Exploring the wilderness: an interview with Amie Wilkinson
Interview by Anna Maria Cherubini
- Born inUnited States
- Studied inUnited States
- Lives inUnited States
Amie Wilkinson is Professor of Mathematics at University of Chicago and the author of fundamental work in dynamical systems.
Can you tell us about your research? You explore dynamical systems: which are the main issues in the field and which questions are most interesting to you?
My research is in the area of dynamical systems, which is the study of the long term evolution of a space subject to a set of deterministic rules. Such rules are given, for example, by a system of ordinary differential equations. I study smooth systems in particular. I’m very interested in considering what happens to a smooth manifold when it is acted upon by a group, the simplest example being the group generated by a single diffeomorphism of the manifold.
In this case you can picture the action of the dynamical system over time to be given by composing this diffeomorphism with itself, and one is interested in the long term behavior of this iterated system. For example, what happens to a point in the manifold when one watches it evolve over time? What structures on this manifold are preserved under the action of this group? What measures on the space of probability measures on the manifold are preserved? What fractal-type structures are preserved?
There are three different types of questions that really interest me. The first is: you take an interesting type of dynamical behavior such as mixing and you ask what mechanisms in the dynamical system will produce mixing, and by mechanisms I mean some rough property of the system that gives you only partial information about the system but will imply properties like ergodicity. And mechanisms typically produce robust behaviors, robust mixing is an example. I look for mechanisms that produce chaotic behavior robustly. Part of my research has been focused on this type of question.
So, you are interested in robust mechanisms, e.g. chaos which is not destroyed by perturbations.
One sees chaotic behavior and if the system has changed a little bit, it’s still chaotic: I search for this kind of mechanisms.
The second aspect I’m interested in is what properties are generic for a dynamical system. This is related to the previous one. For example: the centralizer of a diffeomorphism is the set of all diffeomorphisms it commutes with under composition, we can view the centralizer as representing symmetries of the system. I proved with Bonatti and Crovisier that the generic diffeomorphism in the C^1 sense has a trivial centralizer, meaning that it only commutes with its own iterates, it has no symmetries other than the obvious symmetries. That work answered a question of Smale and opens up a whole world of questions about the symmetries of typical systems.
But, in general, this interests me from two angles. One is the idea of understanding the symmetries of a dynamical system and the other is understanding typical, generic, properties of dynamical systems. There are very few properties that can be generic for all systems, if you think about it, because the set of all dynamical systems is so rich.
I have also studied the property of ergodicity from a generic perspective: with Avila and Crovisier I showed that, generically, a property for all conservative diffeomorphisms, i.e. positive entropy, implies ergodicity. So that is another example of problems about generic diffeomorphisms that I’m interested in.
And then I’m interested in questions of rigidity. There are several angles from which we can approach rigidity questions in dynamics.
One is: suppose that the group acting on a manifold is reasonably complicated, for example not the integers but something more interesting. Can you classify all the dynamical behavior with actions of that group on a given manifold? In this case you’d say that these actions are very rigid. Maybe the most dramatic example is the Zimmer conjecture, recently proved by Brown, Fisher and Hurtado, which says that certain groups just don’t act on any manifolds of sufficiently low dimension.
Another family of rigidity questions is: when can you recover complete information about a dynamical system from partial information? I have some recent work with Danijela Damjanovic and Disheng Xu where we consider rigidity for the centralizer problem and we show that for a certain class of systems if the centralizer is too large then the system is algebraic.
There are similar questions in geometry: there’s a dynamical system that one can associate to a Riemannian manifold called the geodesic flow. And one can ask what dynamical properties of the geodesic flow characterize the manifold on which the flow is defined. There’s some very interesting work of my former student Clark Butler where one can say that some numbers produced by the dynamics of a geodesic flow, the Lyapunov exponents, tell you that the manifold is a hyperbolic manifold.
So, using the complexity of the dynamics and only partial information you can reconstruct everything about the dynamics.
Is there any particular problem that you would like to address?
There remain very interesting open questions within the broader areas I described.
One of the more tempting is to try to prove or disprove a conjecture by Pugh and Shub about robust ergodic or robustly mixing diffeomorphisms. The conjecture says that partial hyperbolicity typically implies ergodicity. Partial hyperbolicity plus another condition: stably accessibility.
Keith Burns and I proved this with one additional assumption on the system and I go back and forth whether one can prove the same without this additional condition, so that is something I really want to understand.
And there’s a broader conjecture, that I think is probably true, which follows from this conjecture but, there are other ways to approach it. It says that generically, in the very strong sense, partial hyperbolicity implies stable ergodicity. I’d love to understand that and I definitely need new ideas to do that.
I regard dynamics as a vast wilderness that one explores. And some areas I find infinitely interesting.
One is what are the typical features of diffeomorphisms acting on a manifold and then the other is to understand the group structures of the space of diffeomorphisms and how the group structure can reflect dynamical features. And that includes questions about random dynamical systems.
How did you get into this particular area?
When I was a student I met some very interesting people who were interested in dynamics who effectively conveyed enthusiasm of the subject. They were graduate students at that time. So, my first exposure to dynamics did not come from professors but came from graduate students. In fact, when I was in college I hadn’t found a very good fit for my interest until I saw dynamics. College was very heavily focused on algebra, which I appreciate now, but no one told me that a group is something that acts. And if I had been taught that I would have loved group theory and probably algebra and everything. But instead a group was a set with an operation. So, I was disappointed in mathematics until I came across dynamics which to me was a subject that’s alive and not static. And I liked that.
When have you decided to become a professional mathematician?
Well, even when I was in high school, before I went to college, I was pretty sure I wanted to be a mathematician. And then I went to college and I wasn’t sure anymore, but by the very end of college I decided again that I at least wanted to go to graduate school. In fact, I took a year after college and I worked. At that point I realized that my idea of what it is to be a mathematician would evolve over time. So, I decided to go to grad school because I loved math and I wanted to see more math and I would take it one year at a time and see how my relationship with mathematics developed.
This is a very healthy approach. Have you received any kind of support during your career as a mathematician from people around you?
Oh, I received a lot of support from people over the years starting with my teachers in high school. I believe that when I was young my talent was recognized and appreciated and I think that was very important.
In college I received less support but there were certain people who were very helpful. But in college I think I really learned independence, I didn’t really have mentors. In graduate school there were two people who gave me tremendous support. One was my advisor Charles Pugh whom I would like to fully recognize. He was one of the most gentle, open, enthusiastic mentors. I always felt that he had a lot of confidence in me and that really helped me grow. And then Lisa Goldberg, who was a professor at CUNY visiting Berkeley when I was a graduate student, did a lot for me when I was beginning, she connected me with people. I did my first research project because of someone Lisa had connected me to.
Those two persons were hugely helpful when I was a student. There’s one other person, Michael Shub who I did a lot of research with when I was young. At some point, I was maybe five or six years out of graduate school, I was going to give a talk on my work with Keith Burns on this conjecture of Pugh and Shub, of which we had solved a big case. I showed my talk to Mike and he said: “This talk is too student-y, it’s what a student would say”. And it was at that time that I realized he was encouraging me, metaphorically, to jump out of the nest. And that was a very important moment. From then on, I had people who were always looking out for me and were helpful, but I didn’t rely on any kind of mentor.
Do you recall any obstacle or any particular difficulty?
College was really a critical point in my evolution as a mathematician. I was a student at Harvard and I knew, even before I went there, that there were many people who were much faster and quicker and knew more math than I did at that age. I was aware of that, but even so, it was difficult to suddenly be in an environment where there were so many people who were very comfortable with much more advanced mathematics than I had really seen before.
So, it was a little bit hard because there were these really smart students, some of them good friends of mine. I didn’t resent these students but I felt discouraged: it made me think that obviously I wasn’t going to be one of the top mathematicians because there were so many people who did better than I did.
For the most part the Harvard department was not discouraging but just indifferent. Some people were encouraging like Dick Gross who suggested that I take a seminar in ergodic theory with the grad students, but I would say most people were uninterested.
But there was one person, Andrew Gleason, who taught me algebra and was the undergraduate advisor. I went to talk to him at the end of my senior year, when I decided I would go to graduate school I asked for advice from him. And he didn’t say “Oh that’s wonderful, glad to have another person doing mathematics. That’s great.” He said, “Well, if you want to be a mathematician, you need to understand that there is a giant pyramid of mathematicians and very few are at the top and most are at the bottom. And if you’re comfortable with the idea that you might be at the bottom then you should do it. But, otherwise…”
Can I say he was sexist? Well, I heard later from several other women that he was equally discouraging and I haven’t heard from men similar stories, but in any case, that was a difficult thing to hear. On the positive side it gave me this attitude that if I was going to be a mathematician I wasn’t going to rely on some older person to push me through it. It gave me some anger, which was good, which helped.
I would say there were maybe one or two people who tried to sabotage my career over the years but basically on the balance people have been very good. Even when I was having a bad experience there was always something that was good. And it kept me interested, kept me going.
So, speaking about sabotaging, any thoughts about the obvious disparity between genders in careers in mathematics? And would you suggest any action to improve this scenario or do you feel personally engaged in promoting equality?
I think that there is no reason that we should not have 50 percent of mathematicians being women at all levels. There’s no difference, I believe, in ability to become successful mathematician between the genders.
I think that there is no reason that we should not have 50 percent of mathematicians being women at all levels. There’s no difference, I believe, in ability to become successful mathematician between the genders.
So, what causes that disparity? I think it is a complex issue. One is the way we are socialized. I think that from a young age we as women are trained much more to look for approval from others in what we do. I’m not saying this is an inherent difference, it’s training, but we are maybe more cautious, more concerned with approval, less confident. And at a young age, at college or even earlier, this can become a big obstacle toward becoming a mathematician. Because…
…if there is someone telling you that you will very probably be at the bottom…
Yes, someone telling you this could have a huge impact.
At the University of Chicago there used to be an honors program within the undergraduate math major, built up as a very competitive thing. To become one of the better students, you had to enter this honors program. And the honors program was brutal and extremely difficult and very competitive. And I think a lot of women just don’t love that kind of atmosphere. I didn’t like that kind of atmosphere.
And I have to say that Harvard has become in many ways more competitive than it was when I was there: when I was there in fact most of my discouragement was internal, I would look around and see these people who were smarter than I was and think I didn’t belong. Whereas I think many of the boys just didn’t care.
I hate to generalize about women because I think we are as diverse as men but I’m only pointing to ways that we are socialized. I think also when we are younger we are socialized to care more about the world, the big, broad world issues. And when I was younger I remember feeling a little bit that math was just these hard problems that you were given. And it was like you were being told to go climb that mountain and you weren’t really told why, what was on the other side. I had the feeling the boys maybe didn’t care, like: “Oh brother, great. I’m going to climb that mountain!“, whereas at some point I just didn’t really understand the purpose. And part of the reason I enjoyed math when I was younger was I got so much good attention for it, and i felt “I’m so good at it”. But at some point, when you’re not getting that attention you have to think about what’s the real interest that you have. Anyway, in some sense it was very good for me… But now I’m talking about myself again. Let’s go back to women
About the women who actually make it through that very competitive atmosphere, they’re very very strong but we lose a lot of women because that kind of atmosphere it’s still the general way. We changed this a lot at the University of Chicago and made the math major much more inclusive and that really helped with the numbers of women.
How have you managed to make it more inclusive?
There’s been more faculty involvement in teaching undergraduate courses and designing undergraduate courses. For a long time, it was directed by one person and he passed away. And then the faculty was really forced to participate in undergraduate education and it really opened it up. It was only since 2012 that there’s been any significant number of senior women, and taking classes from women research mathematicians has helped. I mentor a lot of undergraduate students and I’m the opposite of Andy Gleason. I’m not a cheerleader but I’m realistic, I give good advice, I encourage, I just pay attention. I think it’s really the work of the faculty to care about the issue. The same has been for increasing the number of women graduate students at Chicago.
Even at the level of graduate students, and postdocs, I see a huge confidence gap between men and women. Not a talent gap but a confidence gap.
I see a huge confidence gap between men and women. Not a talent gap but a confidence gap.
I work very hard on training women to assert themselves, to present their work and so on, in a confident way.
That’s very simple but I think another big obstacle for women in mathematics in the United States is a combination of two body problems and the lack of good childcare support from the government. I’ve seen many times women who were in relationships where their partner was not very supportive of them at home. And being forced to move somewhere where they can’t do math or being forced to spend a lot of time raising children doesn’t help. That’s an obvious thing we could do to improve the lives of women in the United States.
Can you tell me something about what you do for outreach?
Well, I am trying to do less now, I need to, but one thing I focused on a lot in the last five years has been trying to communicate what it is to do mathematics and what it is to be a mathematician to non-mathematicians. And also, how to communicate some interesting questions in my field to non-specialist mathematicians. And that is very challenging.
So, part of what I’ve done is giving a lot of talks both to math departments, like lecture series, and also to general audiences. That has improved me as a mathematician because when you’re forced to clarify your ideas to the extent you can explain it to a non-expert then you really understand.
And I wrote an article in The New York Times two summers ago when Maryam Mirzakhani and Marina Ratner died. I was trying to encourage the New York Times to do an obituary on Ratner, I finally convinced the writer to do it and they published it, which was great. But then the writer turned around to me and said: “Well, you’re very passionate about the work of Ratner and why not you write an op ed about her and Mirzakhani’s work?”. So, I did that.
And I typically don’t say “no” when a group of young mathematicians like college students or high school students ask me to talk to them. I kind of view that as my responsibility to give back what I’ve been given.
But I just said “no” to a big talk in the US.
I got a very funny review on a research grant proposal I wrote for NSF. it said something like about my outreach. I think it just said in general that the output of the principal investigator has been outstanding and care should be taken not to burn her out with too many requests. So, even my reviewer was commenting that I I think I’m overworked.
But you got the grant.
Yes, I got the grant.
Congratulations. What recommendation would you give to a young mathematician or a young person wanting to become a mathematician? Or, looking backwards, what advice would you give to yourself, to young Amie?
Not to panic, not to rush. There’s plenty of time to learn math. There’s plenty of time to develop as a mathematician and becoming a mathematician is a process. You can’t really know what you will be doing, whether you’re made to be a mathematician or not. You really can’t know that until you know quite a bit of mathematics and I think that actually goes beyond being a postdoc.
Don’t forget the amount of growth that is ahead of you and the amount of confidence you will gain from your knowledge, and the authority you will gain from that self-confidence.
I guess I would say: I understand it’s not easy, and I’ve been through the struggle you’ve been through and most mathematicians have. I think the best I could do would be just to say that, because young people look at older mathematicians and they think “Well, this person is different, they must be very special to get where they are. They must have always been different, they must not have had problems”.
But we certainly all had problems.
Which are your other passions apart from mathematics and dynamical systems?
Well, I’m passionate about creating things and creative experiments in my life. So that means trying new things. I love to cook, not just when you find a recipe and buy the ingredients, but the kind of cooking where you have ingredients and you have to make something out of them. You have something and you must create within these constraints.
I like visual arts a lot. And just as a creative outlet I design fabric, and make clothes out of these fabrics, I design and I make handbag and purses for all my family members.
When you say you make fabrics you mean you design them and then someone makes them or you weave them?
I do not weave textiles but make images, I usually produce them electronically. I use software like Photoshop and I make patterns. And then I sew them. I have just made a dress.
I used to do a lot of dancing when I was in graduate school. I started belly dancing and I even did that semi-professionally for a while, I danced in restaurants with a group. Actually, for a while dance with my passion.
I’m also passionate about my family. I had two wonderful kids and I love the company of my husband and my children.
And do you have a dream that you would like to share with us?
The dream I had was just a nightmare about teaching a class. It was the mathematician version of the student nightmare where you show up at school and you have to take a final exam but you never went to the class. So, I had a dream that I had office hours for a class. And I had given them homework but I had never looked at the homework. And then the students showed up and I would try to write their names on the board but their names were all really weird and I couldn’t hear them because I was going deaf. And then the way they spelled the names had nothing to do with how they sound, there were letters and Greek symbols. Then more and more students piled in the room and I looked at the problems and I realized that I couldn’t even explain them to the students. So, I tried to do them on the boards but the boards weren’t blackboards they were whiteboards, which is a nightmare in itself, and secondly I had just moved into the office and before the office was used by administrative staff who used the whiteboards to put up Christmas decorations and photographs. So, there were little holes where I could not write. Then someone I opened the refrigerator where there was rotten food and the whole room began to smell. The funny part was the students were very disappointed but they were very very polite and I felt very guilty about the whole thing.
And that was my last dream.
I think one needs to do more than dream of a different world. One has to act.
Well, we are currently living in a nightmare. That is the way the world moves right now, it’s really a nightmare and I think one needs to do more than dream of a different world. One has to act.