con­fer­ence had an in­ter­na­tion­al feel. I was very lucky to do that.

The speak­ers were from Aus­tralia?

Some of them were from over­seas, but some were from Aus­tralia. I re­mem­ber there were people who talked in par­tic­u­lar about the fu­ture of com­put­ing.

Go­ing back to your time at the Uni­versity of Mel­bourne, were there pro­fess­ors there who were spe­cial to you, who en­cour­aged you?

Yes, es­pe­cially as I got in­to my third year. In Aus­tralia, the reg­u­lar bach­el­or’s de­gree was a three-year de­gree. Then you could do an hon­ors year, if you got high enough marks, and would write an hon­ors thes­is as well as take some courses. So I did that. In my second and third years, I had taken a couple of courses in ana­lys­is from a pro­fess­or, Jerry Ko­liha, and really liked them, so I chose to do my hon­ors thes­is with him on pseudo-in­verses of op­er­at­ors. It’s part of func­tion­al ana­lys­is. I really liked his math­em­at­ic­al style. He was very pre­cise but also very en­thu­si­ast­ic about math­em­at­ics. That was a good ex­per­i­ence.

Then I stayed on to do a mas­ter’s de­gree, which at the Uni­versity of Mel­bourne was purely by re­search. I did re­search in dif­fer­en­tial games, with Dav­id Wilson, which I also en­joyed.

What does that mean, dif­fer­en­tial games?

It’s game the­ory, but where the state of the sys­tem that you are look­ing at is dy­nam­ic. It’s gov­erned by a dif­fer­en­tial equa­tion. You have two or more play­ers who have con­trols in those dif­fer­en­tial equa­tions that can in­flu­ence the be­ha­vi­or. A way to think about it is that with op­tim­iz­a­tion or con­trol, you have one play­er who’s try­ing to con­trol the be­ha­vi­or of a dif­fer­en­tial equa­tion. In dif­fer­en­tial games, you have more than one play­er, and they are com­pet­ing against one an­oth­er.

Dif­fer­en­tial games are used for mod­el­ing dy­nam­ic situ­ations that arise in a whole host of ap­plic­a­tions. At the time there was a lot of in­terest in stra­tegic ap­plic­a­tions, but today dif­fer­en­tial games are used in oth­er areas, in en­gin­eer­ing and even in data sci­ence.

Did you pub­lish a pa­per about that re­search?

I pub­lished two pa­pers.

That’s im­press­ive. How did you come to the idea of go­ing for a PhD in math­em­at­ics? Did you al­ways have it in mind, or did your pro­fess­ors en­cour­age you?

I think I al­ways had it in mind as the next nat­ur­al step. Cer­tainly I had in my mind since I was an un­der­gradu­ate that I would like to be­come a uni­versity pro­fess­or. At the time, it was usu­al for Aus­trali­an stu­dents who wanted to pur­sue a PhD to go over­seas. Usu­ally at that time people would go to the UK. I did ap­ply to the UK, but I also ap­plied to a couple of places in the US. Since I’d been do­ing re­search on dif­fer­en­tial games, I wanted to at least have the op­tion of con­tinu­ing that.

So I looked for places that had a strong math­em­at­ics de­part­ment but where there was some­body do­ing dif­fer­en­tial games. I ap­plied to both Stan­ford and Berke­ley and got ad­mit­ted to both. Stan­ford had someone in aero­naut­ic­al en­gin­eer­ing who did dif­fer­en­tial games. I got ad­mit­ted to the math pro­gram, but I would have been able to work with that per­son if I’d wanted to. That’s how I chose Stan­ford. I could have gone to the UK, but if I wanted to con­tin­ue do­ing dif­fer­en­tial games, it was a bit bet­ter to come to the US.

It’s def­in­itely a dif­fer­ent setup go­ing to the US versus go­ing to the UK. In the UK, you fo­cus very quickly on what you want to pur­sue for your PhD, where­as in the US there is a lot more course­work first, and then you choose what you want to do. The Aus­trali­an sys­tem is a lot like the Brit­ish sys­tem. I feel like I got the best of both sys­tems, be­cause I got my early edu­ca­tion in Aus­tralia and then I came to the US and got the be­ne­fits of gradu­ate edu­ca­tion here. It’s very easy here to pick up new sub­jects, be­cause you can of­ten just go and take a course.

You knew you were go­ing abroad and would leave your fam­ily. Was that dif­fi­cult?

Es­pe­cially at that time, when air travel was not so com­mon, it was a big step. It was very ex­pens­ive to travel back home, so I only went back a couple of times to Aus­tralia dur­ing my PhD. But people wrote let­ters in those days, so I would write a let­ter every week, and my par­ents would write a let­ter every week. We used aero­grammes, as they were less ex­pens­ive than a let­ter in a sep­ar­ate en­vel­ope. Phone calls were also very ex­pens­ive and only used rarely. As an in­ter­na­tion­al stu­dent, I did have a host fam­ily, the Rich­erts, who would in­vite me, to­geth­er with some oth­er Stan­ford gradu­ate stu­dents, to events like Thanks­giv­ing. That was a good way to ex­per­i­ence Amer­ic­an fam­ily life. The Rich­erts were very kind to me, and we have stayed in touch over the years.

Was there much of a cul­ture shock to go to Cali­for­nia, to a uni­versity that must have had a very dif­fer­ent style and feel from Mel­bourne?

In some ways yes, al­though Cali­for­nia is quite a bit like Aus­tralia, in terms of the ve­get­a­tion and cli­mate. I think the first thing that struck me was that the lec­tures in Aus­tralia were more form­al than the lec­tures in the United States. At the time, there was not very much re­li­ance on text­books in Aus­tralia. A lec­turer would write on the board very care­fully, and you would take very care­ful notes. So I was used to learn­ing just from whatever some­body presen­ted in a lec­ture, rather than read­ing a text­book. I think that’s a good skill to have, to be able to take notes and listen well.

Of course, the sys­tem as I said is a bit dif­fer­ent too, in that, for a PhD in Aus­tralia, once you start, you’ve already se­lec­ted who you want to do your PhD with. The PhD was short­er, three years, be­cause people were usu­ally sup­por­ted on fel­low­ships. In the US in math­em­at­ics, people are most of­ten sup­por­ted on TA­ships [teach­ing as­sist­ant­ships]. I was for­tu­nate for the first couple of years to have par­tial sup­port from a schol­ar­ship from the Uni­versity of Mel­bourne. That meant I didn’t have to TA quite as much in the first few years, though mainly I was sup­por­ted by a TA­ship from Stan­ford. For the TA­ship, I largely had to teach my own courses, which had about thirty stu­dents. I was re­spons­ible for teach­ing the whole course, in­clud­ing grad­ing and ex­am­in­a­tions. That was a very good ex­per­i­ence.

At that time, in PhD pro­grams in the US, there was course­work in the be­gin­ning that you didn’t have in Aus­tralia or Bri­tain. I took prob­ab­il­ity courses from Kai Lai Chung and oth­er people. One nice thing about Stan­ford was that there were people do­ing prob­ab­il­ity in dif­fer­ent parts of the uni­versity. There were people in stat­ist­ics, in op­er­a­tions re­search, even in the busi­ness school, who were do­ing ap­plied prob­ab­il­ity, who were in­vent­ing new prob­ab­il­ity mod­els and ana­lyses.

My first year I took a game the­ory course as well, be­cause I wasn’t sure if I still wanted to do dif­fer­en­tial games at that point. That was very in­ter­est­ing, be­cause game the­ory was in the eco­nom­ics de­part­ment. They had a sum­mer pro­gram in game the­ory, and they se­lec­ted me to write the notes from the course they were teach­ing. I got to meet some fam­ous people in game the­ory, like Robert Au­mann. That was pretty amaz­ing. I wouldn’t have met those people in Aus­tralia. That’s one of the bo­nuses of hav­ing come to the US.

Even­tu­ally I de­cided to do prob­ab­il­ity rather than game the­ory. And be­ing a prob­ab­il­ity stu­dent at Stan­ford was great, be­cause there were lots of vis­it­ors com­ing through to give a talk or to stay for longer peri­ods of time, and you got to meet lu­minar­ies in prob­ab­il­ity. It was an ex­cel­lent place to be for that.

You and some oth­er wo­men PhD stu­dents took a read­ing course with Samuel Karlin. Can you tell me about that?

We were the three wo­men out of about a dozen PhD stu­dents in my year. We had a lot of our ini­tial classes to­geth­er, but it also turned out that we were all in­ter­ested in prob­ab­il­ity. I think we’d taken the ba­sic gradu­ate prob­ab­il­ity course to­geth­er. Sam Karlin was writ­ing a book on stochast­ic pro­cesses — ac­tu­ally a two-volume set — with Howard Taylor [of Cor­nell Uni­versity]. Stochast­ic pro­cesses wer­en’t em­phas­ized so much in the ba­sic prob­ab­il­ity course, so we wanted to learn more about that. We asked Karlin if he would do a read­ing course with us.

The three of us would meet with him once a week and go over the read­ing we’d been do­ing and ask ques­tions. There were some long chapters, in­clud­ing one of more than 100 pages on dif­fu­sion pro­cesses, and we worked our way through that. That was a great ex­per­i­ence, both learn­ing to­geth­er — we would talk with one an­oth­er about what we were read­ing — but also in­ter­act­ing with Sam Karlin. He was al­ways very en­thu­si­ast­ic about whatever he was work­ing on and would tell us about his re­search. I got to meet some of the people vis­it­ing Stan­ford who were work­ing with him, like Si­mon Tav­aré. There were al­ways prob­ab­il­ity vis­it­ors com­ing through, either vis­it­ing Chung or Karlin, who were the two main prob­ab­il­ists in the math­em­at­ics de­part­ment.

What was Karlin like as a per­son?

He was al­ways very nice to us. He was very hard work­ing and com­mit­ted to what he was do­ing. You’d go in­to his of­fice, and im­me­di­ately he would start talk­ing about math. He was so en­thu­si­ast­ic about it, it was in­fec­tious.

He was also very pro­lif­ic. He star­ted col­lab­or­at­ing with bio­lo­gists at some point in his ca­reer, so there were al­ways a lot of people com­ing through, con­sult­ing with him about prob­lems they needed to have solved. I re­mem­ber him say­ing at one point that when he made the trans­ition to work­ing with bio­lo­gists, he learned you needed to have ex­per­i­ments. I al­ways re­membered that, that it’s im­port­ant to be con­nec­ted to the ex­per­i­ment­al side of bio­logy if you want to work with bio­lo­gists.

We three wo­men who were in the read­ing course all went in­to prob­ab­il­ity. I worked with Kai Lai Chung, Amy Rocha worked with Joe Keller, who had come fairly re­cently to Stan­ford, and Marge Foddy worked with Sam Karlin.

Three wo­men out of an in­com­ing class of about a dozen PhD stu­dents — that’s a large per­cent­age of wo­men.

It was a big per­cent­age. In oth­er years, there were no in­com­ing wo­men stu­dents. It was maybe a bit un­usu­al. But it was good to have oth­er wo­men in the class, es­pe­cially since we were all in­ter­ested in prob­ab­il­ity.

Was it im­port­ant to you that oth­er wo­men were there? Did you feel any of what has been called the “chilly cli­mate” that wo­men in PhD pro­grams have some­times felt — that they felt out of place or were not taken ser­i­ously?

I don’t think I really felt that, but it’s al­ways good to not be too isol­ated. It was good that there were oth­er wo­men in the pro­gram. At that time, in sci­ence and en­gin­eer­ing in gen­er­al, there wer­en’t so many wo­men. The Dean of Gradu­ate Af­fairs, Jean Fet­ter, was cog­niz­ant of this, and or­gan­ized the mak­ing of a little book­let about wo­men in sci­ence and en­gin­eer­ing at Stan­ford. In my fi­nal year, her of­fice helped to ini­ti­ate a lec­ture series for wo­men in sci­ence and en­gin­eer­ing, where wo­men fac­ulty or wo­men in in­dustry would come and give a short present­a­tion about their re­search and there would be a re­cep­tion af­ter­ward. I was the co­ordin­at­or for that. The series was a good way to con­nect with people who had gone on to suc­cess­ful ca­reers in sci­ence and en­gin­eer­ing. It was also a way to meet wo­men gradu­ate stu­dents from oth­er de­part­ments.

It sounds like you ad­jus­ted very well to Stan­ford. You found a good place for your­self.

I think it worked out well, es­pe­cially when I found prob­ab­il­ity. For every gradu­ate stu­dent there is this time when you are look­ing for an ad­visor: Who can I work with?

A peri­od of un­cer­tainty.

Yes. Kai Lai Chung was my ad­visor in math, but I got my thes­is prob­lem through tak­ing courses in oth­er de­part­ments — ac­tu­ally a course in the busi­ness school taught by Mike Har­ris­on.1 So I really had two ad­visors, Mike Har­ris­on and Kai Lai Chung. Stan­ford was a good place to do that be­cause they had a scheme where you could have ad­visors from dif­fer­ent de­part­ments. At some point [S.R.S.] Varadhan vis­ited Stan­ford, and I met with him. He gave me help­ful in­put on the prob­lem I was work­ing on and in­tro­duced me to sub­martin­gale prob­lems as a way to char­ac­ter­ize dif­fu­sion pro­cesses.

What are your memor­ies of Kai Lai Chung?

He was en­thu­si­ast­ic about math­em­at­ics but also had a me­tic­u­lous and el­eg­ant style. I liked that. It suited me well. Chung and I wrote a book to­geth­er on stochast­ic in­teg­ra­tion, which grew out of a top­ics course he taught while I was a PhD stu­dent.2

In an­oth­er ad­vanced top­ics course I took with Chung, he men­tioned an open prob­lem about the fi­nite­ness of the Feyn­man–Kac gauge re­lated to solu­tions of the re­duced Schrödinger equa­tion. A little bit later I was in a PDE course, and I real­ized that some tech­nique from the PDE course could be used to look at the prob­lem — it was ac­tu­ally a con­jec­ture. Part of the prob­lem was that it wasn’t clear wheth­er the con­jec­ture was true or not. I could use the PDE res­ult to show that, at least un­der some smooth­ness con­di­tions, the an­swer was yes, the con­jec­ture was cor­rect. That led to the first pa­per I wrote as a PhD stu­dent, be­fore I was ac­tu­ally work­ing on my thes­is prob­lem prop­er, which was on a dif­fer­ent top­ic.

When you make con­nec­tions like that, you know you are really a math­em­atician.

Yes. There’s a real thrill in dis­cov­er­ing something new and prov­ing that it’s cor­rect. It’s a bit hard to de­scribe. You work on a prob­lem really hard, and you try this way and that way, and it doesn’t work out. But then even­tu­ally you get some idea, and it just clicks. It’s really a won­der­ful feel­ing.

Of course the first thing you do is you check it many times to make sure you’re not fool­ing your­self!

Feyn­man said that the easi­est per­son to fool is your­self.

Yes, I think you have to be your own worst crit­ic. That’s a use­ful tal­ent to ac­quire, to be able to find your own mis­takes.

So you met Mike Har­ris­on through a course in the busi­ness school?

Yes. He offered a course on stochast­ic cal­cu­lus and its ap­plic­a­tions. It was a great course, and even­tu­ally he made a book out of it, which was very well re­ceived. It was pretty stand­ard at Stan­ford that gradu­ate stu­dents from oth­er de­part­ments would take prob­ab­il­ity courses in the math de­part­ment, and also prob­ab­il­ity stu­dents would take prob­ab­il­ity-re­lated courses in oth­er de­part­ments.

Dur­ing the course Mike would some­times put some prob­lems on the black­board that were al­most re­search prob­lems, or little pieces of something re­lated to re­search prob­lems. So I got a taste of the kind of things he was in­ter­ested in. It was a blend of ap­plic­a­tions and rig­or­ous math­em­at­ics. After that course, I talked to him about wheth­er there might be some re­search ques­tions there that I could work on. He sug­ges­ted something, and I went away and thought about it, came back, made a little pro­gress on it — that’s how it evolved to me work­ing on that prob­lem with him as a coad­visor, and Kai Lai Chung as my ad­visor in math. So it all worked out in the end. I just fol­lowed the math to where I thought the in­ter­est­ing prob­lems were.

That’s part of be­ing a math­em­atician too, know­ing what you find in­ter­est­ing and trust­ing that. Also hav­ing a taste for prob­lems.

I think some of that you get with ex­per­i­ence too. Maybe I was lucky that I already had some re­search ex­per­i­ence, so I was on the lookout for prob­lems. I didn’t really know prob­ab­il­ity at the gradu­ate level be­fore I went to Stan­ford, so that was something new for me. The first year I was tak­ing be­gin­ning courses, and the second year I took a prob­ab­il­ity course and then star­ted tak­ing more ad­vanced courses. So it took a little while to find prob­ab­il­ity, but it was what I was look­ing for. I really like it as a field be­cause it touches oth­er areas of math­em­at­ics, both pure and ap­plied, but also it in­ter­faces with a lot of ap­plic­a­tions. It’s the the­or­et­ic­al basis for stat­ist­ics, it in­ter­sects with op­er­a­tions re­search and with ap­plic­a­tions in lots of dif­fer­ent fields in sci­ence and en­gin­eer­ing. Re­cently I’ve been work­ing with bio­lo­gists. Prob­ab­il­ity is a per­fect field for me.

There are dif­fer­ent parts to prob­ab­il­ity. I tend to work more on the con­tinu­ous side of prob­ab­il­ity. There is also the dis­crete side, which tends to be more com­bin­at­or­i­al and al­geb­ra­ic. I am more on the side that is closer to ana­lys­is. There are a lot of con­nec­tions with ana­lys­is, but it’s built on top of ana­lys­is. If you just know ana­lys­is, it doesn’t mean you can do prob­ab­il­ity. There’s ex­tra struc­ture there, dif­fer­ent prob­lems. And there is ran­dom­ness every­where in the world, so there are lots of dif­fer­ent things to work on.

Do you re­mem­ber the so-called Monty Hall Prob­lem that was all over the news years ago? It came from a game show where con­test­ants had to choose which door to open, and one door would have a nice prize and the oth­er doors would have a gag prize like a goat. A news­pa­per colum­nist wrote about the cor­rect strategy to use in some par­tic­u­lar case, and many math­em­aticians wrote let­ters com­plain­ing that she’d got­ten it wrong. In fact the colum­nist was right. Do you re­mem­ber this?

I seem to re­call that the prob­lem de­pends on ex­actly what you know at what point, but also on how you phrase the ques­tion.

Yes. The math­em­aticians who wrote let­ters didn’t un­der­stand the prob­ab­il­ity as­pect of the prob­lem. They didn’t have the in­tu­ition of prob­ab­il­ists, which is a dif­fer­ent kind of in­tu­ition, is that right?

I agree that prob­ab­il­ity has dif­fer­ent or ad­di­tion­al in­tu­ition. I think of it as an ex­tra lay­er that you are put­ting on everything else that you know. It doesn’t mean that you can’t use all of the know­ledge you have from math­em­at­ics, but there is an ex­tra lay­er that ran­dom­ness brings.

Can you talk about that lay­er? What kind of in­tu­ition do you need to un­der­stand ran­dom things?

Of­ten as hu­man be­ings we are con­di­tioned to think de­term­in­ist­ic­ally; we want to think de­term­in­ist­ic­ally. Some­times people’s in­tu­ition leads to cer­tain as­sump­tions, for ex­ample, that things are equally likely when they are not, or that they are in­de­pend­ent when they are not. I’ve seen this with dy­nam­ic­al sys­tems, for ex­ample. People are very used to ask­ing, What is the long-term be­ha­vi­or of this dy­nam­ic­al sys­tem? If that sys­tem has noise in it, you have to phrase such ques­tions dif­fer­ently. A dif­fu­sion pro­cess, for ex­ample — which is a solu­tion of a stochast­ic dif­fer­en­tial equa­tion — could ba­sic­ally go every­where. People are used in dy­nam­ic­al sys­tems to ask­ing: Does it con­verge to zero? You have to phrase the prob­ab­il­ity ques­tions dif­fer­ently, like: How long does it take to get to a neigh­bor­hood of the ori­gin? And even­tu­ally when it gets there, it might still vis­it every­where else. If you have in­tu­ition that is re­lated to de­term­in­ist­ic mo­tion, you don’t nat­ur­ally think of those kinds of things.

So prob­ab­il­ity is great — it’s like an ex­tra as­pect that can be there, and it provides ad­di­tion­al tools for try­ing to solve things.

You men­tioned the course with Mike Har­ris­on on stochast­ic cal­cu­lus and ap­plic­a­tions. Can you give an ex­ample of an ap­plic­a­tion there?

He was very in­ter­ested in queueing and in­vent­ory con­trol. It comes up in ap­plic­a­tions in man­u­fac­tur­ing, but also the same kinds of mod­els come up in tele­com­mu­nic­a­tions and even in trans­port­a­tion. These were queueing net­work mod­els. That was my first in­tro­duc­tion to the con­nec­tion between queueing net­works and dif­fu­sion pro­cesses called re­flect­ing Browni­an mo­tions. That was a nice con­nec­tion to learn about.

Stochast­ic cal­cu­lus has as many ap­plic­a­tions as reg­u­lar cal­cu­lus has! And prob­ably more. Whenev­er you have dy­nam­ic­al sys­tems sub­ject to noise, stochast­ic cal­cu­lus is a tool for try­ing to ana­lyze those sys­tems. It comes up in phys­ics, en­gin­eer­ing, eco­nom­ics, and fin­ance. The Black–Scholes equa­tion in fin­ance is an ex­ample of a stochast­ic dif­fer­en­tial equa­tion, but you can have much more com­plic­ated fin­ance mod­els than that. Stochast­ic cal­cu­lus was in­ven­ted by Kiy­oshi Itô and is some­times called Itô cal­cu­lus. After Itô, there was a lot of de­vel­op­ment of gen­er­al the­ory, es­pe­cially by the French school of prob­ab­il­ity.

After you got your PhD you went to the Cour­ant In­sti­tute and worked with Varadhan.

I had already had some in­ter­ac­tion with him about my thes­is, as I men­tioned. Some years be­fore that, he had had a sab­bat­ic­al at Stan­ford and had talked to Mike Har­ris­on about the re­flect­ing Browni­an mo­tion prob­lems. Then when I was a PhD stu­dent at Stan­ford, Varadhan vis­ited Stan­ford for a day or so, and I met him through Mike Har­ris­on. Varadhan and I dis­cussed the re­flect­ing Browni­an mo­tion prob­lem I was work­ing on, and even­tu­ally that led to a joint pa­per.3 When I was at Cour­ant, I talked to him about vari­ous re­flect­ing Browni­an mo­tion prob­lems.

Cour­ant was a great place to be. On the top floor of the build­ing they had a tearoom where people would go for af­ter­noon tea and lunch. The fac­ulty and stu­dents would gath­er there and talk math­em­at­ics. There was a very friendly at­mo­sphere for postdocs and stu­dents. It was pretty easy to talk to the fac­ulty. You’d sit down at lunch with Cath­leen Mor­awetz and oth­ers.

How was it for you to meet Cath­leen Mor­awetz?

I knew of her, but at Cour­ant was the first time I met her. Of course she was to me a gi­ant in math­em­at­ics, though we are in dif­fer­ent fields. I think the first time I sat in the tearoom for lunch, she was talk­ing about the re­search mono­graph she’d been work­ing on over the sum­mer. It was great to be able to meet her in this in­form­al at­mo­sphere.

After I left Cour­ant, I in­ter­ac­ted a bit with Cath­leen when she was in­volved with the AMS [Amer­ic­an Math­em­at­ic­al So­ci­ety], in­clud­ing when she was AMS pres­id­ent. She was a very im­press­ive per­son and al­ways very nice to me.

There had been no wo­men on the math fac­ulty at Stan­ford, right?

There was Mary Sun­seri, but she only taught cal­cu­lus. I don’t think she did re­search. The whole time I was at Stan­ford there were no wo­men re­search­ers on the fac­ulty.

Was Cath­leen Mor­awetz the first wo­man math re­search­er you met?

No. I went to a few con­fer­ences when I was a gradu­ate stu­dent in my last year and met some wo­men re­search­ers there. I met Cath­er­ine Doléans-Dade at one of the con­fer­ences in the Mid­w­est. She was at the Uni­versity of Illinois, a very good prob­ab­il­ist. I also met Cindy Green­wood, who was at the Uni­versity of Brit­ish Columbia, and I met a few oth­er wo­men prob­ab­il­ists when I was on the job mar­ket and vis­ited some places.

Since our last con­ver­sa­tion, I googled “his­tory of com­put­ing in edu­ca­tion in Aus­tralia” and pretty quickly found your fath­er. Westy Wil­li­ams was his name?

Yes.

He was a vis­ion­ary, as you said, and he had a big ef­fect.

There’s been a little bit of his­tory writ­ten in Bendigo about what he did, in­clud­ing a book pub­lished re­cently about the his­tory of the col­lege he was at. There was a nice blurb in that book about what he did there, and they in­ter­viewed me for that. The col­lege — it’s now called La Trobe Uni­versity — named the com­puter cen­ter after my fath­er.

One art­icle I found on the web noted how hard it was at the time to move com­puters around, be­cause they were huge and heavy. They took up a whole room. You said the com­puters your fath­er got were bought from the UK?

He was in­flu­en­tial in ac­quir­ing sev­er­al com­puters over sev­er­al years. The first few com­puters I think were from Eng­land, from ICL [In­ter­na­tion­al Com­puters Lim­ited]. Later some were from CDC [Con­trol Data Cor­por­a­tion] in the US. Cer­tainly there was noth­ing be­ing man­u­fac­tured in Aus­tralia; they had to come from over­seas. The ma­chine I used was huge and needed spe­cial air con­di­tion­ing, which at that time was kind of a rar­ity. You had to try to keep the room dust-free, so ini­tially any­body who went in the com­puter room had to wear little booties. But it was trans­form­at­ive to have ac­cess to com­put­ing power like that.

When we talked last time we left off when you were at Cour­ant. Then you went to UC­SD and have stayed in San Diego your whole ca­reer. So that was a good fit for you. When did you go to San Diego? Was that 1986?

No, 1984 was when I phys­ic­ally ar­rived. The fi­nal year of my PhD, I was look­ing for jobs. I got the ten­ure-track of­fer from San Diego and the postdoc of­fer from Cour­ant. San Diego let me ac­cept its of­fer and take leave for the first year, to do the postdoc at Cour­ant. After that I went to San Diego. I’ve been very happy here, it’s been great. I was at­trac­ted to UC­SD’s strong prob­ab­il­ity group. Ron­ald Getoor and Mi­chael Sharpe were here, real ex­perts in stochast­ic pro­cesses, es­pe­cially Markov pro­cesses. So that was at­tract­ive for me. Of course San Diego has a great math de­part­ment too.

It was a tough time when I fin­ished my PhD. I thought ini­tially I would go back to Aus­tralia, but there were no aca­dem­ic jobs there. There were few in the US, so I was for­tu­nate to get both a postdoc and a ten­ure-track po­s­i­tion.

Yes, the 1980s was not an easy time for the math­em­at­ics job mar­ket in the US — but no aca­dem­ic jobs in Aus­tralia!

If I’d gone back to Aus­tralia, I could have be­come an ac­tu­ary, which is not really what I wanted to do! Not that there’s any­thing wrong with be­ing an ac­tu­ary, but I did have in my mind for a long time that I would like to be a uni­versity pro­fess­or. I like do­ing re­search and teach­ing the next gen­er­a­tion. For­tu­nately it worked out, but it wasn’t al­ways clear that it was go­ing to work out.

I’d like to talk now about your math­em­at­ic­al work. A lot of it has centered on ques­tions com­ing from prob­lems around queueing net­works. Can you say what these queueing net­works are and give a con­crete ex­ample?

I would say that my in­terests broadly are in stochast­ic pro­cesses and their ap­plic­a­tions. Ini­tially, I worked on re­flect­ing Browni­an mo­tion be­cause it was an in­ter­est­ing stochast­ic pro­cess, and there wasn’t a lot of the­ory de­veloped for it. It arose as an ap­prox­im­a­tion to a queueing net­work, but there wasn’t a lot of the­ory de­veloped in gen­er­al for those kinds of re­flect­ing Browni­an mo­tions.

Typ­ic­ally the state space was a poly­hed­ron and re­flec­tion at the bound­ary was ob­lique, where the dir­ec­tion of re­flec­tion had a dis­con­tinu­ity at the in­ter­sec­tion of bound­ary faces. This made the prob­lem non­stand­ard. In the 1980s and early 1990s, I worked a lot on de­vel­op­ing a found­a­tion­al the­ory for such re­flect­ing Browni­an mo­tions, es­pe­cially on ex­ist­ence and unique­ness and char­ac­ter­iz­ing their be­ha­vi­or. This in­cluded work with my first PhD stu­dent, Lisa Taylor. So I didn’t work on the queueing net­work side ini­tially; I worked more on the stochast­ic pro­cess side. Then over time, it be­came ap­par­ent that in mak­ing that con­nec­tion between the re­flect­ing Browni­an mo­tions and the queueing net­works, there were some in­ter­est­ing out­stand­ing prob­lems.

I’ll de­scribe what a queueing net­work is in a mo­ment! But in the early 1990s, people real­ized that these queueing net­works, when they were het­ero­gen­eous — i.e., they pro­cessed dif­fer­ent types of jobs — they could be un­stable, while the ana­log­ous ho­mo­gen­eous net­works were stable. That was a sur­prise. So try­ing to fig­ure out when they were stable, and when they could be ap­prox­im­ated by re­flect­ing Browni­an mo­tions, be­came an im­port­ant prob­lem. For ho­mo­gen­eous net­works, the ap­prox­im­a­tion had already been de­veloped by Mike Har­ris­on and Mar­tin Re­iman. But for what were called mul­ti­class queueing net­works — mean­ing the het­ero­gen­eous case — it hadn’t been fully de­veloped. There were two dif­fer­ent as­pects to that. You needed a the­ory for the re­flect­ing Browni­an mo­tions, but you also needed to prove lim­it the­or­ems that show that you could ap­prox­im­ate the mul­ti­class queueing net­works by one of these re­flect­ing Browni­an mo­tions.

To come to what queueing net­works are: You have en­tit­ies that come in­to a net­work — they could be cus­tom­ers, or jobs, or pack­ets of data, or even mo­lecules in a bio­lo­gic­al ap­plic­a­tion. These en­tit­ies need pro­cessing at vari­ous nodes or sta­tions in the net­work. They might need to queue up to wait for pro­cessing, and that’s where the queueing as­pect comes in. They might need to vis­it more than one sta­tion in the net­work be­fore they leave, and that rout­ing in the net­work can be ran­dom. So there are dif­fer­ent sources of ran­dom­ness: The times between ar­rivals can be ran­dom, the times to pro­cess things can be ran­dom, and the rout­ing in the net­work can be ran­dom.

Queueing net­works can be used to mod­el dif­fer­ent ap­plic­a­tions in sci­ence and en­gin­eer­ing — things like man­u­fac­tur­ing pro­cesses, tele­com­mu­nic­a­tions, also ori­gin­ally tele­phone net­works, later on the In­ter­net and wire­less net­works. Cus­tom­er ser­vice sys­tems and big call cen­ters have been a source of in­terest in more re­cent times. But you can use queueing net­works to mod­el even things like bio­lo­gic­al net­works, which is something I have been very in­ter­ested in re­cently.

Stochast­ic net­works — or stochast­ic pro­cessing net­works — are a more gen­er­al ver­sion of queueing net­works. A lot of the the­ory of queueing net­works star­ted to be de­veloped in the 1950s — well, it goes back even earli­er than that, but there were a lot of de­vel­op­ments in the 1950s and 1960s, though largely for ho­mo­gen­eous net­works. A lot of the work was on ex­act ana­lys­is of the net­works. Then in the 1960s, ap­prox­im­a­tions star­ted to be de­veloped, both by the Rus­si­an school, es­pe­cially Al­ex­an­dr Borovkov, and also by [Don­ald] Ig­le­hart and [Ward] Whitt in the US.4

You men­tioned ex­act ana­lys­is, and I won­der why that doesn’t al­ways work. En­tit­ies come in­to the net­work, they get pro­cessed, they move on — it’s a step-by-step sys­tem, so if you think na­ively, it seems you could cre­ate a map of everything that could hap­pen. But it’s not that simple?

There are two things that make it more com­plic­ated. One is that there is a lot of ran­dom­ness in vari­ous as­pects of the mod­el, like times between ar­rivals and pro­cessing times. These are usu­ally ran­dom vari­ables; they are not usu­ally de­term­in­ist­ic. The oth­er thing is that the rout­ing in the net­work could also be prob­ab­il­ist­ic, rather than de­term­in­ist­ic, and there can be com­plex feed­back pat­terns in the rout­ing.

You might think, well, we know the av­er­age rate at which things are ar­riv­ing, the av­er­age rate at which things are pro­cessed, and so forth. So we should be able to fig­ure out un­der what con­di­tions we can make it so that everything just flows smoothly through the sys­tem. If everything were de­term­in­ist­ic, you could do that. But be­cause of the ran­dom­ness in things like the pro­cessing times, you will of­ten get queueing. For ex­ample, there will of­ten be in­stances when the amount of time it takes to pro­cess something is great­er than the in­ter­val un­til the next ar­rival, so the next ar­rival is go­ing to have to wait for ser­vice.

This is an un­der-ap­pre­ci­ated as­pect of pro­cessing net­works, that ran­dom­ness can cause queueing. People like to think de­term­in­ist­ic­ally, but the ran­dom­ness in the sys­tem is im­port­ant, es­pe­cially if you try to bal­ance the sys­tem, which is a nat­ur­al thing to try to do. Man­agers of sys­tems try to make sure that they don’t un­der-util­ize re­sources, so they typ­ic­ally try to keep these sys­tems in what’s called “heavy traffic”, that is, where there is full util­iz­a­tion of the re­sources. But then you can get sub­stan­tial queueing and bot­tle­necks in the net­work. Ran­dom­ness in any­thing — in­terar­rival times, ser­vice times, and so on — af­fects queueing. So people try to design sys­tems to use the re­sources op­tim­ally and also to un­der­stand what is caus­ing con­ges­tion and how it might be al­le­vi­ated.

That’s where math­em­at­ic­al mod­els can help a lot, be­cause these sys­tems can be quite com­plex, and they can have feed­back. An en­tity might go around sev­er­al times to the same sta­tion, per­haps be­cause something has to re­peat pro­cessing or be­cause it’s a ma­chine that puts down many lay­ers of a ma­ter­i­al, such as in semi­con­duct­or wafer fab­ric­a­tion. Once there is feed­back in the sys­tem, it can be dif­fi­cult to tell un­der what con­di­tions the sys­tem will be stable, in the sense that the av­er­age queue lengths don’t grow without bound. In queueing there are two as­pects. One is to un­der­stand how a sys­tem be­haves and to study its per­form­ance, to meas­ure things like queue lengths or idle time in the sys­tem. An­oth­er is to fig­ure out what good con­trols are for the sys­tem. One of the things that I’ve worked on is In­ter­net con­ges­tion con­trol — for ex­ample, what kinds of policies make the sys­tem be­have well. It can be very com­plex and dif­fi­cult to un­der­stand what’s caus­ing con­ges­tion and in­stabil­ity.

I men­tioned the ex­amples that came out in the early 1990s, where queueing net­works that people thought should be stable un­der stand­ard con­di­tions turned out to not be stable, in the sense that av­er­age queue lengths could grow without bound. Even quite simple sys­tems, once they have this feed­back phe­nomen­on and cer­tain kinds of dis­cip­lines for serving or pro­cessing en­tit­ies in the sys­tem, can be­come un­stable. We un­der­stand that bet­ter now, be­cause people have worked on it since the early 1990s.

You might ask, Why wasn’t that un­der­stood earli­er? These sys­tems have been used in man­u­fac­tur­ing and tele­com­mu­nic­a­tions for a long time. Part of the reas­on is that, for ex­ample, in man­u­fac­tur­ing, there were man­agers run­ning around try­ing to re­solve con­ges­tion when they found it. That’s a kind of con­trol fea­ture that can mask an un­der­ly­ing source of in­stabil­ity in the sys­tem, when there might have been more sys­tem­at­ic ways to deal with it. That’s an ex­ample where math­em­at­ic­al mod­els really helped people to un­der­stand these mul­ti­class, or het­ero­gen­eous, queueing net­works.

You star­ted work­ing on re­flect­ing Browni­an mo­tions in the con­text of queueing net­works. Browni­an mo­tion comes from stat­ist­ic­al mech­an­ics, which makes me think of atoms col­lid­ing with each oth­er. It seems coun­ter­in­tu­it­ive that something like that would ap­ply to queueing net­works.

One of the nice things about work­ing on a the­ory for a math­em­at­ic­al ob­ject is that you might be mo­tiv­ated to study it for one ap­plic­a­tion, but then it turns out to have oth­er ap­plic­a­tions. Re­flect­ing Browni­an mo­tions arise in phys­ics and also in fin­ance. But let me ex­plain why it’s nat­ur­al for them to come up in queueing sys­tems.

It’s true that Browni­an mo­tion is a mod­el for mo­lecules col­lid­ing with one an­oth­er. But also, in stat­ist­ics, you can get Browni­an mo­tion as a scal­ing lim­it of a ran­dom walk. A ran­dom walk is a stochast­ic pro­cess that takes dis­crete steps, where those steps are giv­en by a se­quence of in­de­pend­ent, identic­ally dis­trib­uted ran­dom vari­ables. A simple ex­ample of a ran­dom walk is, you are walk­ing on a line, and every time you take a step you flip a coin. If it comes up heads, you take one step for­ward, if it comes up tails, you take one step back. If you speed up time, like you’re watch­ing a movie, and shrink the size of the steps in the right way, in the lim­it of the ran­dom walk you get Browni­an mo­tion.

Sums of in­de­pend­ent, identic­ally dis­trib­uted ran­dom vari­ables like that come up every­where in stat­ist­ics. A won­der­ful as­pect of Browni­an mo­tion is giv­en by Don­sker’s the­or­em, which says that in the ap­prox­im­a­tion of a ran­dom walk, all the ran­dom steps need to have is fi­nite mean and vari­ance, and then the stat­ist­ics of the Browni­an mo­tion that you get in the lim­it de­pend only on those two stat­ist­ics. It’s in­sens­it­ive to the rest of the dis­tri­bu­tion of those steps. That’s called an in­vari­ance prin­ciple, in the sense that you can ap­prox­im­ate quite gen­er­al ran­dom walks with gen­er­al dis­tri­bu­tions for the step sizes, by this uni­ver­sal ob­ject, the Browni­an mo­tion.

There are oth­er ex­amples of that kind of thing in stat­ist­ic­al phys­ics, things like KPZ [the Kardar–Par­isi–Zhang stochast­ic par­tial dif­fer­en­tial equa­tion], where you get an in­vari­ant ob­ject ap­prox­im­at­ing a lot of in­ter­act­ing particle sys­tems. That’s very use­ful. I’m build­ing to the con­nec­tion with queueing net­works!

Let me just talk about a single queue. In a single queue, sup­pose you have a Pois­son ar­rival pro­cess and a se­quence of in­de­pend­ent, identic­ally dis­trib­uted ex­po­nen­tial pro­cessing times, where the av­er­age ar­rival rate is equal to the av­er­age pro­cessing rate. Then the queue is in heavy traffic. When the queueing sys­tem has jobs in the queue and there is a new ar­rival, that’s like tak­ing a step up; the queue length goes up by one. If a job fin­ishes be­ing pro­cessed, the queue length goes down by one. When the queue is nonempty, it is fol­low­ing a (con­tinu­ous time) ran­dom walk, go­ing up by one and down by one. You can ap­prox­im­ate that by a Browni­an mo­tion. This ap­prox­im­a­tion can be gen­er­al­ized to where the in­terar­rival and pro­cessing times are not just ex­po­nen­tially dis­trib­uted, via an in­vari­ance prin­ciple. Now, when you get to the bound­ary — that is, when the queue length reaches zero — you need to keep the queue length non­neg­at­ive. That’s where the re­flec­tion comes in, in re­flect­ing Browni­an mo­tion. It’s not really a mir­ror re­flec­tion; an­oth­er term people have used is reg­u­lated Browni­an mo­tion. It’s a kind of con­trol, or reg­u­la­tion, at the bound­ary, which keeps the pro­cess non­neg­at­ive.

In the one-di­men­sion­al case, it’s very easy to de­scribe how to keep the queue length non­neg­at­ive. But once you go to mul­tiple di­men­sions, the con­straint at the bound­ary means that you might have one queue that be­comes empty, but it’s feed­ing in­to an­oth­er queue, and then there is lost po­ten­tial flow to the next queue. In terms of the ef­fect at the bound­ary, you get an ob­lique dir­ec­tion of re­flec­tion on the bound­ary rather than nor­mal re­flec­tion, be­cause of the struc­ture of the queueing net­work. So these re­flect­ing Browni­an mo­tions have what’s called ob­lique re­flec­tion at the bound­ary of, say, an or­thant in $d$-di­men­sion­al space, where $d$ is the num­ber of sta­tions or nodes in the net­work. They are dif­fer­ent from many of the re­flec­ted dif­fu­sions that were stud­ied earli­er, which were of­ten in smooth do­mains with nor­mal re­flec­tion.

When I star­ted to get in­volved in this area, there was a little bit of the­ory for these ob­liquely re­flec­ted Browni­an mo­tions as­so­ci­ated with ap­prox­im­a­tions to ho­mo­gen­eous queueing net­works. However, there wasn’t a the­ory to cov­er oth­er ap­prox­im­a­tions, in par­tic­u­lar, those com­ing from most het­ero­gen­eous queueing net­works. So one of the things I was in­volved in early on was de­vel­op­ing a the­ory for these re­flect­ing Browni­an mo­tions that would cov­er more of a full spec­trum of the stochast­ic pro­cesses you would ex­pect to get. I al­ways had in mind the queueing ap­plic­a­tions, but I didn’t really start to work on prov­ing queueing lim­it the­or­ems un­til a bit later on, after I had de­veloped the­ory for the re­flect­ing Browni­an mo­tions. It was in the 1990s that I star­ted work­ing on prov­ing lim­it the­or­ems. Be­fore that, I was work­ing a lot on the fun­da­ment­al the­ory for the re­flect­ing Browni­an mo­tions.

Re­gard­ing lim­it the­or­ems, in the late 1990s, Maury Bramson and I co­ordin­ated in de­vel­op­ing a mod­u­lar ap­proach5 to prov­ing heavy traffic lim­it the­or­ems for mul­ti­class queueing net­works. This in­volved us­ing the asymp­tot­ic be­ha­vi­or of hy­dro­dynam­ic lim­its, called flu­id mod­els, to prove a di­men­sion re­duc­tion on dif­fu­sion time scale, called state space col­lapse, which then fed in­to prov­ing a dif­fu­sion ap­prox­im­a­tion for the queueing net­work.

You’ve had many col­lab­or­at­ors over the years. Can you tell me about what col­lab­or­a­tion means to you?

I have been for­tu­nate to have many won­der­ful col­lab­or­at­ors. These range from PhD stu­dents and postdocs to col­leagues at UC­SD and around the world. These col­lab­or­at­ors have in­cluded math­em­aticians, as well as re­search­ers from fields of ap­plic­a­tion such as bio­logy, cog­nit­ive sci­ence, con­trol the­ory, and op­er­a­tions man­age­ment. While some col­lab­or­a­tions have been with stu­dents, postdocs, vis­it­ors and col­leagues at UC­SD, a sub­stan­tial num­ber have come from ex­ten­ded vis­its I have made to pro­grams at re­search in­sti­tutes, and also on sab­bat­ic­al vis­its to vari­ous uni­versit­ies around the world. I found such vis­its es­pe­cially use­ful for learn­ing of new re­search prob­lems and mak­ing con­nec­tions with new col­lab­or­at­ors to work with. Sup­port I re­ceived from UC­SD, as well as from vari­ous fel­low­ships and in­sti­tu­tions, was in­valu­able in mak­ing such vis­its pos­sible. This was es­pe­cially im­port­ant early in my ca­reer and has con­tin­ued to be an im­port­ant way for me to broaden my re­search.

One of my col­lab­or­at­ors is my hus­band of 30 years, Bill Helton, who is also a math­em­atician at UC­SD. Bill has a great sense of hu­mor and has been very sup­port­ive of me in my ca­reer. Al­though he is in a dif­fer­ent field, we wrote a couple of pa­pers to­geth­er. One of them, with Ghe­orghe Cra­ciun, was on ho­mo­topy meth­ods for count­ing re­ac­tion net­work equi­lib­ria and grew out of an IMA [In­sti­tute for Math­em­at­ics and Its Ap­plic­a­tions] work­shop.6 The oth­er pa­per I have with Bill was writ­ten with two oth­er col­lab­or­at­ors, Frank Kelly and Ilze Zied­ins, both ex­perts on stochast­ic net­works. That pa­per was on ana­lys­is of a traffic net­work with in­form­a­tion feed­back and on­ramp con­trols.7

You men­tioned the bio­lo­gic­al ap­plic­a­tions you have been work­ing on re­cently. Can you tell me about that?

Since the 1990s, I worked on queueing in more gen­er­al stochast­ic net­works and prov­ing lim­it the­or­ems that jus­ti­fy vari­ous con­tinu­ous stochast­ic pro­cesses as ap­prox­im­a­tions, in­clud­ing cer­tain meas­ure-val­ued (in­fin­ite di­men­sion­al) pro­cesses. I also got in­ter­ested in con­trol prob­lems for these stochast­ic net­works. In the mid-2000s, in­spired by an­oth­er work­shop at the IMA, I be­came in­ter­ested in mod­els that came out of bio­lo­gic­al net­works. I thought that was a very in­ter­est­ing dir­ec­tion for ap­plied prob­ab­il­ity. I star­ted in­vest­ig­at­ing what are called stochast­ic chem­ic­al re­ac­tion net­works. And one of the things I real­ized pretty quickly was that to work on ap­plic­a­tions re­lated to bio­logy, it was good to be con­nec­ted to people who did ex­per­i­ments, as Sam Karlin had told me.

I found there was a group here at UC­SD work­ing in syn­thet­ic bio­logy, led by Jeff Hasty and Lev Tsim­ring. They were us­ing some stochast­ic and de­term­in­ist­ic mod­els to ap­prox­im­ate small ge­net­ic cir­cuits. That seemed very suit­able as a po­ten­tial ground for stochast­ic mod­el­ing and ana­lys­is. At that time NSF [Na­tion­al Sci­ence Found­a­tion] had a nice pro­gram called In­ter­dis­cip­lin­ary Grants in the Math­em­at­ic­al Sci­ences, which aimed to en­cour­age math­em­aticians to spend a year vis­it­ing a group in an­oth­er dis­cip­line in which they had nev­er worked be­fore. This was ideal for me, be­cause I hadn’t col­lab­or­ated with bio­lo­gists be­fore. I ap­plied for one of those grants and got it, and I went to vis­it the Hasty–Tsim­ring lab for a year.

We ended up work­ing on some stochast­ic mod­els of en­zymat­ic pro­cessing, which you can think of as queueing-like mod­els. As usu­ally hap­pens, ini­tially you can use some ex­ist­ing the­ory, but then you have to ex­tend it in dif­fer­ent ways to ad­apt to the dif­fer­ent situ­ations. That was a great col­lab­or­a­tion that went on for sev­er­al years with the lab and got me my start in col­lab­or­at­ing with bio­lo­gists, es­pe­cially with people who were do­ing cel­lu­lar and mo­lecu­lar bio­logy.

The vis­it I made to the Hasty–Tsim­ring lab was partly a sab­bat­ic­al that I had from UC­SD. In 2019 I had an­oth­er sab­bat­ic­al, this time in Bo­ston, vis­it­ing the Cen­ter of Math­em­at­ic­al Sci­ences and Ap­plic­a­tions at Har­vard, and also MIT. While there I con­nec­ted with Dom­itilla Del Vec­chio, who is also a re­search­er in syn­thet­ic bio­logy. We have an on­go­ing col­lab­or­a­tion about mod­el­ing epi­gen­et­ic cell memory. I find it very in­triguing to work on stochast­ic net­work mod­els like this, where there is a need to de­vel­op some new the­ory. Also, of­ten I find if you work on an ap­plic­a­tion, ques­tions come up that haven’t already been answered by the ex­ist­ing the­ory. It’s a good way to get nat­ur­al ques­tions and make sure you are ask­ing good, chal­len­ging ques­tions. I find this col­lab­or­a­tion very ex­cit­ing.

That’s amaz­ing you jumped in­to this new area. Was it dif­fi­cult to get up to speed to be able to talk to these people who were do­ing such a dif­fer­ent kind of work?

I had col­lab­or­ated be­fore with en­gin­eers, in mech­an­ic­al, in­dus­tri­al, and elec­tric­al en­gin­eer­ing. But I found that col­lab­or­at­ing with bio­lo­gists, I had to stretch fur­ther. Bio­phys­i­cists were good in­ter­me­di­ar­ies. They of­ten use math­em­at­ic­al lan­guage but also know the ex­per­i­ment­al side. Lev Tsim­ring is a bio­phys­i­cist, and Jeff Hasty trained as a bio­phys­i­cist, al­though he is very in­volved in ex­per­i­ments now as well. There was also a tal­en­ted bio­phys­ics postdoc, Will Math­er, whom I talked with a lot when I went to the Hasty–Tsim­ring lab. That helped a great deal. Gen­er­ally in in­ter­dis­cip­lin­ary col­lab­or­a­tions, it’s a real team ef­fort.

But I found ini­tially that read­ing bio­logy pa­pers was like learn­ing a dif­fer­ent lan­guage. In a math­em­at­ics pa­per, of­ten there is a suc­cinct for­mula, and if you un­der­stand the for­mula, you un­der­stand a lot about what is go­ing on. The bio­lo­gic­al pa­pers used a lot more words and few­er for­mu­las. When I gave talks, I found that even though there might be a beau­ti­ful for­mula that you could show, it was im­port­ant to do things like show a graph of an in­stance of the for­mula — even though one for­mula is worth a thou­sand graphs! It’s just a dif­fer­ent way of present­ing things. I def­in­itely felt that there was a steep learn­ing curve in start­ing to col­lab­or­ate with bio­lo­gists, but I had very good people I could talk to. Of­ten it’s the postdocs and the gradu­ate stu­dents who you can talk to on a daily basis and learn many things from.

In en­gin­eer­ing of­ten there is already a mod­el, but in bio­logy of­ten there’s not. You can make a mod­el that’s so com­plic­ated that you wouldn’t be able to do any­thing with it. So you have to ask a lot of ques­tions about what’s im­port­ant to put in­to the mod­el. Also, there are of­ten para­met­ers that aren’t known very pre­cisely. For all of that it was very help­ful to be em­bed­ded in the lab.

After you de­veloped these bio­lo­gic­al mod­els, how were they then used?

In the Hasty–Tsim­ring lab, and also in the work that I’m do­ing now with Dom­itilla Del Vec­chio at MIT, the mod­els help to guide the ex­per­i­ments. As is typ­ic­al in ap­plied math, there is a feed­back loop: You do an ex­per­i­ment, and you find that maybe the res­ults are a bit dif­fer­ent from what the mod­el is pre­dict­ing, so you go back and re­fine the mod­el. The mod­el is also very help­ful in ex­plor­ing dif­fer­ent para­met­er re­gimes that you might not be able to fully ex­plore with ex­per­i­ments, which can be very costly. Some of the mod­els I’ve worked on might be even­tu­ally used to help com­bat dis­ease, al­though what I am do­ing at the mo­ment is more ba­sic sci­ence.

What do you do when you do math­em­at­ics? Do you go out for a walk and look at the trees? Do you lie on your back, close your eyes, and enter some oth­er world? How does it work?

I mostly sit with a pad and pa­per, but I al­ways find it’s help­ful to take a walk and think. I have to ad­mit that math is in my head a lot of the time! It’s kind of a nat­ur­al thing. When you are work­ing on a prob­lem, you first have to learn enough about it so that you can carry it around in your head and think about it. But it doesn’t take a hol­i­day — it’s al­ways in your head!

The prob­ab­il­ist Chris Burdzy had a Warschawski Vis­it­ing As­sist­ant Pro­fess­or po­s­i­tion at UC­SD some years ago. We had a prob­ab­il­ity sem­in­ar where Chris was giv­ing a talk one day. He men­tioned an open prob­lem and said that, if some­body solved it, that per­son would get an ice cream cake. Well, I was in­ter­ested in the prob­lem, not ne­ces­sar­ily the ice cream cake! But later that week I was tak­ing a walk across a park in San Diego, and I had an idea about how to solve this open prob­lem. That was an “Aha!” mo­ment. It turned out it was a good idea, and Chris and I wrote a pa­per to­geth­er.8 At the de­par­ture party for Chris when he went on to his next po­s­i­tion, the ice cream cake ap­peared.

He made good on his prom­ise.

He did, yes. Math is in my head a lot of the time. That’s just the way it is. It’s a pas­sion.

You said you have to know enough about the prob­lem to be able to carry it around in your mind. Can you talk about what that thing is that you carry in your mind? Is it a pic­ture? Equa­tions? Shapes? Or some pro­cess that un­folds? What does that look like?

I’m a visu­al thinker. A lot of the prob­lems I work on in­volve stochast­ic pro­cesses, and I am of­ten in­ter­ested in what people would call the sample path be­ha­vi­or. For ex­ample, con­sider re­flect­ing Browni­an mo­tion in three di­men­sions, which lives in the pos­it­ive or­thant, or in two di­men­sions, which lives in the pos­it­ive quad­rant. It runs around like Browni­an mo­tion, but it’s got this re­flec­tion at the bound­ary. So I have those pic­tures in my mind, of how the sample paths be­have. Of­ten you are in­ter­ested in ques­tions like, does a sample path hit the corner of the quad­rant? How long does it take to get there? Then some­times equa­tions or con­structs come to mind about how you might prove that.

I don’t really see line-by-line proofs when I visu­al­ize things, it’s more try­ing to get an idea about how to solve the prob­lem. I’m a geo­met­ric thinker. Even with stochast­ic pro­cesses, I think about the geo­metry of what’s go­ing on, es­pe­cially how these sample paths be­have, which has con­nec­tions with ana­lys­is.

A sample path for re­flect­ing Browni­an mo­tion is ba­sic­ally a con­tinu­ous path, so you are fol­low­ing this ran­dom con­tinu­ous func­tion as it moves around in a state space. The bound­ary be­ha­vi­or is a bit sin­gu­lar, so things are not usu­ally ab­so­lutely con­tinu­ous, and that’s a little bit tricky, but it’s a kind of stochast­ic dif­fer­en­tial equa­tion with state con­straints. To con­nect with ana­lys­is, a use­ful tool is stochast­ic cal­cu­lus, or Itô cal­cu­lus, which I men­tioned be­fore. When I think about how to ap­ply those tools, usu­ally then I would want to get my pen­cil and pa­per out and start writ­ing things down, be­cause it’s easy to fool your­self at that stage!

Do you do any pro­gram­ming now, say to carry out little ex­per­i­ments that might oc­cur to you?

I of­ten have some gradu­ate stu­dents, or oc­ca­sion­ally a postdoc, do some pro­gram­ming, es­pe­cially in Math­em­at­ica. So for ex­ample with the bio­logy mod­els at the mo­ment I have a very good gradu­ate stu­dent help­ing me with that, Yi Fu. A won­der­ful thing that’s changed since I first wrote com­puter pro­grams is that there are a lot more high-level lan­guages now, things like Math­em­at­ica and Mat­lab. So some­times I’ll write a little Math­em­at­ica code, maybe to il­lus­trate a the­or­em or test a con­jec­ture. If stu­dents write the code, some­times I’ll tinker with it and pro­duce ex­amples and test dif­fer­ent scen­ari­os.

Also, I coteach a com­pu­ta­tion­al fin­ance course in the Rady School of Man­age­ment, where we teach some al­gorithms, though for most of the cod­ing, we have a TA who helps.

Have you also done re­search in fin­ance, or this is just a course that you en­joy teach­ing?

I’ve al­ways had a little bit of an in­terest in fin­ance, be­cause it’s an ap­plic­a­tion of stochast­ic cal­cu­lus. So I de­veloped a course at UC­SD on math­em­at­ic­al fin­ance and wrote an in­tro­duct­ory book, which is pub­lished by the AMS.9 At UC­SD we have both an un­der­gradu­ate- and a gradu­ate-level math­em­at­ics course based on that book. In ad­di­tion, I am coteach­ing the com­pu­ta­tion­al fin­ance course with an­oth­er math­em­at­ics pro­fess­or. Fin­ance is a good ap­plic­a­tion of prob­ab­il­ity, and also stu­dents of­ten want to get some back­ground in it, to en­hance their job op­por­tun­it­ies. We have stu­dents from oth­er de­part­ments who take our math­em­at­ic­al fin­ance courses too, es­pe­cially stu­dents from eco­nom­ics, also some­times from phys­ics and en­gin­eer­ing. It’s a pop­u­lar top­ic.

It sounds like you have a lot of di­versity in your work.

Yes, I feel that, and I also feel that what I have worked on has changed with time. I’m al­ways look­ing to find new op­por­tun­it­ies. I like work­ing with oth­er people, and I like work­ing on prob­lems and de­vel­op­ing new math­em­at­ics. I have an open mind to that. I am bet­ter at cer­tain kinds of math­em­at­ics than oth­ers, so I tend more to the ana­lys­is side.

You said that in Aus­tralia you had to learn how to fol­low lec­tures and listen to people. Do you see stu­dents today de­vel­op­ing that skill? Every­one is watch­ing video lec­tures and read­ing ma­ter­i­al on the In­ter­net.

I think there is a lot of con­cern about that. I’m by no means an ex­pert, but the abil­ity to fo­cus and listen to some­body when they are speak­ing or giv­ing a lec­ture, and tak­ing notes as well, are im­port­ant skills that not every­body has these days. I think there is a tend­ency with videos for people to just skip through them and not really ab­sorb a lot. That ac­tu­ally can make it more dif­fi­cult to learn the math­em­at­ics.

There is something about writ­ing on a black­board, and the pace that comes with it, that seems to be very help­ful for people to be able to ab­sorb things. Math­em­at­ics it­self is a kind of short­hand lan­guage, and in a lec­ture you are try­ing to ab­sorb that lan­guage as well as new con­cepts. If you just fly through it in a video, you might not ab­sorb it very well. And we’ve seen that if stu­dents have videos to watch, they don’t ne­ces­sar­ily do as well as if they come to in-per­son lec­tures. I’ve seen that in my own classes, and also oth­er people have re­por­ted it. It’s a chal­len­ging is­sue. Learn­ing new math­em­at­ics takes time and pa­tience and prac­tice.

When we talked about your early life in Aus­tralia, you said you didn’t en­counter the at­ti­tude that “fe­males can’t do math­em­at­ics.” In fact, you were very much en­cour­aged by your par­ents and teach­ers. How was it later in your ca­reer? Did you feel that be­ing a wo­man in math­em­at­ics was an is­sue, or caused any prob­lems, or set up any obstacles?

One thing I no­ticed is that the fur­ther you go up, the few­er wo­men there are. I’ve tried to en­cour­age people who were more ju­ni­or than my­self. I find that I’m most help­ful at the in­di­vidu­al level. I think the most valu­able thing is if you can provide help and ad­vice that is ad­ap­ted to the in­di­vidu­al, wheth­er a man or a wo­man — things like re­view­ing a per­son’s grant pro­pos­al or com­ment­ing on a draft of their re­search state­ment. That all takes time, but those are the kinds of valu­able things that really help.

A dif­fer­ent way of help­ing is through pro­fes­sion­al ser­vice, and I have done a lot of that with vari­ous or­gan­iz­a­tions.10 I find it is one way for me to give back to the or­gan­iz­a­tions that have be­nefited me in my ca­reer and that foster math­em­at­ics and sci­ence re­search. Their role in help­ing ju­ni­or re­search­ers is es­pe­cially im­port­ant.

Go­ing back to the is­sue of be­ing a wo­man in math­em­at­ics — it’s some­times said that wo­men have less con­fid­ence than men, that wo­men lack con­fid­ence. As you spoke about your life, that didn’t seem to come up. You just were in­ter­ested in math and sci­ence, and then the ques­tion of con­fid­ence, the ques­tion of “Can I do this?”, got sub­sumed by the de­sire to learn and un­der­stand. To as­sume that you have to be ex­tremely con­fid­ent that everything will work out is maybe not ne­ces­sary or real­ist­ic.

When you are work­ing on a math­em­at­ics prob­lem, you don’t know wheth­er it’s go­ing to work out! I feel that I just have to try and fol­low what I think is in­ter­est­ing. But every­body wor­ries, “Will I be able to solve this prob­lem? Will I get my PhD? Will I get ten­ure?” I think that’s nat­ur­al. In work­ing on math­em­at­ics, there’ll be times when it’s tough, when a prob­lem doesn’t seem like it’s yield­ing, and you just have to try and push through, or step back and take an al­tern­at­ive ap­proach. Maybe with ex­per­i­ence one gains that per­spect­ive, which is a bit harder to have in the be­gin­ning. But I don’t think be­ing over­con­fid­ent is good!

You said about a pro­fess­or you had in Aus­tralia, “I liked his math­em­at­ic­al style.” Even at that early stage you were look­ing out for what was in­ter­est­ing to you, what you were drawn to, what ap­pealed to you, in­stead of wor­ry­ing, “Can I do it?”

That was def­in­itely true in my un­der­gradu­ate stud­ies. When I did my PhD, it took me a little while to find prob­ab­il­ity as the right field. I just tried dif­fer­ent things and kept look­ing. It was per­sist­ence, or stoicism — Aus­trali­ans tend to be sto­ic! Work­ing hard helps. And def­in­itely there were people who helped me along the way, and I am very grate­ful to them. I’ve been very lucky.