Federico Ardila is Associate Professor of Mathematics at San Francisco State University and Adjunct Professor at Universidad de Los Andes, Bogotá, Colombia. His work focuses on combinatorics, with applications to algebra, geometry, topology, phylogenetics, and optimization. He is currently spending the semester at Mathematical Sciences Research Institute as a Simons Research Professor. Federico was recently announced as a Fellow of the American Mathematical Society and has held a National Science Foundation CAREER Award.
I first became aware of Federico through his AMS Notices article Todos Cuentan: Cultivating Diversity in Combinatorics, where he talks about his experience cultivating diversity in mathematics. We’ll learn more about this and his community building between the US and Colombia in the interview.
AB: What was your first mathematical memory?
FA: I remember some math Olympiad tests arrived at my school in Bogotá, Colombia. I didn’t really like math a lot in school, but I remember these tests had some innovative questions that I didn’t know were math. I remember being excited about solving those. I was young, maybe eight or nine years old.
At the time, I think both my sister Natalia and my cousin, Ana María, had participated in these kinds of math Olympiads; I was the curious little brother. I was lucky and privileged to have this material arrive in front of me at that age. It took me a long time to accept that I wanted to be a mathematician, but definitely, the seeds were planted pretty early on.
AB: Was there a person or a teacher who you say who’d influenced you to study mathematics as a career before you entered university?
FA: The math Olympiads were a crucial experience for me. I ended up going to the summer camps and eventually making it to the Colombian national team. Later I coached the team for a number of years. I found a good community there: I had peers that loved this stuff, even if it was a small group. It was a rare chance to interact with kids from all over the country, and eventually from other countries.
The person who founded and ran the Olympiad program was Mary Falk de Losada, an American woman who moved to Colombia in the 1970’s. She had a big influence on me and many of the mathematicians of my generation in Colombia. She was also the first person I heard speak about discrimination in mathematics.
AB: How did you end up at MIT for your PhD and how did you come to work with your advisor, Richard Stanley?
FA: I arrived there as an undergraduate. I had never heard of MIT when I applied, and I didn’t have the kind of grades that should allow me to get in. But a friend of mine wanted to go there and told me that they offered full scholarships. That’s all I knew about it. I also remember their brochures had a kind of a punk rock aesthetic that intrigued me.
When I arrived in the US, the first class that I took was with Richard Stanley and with Hartley Rogers. They ran this seminar in preparation for the Putnam competition, and I have to say that I had little concept of what a math major was about. But I knew that I liked math competitions, and so I enrolled in them. I ended up being on the Putnam team of MIT, which they led. I was lucky to meet Stanley early on.
It’s interesting because I hated combinatorics. I thought it was just terribly hard. I always felt like I was good at the other stuff, but bad at combinatorics. I think there was a kind of stubbornness there; I wanted to master it.
As a senior, I took a couple of courses with Stanley and Sergey Fomin. And those two courses were just amazing, and they got me to understand that combinatorics was not just a bunch of clever tricks, but that there was a structure to it and a theory to it. That’s when I fell in love with combinatorics in my senior year at MIT.
When I applied to graduate programs, I was still clueless. It didn’t even occur to me to look at what a ranking of schools was. I asked Stanley for names of people I might work with. He mentioned four people, and I applied to those four schools. I was accepted to MIT, and everybody told me that that was the best place for algebraic combinatorics. I ended up staying there and working with Stanley. I also learned a lot from Gian-Carlo Rota, who was still alive at the time, and younger faculty like Sara Billey and Anne Schilling.
AB: Your research now is in combinatorics, with connections to topics like algebra, geometry, and even robotics. Would you tell us broadly, what combinatorics is? What are some of the main goals of your research?
FA: I like the way my partner May-Li puts it; she’s a designer who works in technology and education, and she loves mathematics. She calls combinatorics “the science of possibilities.”
Many people say combinatorics is about counting. It’s true that we do count things sometimes, but the way that I see it, we spend most of the time studying their structure. To give you a very simple example, we do not count the cells of a chessboard one by one: 1, 2, …, 64. We first realize they have the structure of a grid, and then we use that structure to count them 8 x 8 = 64.
Of course, I study more complicated objects, and most of my efforts are spent studying their inherent structure; sometimes those objects are discrete, and sometimes they’re continuous. Once I’ve understood that structure well enough, then I can count them, or measure them, or prove other things about them.
As it happens in all of mathematics, there are some problems that are extremely hard but are isolated from the rest of mathematics. I always look for combinatorics that arises in or is inspired or motivated by other fields of mathematics. I think that makes for deeper and more connected mathematics.
Most of my work is related to Lie theory, to representation theory, or to algebraic geometry. I’m always trying to talk to lots of people in different fields to see how I can be of service as a combinatorialist. I like studying the combinatorial structure of other people’s objects, and seeing what new things we can find out.
AB: What research topics are you working on now?
FA: I’m spending the semester at MSRI, and it’s such a luxury that many of my mathematical heroes and mathematical friends and students are living in Berkeley this semester, and thinking about geometric combinatorics together. It’s been a very productive semester, talking to lots of people, and sprinkling some seeds for future projects. The biggest challenge has been to balance many different projects.
One big project that I’ve been busy with lately, that I’m very excited about, is a project with June Huh at the Institute of Advanced Studies, and with Graham Denham at the University of Western Ontario. It seems we can prove a series of conjectures from the 1970s and 80s on the unimodality of some combinatorial sequences. In combinatorics, there are many sequences that are unimodal, meaning they start increasing, reach a peak, and then they come back down. When you study combinatorics, you’re used to the fact that most of the sequences that we encounter have this shape. It’s kind of an amusing thing to observe, but it’s often incredibly hard to prove.
Some ways of proving these kinds of unimodality conjectures use algebraic geometry (using the hard Lefschetz theorem or the Hodge-Riemann relations) or representation theory (using the representations of the Lie algebra sl_2). What has happened often is that these kinds of conjectures are solved by taking these combinatorial objects that are easy to define, and revealing that in fact, they have a much deeper algebraic or geometric structure.
We are proving some unimodality conjectures about matroids. Matroids are a combinatorial model of “independence”, that unifies aspects of linear independence, algebraic independence, graph theory, matching theory, optimization, and Lie theory. I love matroids; they’re one of these objects that are very connected to lots of fields of mathematics.
June Huh, first in his Ph.D. thesis and later joined by his collaborators Eric Katz and Karim Adiprasito was able to solve some very old unimodality conjectures for matroids using these kinds of techniques from algebraic geometry. For my taste, this was one of the most exciting recent developments in mathematics.
June, Graham, and I are further developing this Hodge theory of matroids to prove a stronger series of conjectures. Along the way, we have discovered a lot of interesting new structure, within and outside of matroid theory. We’re at the writing stages, and everything is working out very beautifully.
AB: You’re a professor at San Francisco State University, but you’re also at Universidad de los Andes in Colombia. How does your work as a professor span the two institutions?
FA: Even though I had access to mathematics as a kid, I did feel foreign to mathematical culture, especially in the US. One thing that I always felt in US academic spaces is that they’re very narrow in their conception of what’s valued. We are advised that it’s your theorems that you’re going to be judged by, so they’re the only thing that is important. I reject that, and I always have, and I remember thinking already when I was a freshman, “I’m going to have to do this my way because I don’t believe in this idea of just being an academic, period.”
I was raised to have an interest in the community, in equity, in trying to make a positive difference. Early on, I knew that I loved mathematics, but that I also wanted to have some bigger involvement with society. I don’t want that to sound pretentious, but I just wanted to help a bit with whatever tools I had.
When it came time to decide where to place myself academically when I finished my PhD, I was intentional. Some of my mentors and peers disagreed with my choices. I wanted to be in a position where I could do high-quality research, and at the same time, I could have an effect outside of research mathematics. I wanted to be in a place where I wanted to live, in a diverse city. I wasn’t willing to move just anywhere because of academia. That’s how I ended up at San Francisco State University. At the same time, I always wanted to do something with Colombia.
SFSU is a very interesting place research-wise for me. One thing that was also important to me, and shaped my view of what an academic mathematician does, is the community that it serves. More than half of our students are first-generation college students, more than half of them are first-generation Americans, and over 75% come from ethnic minority groups. When mathematicians visit me, they’re often struck by our student population, because that’s not what they’re used to seeing on a US campus. But I don’t think that’s accurate; our campus is probably more similar to the average US campus than the typical R-1 university is. I think academic mathematics focuses on a very narrow view of who should be doing mathematics.
I love the Bay Area, and that is one important reason that I’m here. Also, I’m Latino, and my partner is Asian, and few places have such a strong Latinx and Asian culture. When I found myself becoming comfortable in San Francisco, I decided that I wanted to set something up with Colombia so that I could stay in touch, and give back to the community that gave a lot to me.
I noticed that there wasn’t anybody actively doing research in combinatorics in Colombia and there weren’t courses being offered. I saw a need that maybe I could fill. I started offering joint graduate classes that were taught simultaneously at San Francisco State and in Colombia. This started as kind of a wild experiment; it was before all these online courses became a fad, and this was a very do-it-yourself setup.
I think it’s a nice, productive bridge we created. The students not only took the course together, but they truly collaborated. For example, many of the final course projects were done internationally between a student in Colombia and a student in the US, and several became published research projects.
Eventually, I was awarded an NSF CAREER grant that funded my research and allowed me to make this a more systematic program. With that grant, I was able to keep offering these courses and to bring American students to Colombia and Colombian students to the US to do research together. It also allowed me to start the international Encuentro Colombiano de Combinatoria, which meets biannually.
I think that if you want to see minority mathematicians succeed, it’s going to be easier to see ten of them succeed than to see just one. You can’t just take one person, put them in a sea of otherness, and expect them to thrive easily if there’s not a systemic change in the way things work. I wanted to create a wider environment where things were done differently, and there was a deep sense of community.
It’s very exciting to me to see that my students in Colombia and the US are friends now, and they collaborate, and they’re professors in many places, both in Colombia and in the US. It’s become a wonderful community. In mathematics, we’re measured by our theorems, and the mathematics needs to be on point. That’s crucial. I’m very excited to see the mathematics that this community is doing. They’re doing some beautiful stuff.
At the same time, I also like that they’re not just doing mathematics and that there is this sense of activism in the community. We are finding ways to be involved with lots of different universities and even people in nearby countries. We are trying to create a different space where many people can thrive and do great mathematics.
AB: Last year you wrote an influential AMS notices article “Todos Cuentan” or “Everybody Counts.” Would you tell us about the article and its message?
FA: Fundamentally, one thing that I believe very deeply is that mathematics is for everybody, and that’s not how we have traditionally behaved as a mathematics society. Many of our practices are designed to select and support the “best” people. I try to take a different point of view. If a student shows up in my classroom, then there’s a very good reason that they’re there, and it’s my job to support them.
Society has deep inequities. If we don’t address those inequities very mindfully, then they’re just going to be reflected in our classroom. That’s one of the fundamental reasons why we struggle as a mathematics community to truly welcome diverse perspectives. We have to do our homework and learn about what are those inequities and what we can do about them.
I think most math professors entered the university with absolutely no training on what it means to educate our society. We’re trained on how to prove theorems, and we’re often told that the educational part is not so important, and we shouldn’t focus on it because it won’t help us get the best job. So I think I’m just trying to learn about pedagogy, and about the structural inequities that have taken us to where we are today.
A lot of it is about being thorough and scientific in the same way that we are about our science. What is the scholarship on really trying to make sure that we give equitable access to everybody? How do we move from just getting faces that look different in a classroom, to truly welcoming diverse perspectives?
There are many scholars that I’m very indebted to, who inspired a lot of what I wrote in that article; people like Audrey Lorde, bell hooks, Paulo Freire, Estanislao Zuleta, Rochelle Gutierrez, and Bob Moses. They have researched education and inequity more deeply than mathematicians have. I have also been blessed to be surrounded by wonderful people, doing work of a similar spirit in very different fields: my mother in violence prevention, my father in human management, my partner May-Li Khoe in design and education, my sister Natalia in music pedagogy, my SFSU colleagues in science and in ethnic studies, my dear friends Sita Bhaumik in art education and food activism and Dania Cabello in sports as a tool for social change, to name a few. As in my mathematical research, I am always trying to learn from the practices of people in other disciplines.
One very important principle for me is that science is very powerful, and really shouldn’t be concentrated in small sectors of the population. And I think science also brings a lot of joy and empowerment that should also be spread widely among our communities. I really want to encourage mathematicians to constantly ask ourselves what we can do to make mathematics a tool towards a more equitable society.
AB: How did you converge to the agreement you include in your course outlines? What’s the effect that it’s had on students?
FA: I have always wanted to make the classroom a human place, where everyone is welcome, and where we don’t only talk about mathematics. The obstacles to student learning are often not mathematical. I try to make my classroom feel like a very comfortable place, where people are welcomed to bring their full perspectives, and they feel safe taking risks and sharing their ideas.
At the same time, I hope to make clear to my students that they’re in the class not only for their own self but also to help and support others. This is good for the classroom and it is important for their education. That’s what’s going to help them in society as they go and work in teams, and find out that they are valued not only for how they do but also for how they lift up the people around them.
I work to make sure that this doesn’t feel like something I’m imposing on them; it is an agreement among all of us. We take the time for students to discuss what the agreement means to them, what they might add or improve on. I am always impressed by their openness and thoughtfulness in these discussions.
One central principle for me as an educator is to treat students with respect and communicate clearly and openly with them. Many of my better practices as an educator have come from really listening to them.
AB: What advice would you give to young people thinking about studying mathematics?
FA: One thing that I find very important is to recognize the joy of doing mathematics, to constantly seek that joy. At the same time, mathematics can be difficult and frustrating. And it’s very important for a young person to know that it’s not just them; even Field Medalists find math difficult and spend most of their time struggling. That’s the nature of what we do. We’re very curious, and we’re never satisfied with what we already know. We’re always looking for the next thing.
It’s important to remember that the joy and the learning are yours. There’s an expression that I like a lot in Spanish that I share with my students, “Nadie te quita lo bailado“, and that translates to, “Nobody can take away what you’ve danced.” To me, one thing this means is that no one can take away the joy with which you have done things. If you know that you love mathematics, then you can’t let a professor or a low grade take that away from you. If you feel the joy and the power of understanding something new, then that joy and that power are yours, and they are real.
You are a good gauge of what you’re learning, and sometimes the way that institutions measure you does not accurately reflect your potential as a mathematician. I can’t deny that it is useful to learn how to test as well as you can. But I think it’s also important to pursue knowledge with the purity of recognizing that you’re learning something, that you’re enjoying it, and that you’re becoming a richer person for it.
AB: I always finish the interviews by looking forward. What would you say are some of the major directions in mathematics?
FA: I find it very hard to say. I know some mathematicians have very ambitious goals of solving big open problems. For me, I’ve more been driven by walking around the world, and seeing what I see, and trying to uncover something cool. If I see something interesting, I want to open that door and see what’s there. Even though it is important to have these long-term goals for mathematics, I also believe that many of the most interesting developments and research directions didn’t come that way. They came from unexpected places.
One thing that I think is important is disrespecting every border that people have tried to draw in mathematics. An ambitious student who is just starting out might try to find two fields that people think are unrelated, and discover the relationship between them. Mathematics is interconnected in unexpected ways. I think the most interesting work comes from taking two islands in mathematics, and showing how they’re connected. That’s a constant pursuit of mine. For me, it’s centered around combinatorics, but I’m always trying to see how this field relates to something that it doesn’t seem to be related to. I think that always leads to very interesting mathematics.
When it comes to the mathematical pursuit as a whole, one very big question I already mentioned is this: How do we make it possible for every community to participate in mathematics, benefit from mathematics, enjoy it, and use its power? This is a very important question.
In my mind, these two questions of mathematics and inclusion are related. Asking ourselves what are the most original mathematical developments of the future is closely related to asking ourselves who will make those developments. I want to see the most diverse group of people possible tackling the deepest mathematical questions. If you care about diversity and inclusion, you recognize that minoritized populations are often judged by higher standards, and you should tie your outreach efforts to the most interesting mathematics possible. If you care about the development of mathematics, you recognize that many of the most interesting discoveries have come from people who have not been indoctrinated into the ways that most mathematicians think, or who have dared to question those accepted ways of thinking.
My job gives me the opportunity to work with many students who have not been conditioned to think like most mathematicians. When they engage with deep mathematics, I often find that they think very differently from me, and ask very interesting questions that I hadn’t asked myself.
I think it makes sense to think that mathematical research also works this way. New groups of people bring new perspectives and ways of thinking, and they might be the ones who see what everyone else has missed.