“Nadie te quita lo bailado.”( No one can take from you what you’ve danced .)

For Federico Ardila, this Latin American expression epitomizes his approach to life and mathematics. It’s the driving force behind the working party he DJs in venues across the San Francisco Bay Area, where people dance till morning to the beat of his native Colombia. The dance floor is a place “where you have your freedom and you have your power, and nobody can take that away from you, ” Ardila said.

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Original story reprinted with permission from Quanta Magazine, an editorially independent publishing of the Simons Foundation whose mission is to enhance public understanding of science by encompassing research the progress and recent developments in maths and the physical and life sciences.

He taught the expression to his students at San Francisco State University, where he is a math professor, after devoting them a punishingly hard exam. San Francisco State has a highly diverse student torso, and Ardila, who only turned 40, is a prominent voice in the maths community about how to make students from underrepresented groups — such as women and people of color–feel that they belong. But on this occasion, as he appeared around at his students’ demoralized faces, he knew “hes having” missed the mark.

“Nadie te quita lo bailado, ” Ardila told his students.

“I think that’s a very powerful message–that nobody can take away from you the joy that you’ve had doing maths, ” he told Quanta Magazine in an interview last month. “And people can give you grades, but that’s not going to take away the freedom that you felt and the fulfillment that you felt.”

The expression also applies to Ardila’s research, though not always in ways he would have chosen. Four years ago in Portland, Oregon, a burglar smashed his vehicle window and made off with a backpack containing, as luck would have it, five years’ worth of work–all of Ardila’s notes from a sweeping new paper he was developing. Proofs, examples, counterexamples and conjectures were all gone.

But the robber couldn’t steal the maths Ardila had “danced” in his mind. Over the past few years, Ardila and his coauthor, Marcelo Aguiar of Cornell University, have painstakingly reconstructed their work merging the geometrical and algebraic sides of combinatorics–the study of discrete arrangements like a social network, a sudoku puzzle, or a phylogenetic tree. They ultimately posted their 113-page newspaper online in September, and in January Ardila will be presenting their work in an invited address at the Joint Maths Meetings, the most difficult annual math conference in the United States.

Quanta spoke with Ardila at the Mathematical Sciences Research Institute in Berkeley, California, where he is visiting for the fall semester, about the mathematics he has danced and taught. The interview has been condensed and edited for clarity.

Your mathematical flair was identified quite early–in fourth grade, you got the highest score in your age group in their own nationals math rival in Colombia.

It was actually my sister, Natalia, who first proved great promise in maths. I was just the little brother. She and my cousin Ana Maria, they both performed really, really well in this national math rival. And I believe the organizers likely said, “OK, these two women are very good, and then here’s the little brother who’s coming along to the welcoming ceremony. Perhaps he’s OK also.”

I feel like from a young age, the latter are attaches great importance to me. I never enjoyed mathematics in school very much, but my own experience through the Math Olympics was much more creative and much more playful.

Federico Ardila as small children in Colombia with his mother, Amparo, and his sister, Natalia.

Jorge E. Ardila

And it turned out that it was, as many of these spaces are, a very male-dominated space, and eventually both my sister and my cousin felt uncomfortable with this space. I mean, they’re doing amazing things now; my cousin is an engineer and my sister is a music pedagogy prof. But I do think it’s kind of interesting–that was a space where I seemed very comfortable and that felt very fostering to me, and it didn’t feel so to other people. It was a space that was very “othering” for them. I think that’s always served to remind me of the role of a mathematician, of an instructor, in curating different cultures of a place. That’s why that’s been such a theme in my work.

You’ve said that you were surprised to get into the Massachusetts Institute of Technology, where you did your undergraduate and doctoral analyzes. What’s the tale there?

I had never heard of MIT. And it hadn’t traversed my head to learn abroad. I was already enrolled in the local university. But my classmate told me MIT had awesome financial aid and said here math there was really good. I wanted to learn more math, so I decided to play along and apply.

At that instant I was failing most of my class in high school. It was not clear that I was going to graduate. I had a little bit of an attitude trouble. I was very interested in a lot of things but I did not like being told, “Read this” or “Think this way.” I simply kind of want to get do things on my own terms.

I was failing, I guess, six out of eight topics. Had I known what MIT was, I should have known not implementing. There is no I should have applied with that kind of transcript.

I like telling this story to my students because I think we often close entrances to ourselves by thinking that we’re not eligible or that we’re not good enough. And specially if you’re somebody who feels “othered” in your discipline or who feels like you’re lacking confidence, it’s easy to close entrances on yourself. There’s a lot of people in life who are ready to close doorways for you, so you can’t do it for yourself.

When you came to the United States, as an undergraduate at MIT, it was your turn to feel like the “other.”

It’s not that anybody did anything to mistreat me or to doubt me or to explicitly make me feel unwelcome, but I definitely seemed very different. I mean, my mathematical education was outstanding and I had fantastic access to profs and really interesting material, but I simply recognized in retrospect that I was highly isolated.

There’s a system in place that stimulates certain people comfy and others uncomfortable, I imagine simply by the nature of who’s in the space. And I say that without wishing to point thumbs, because I think you can be critical about the spaces that “other” you, but you also have to be critical about the ways in which you “other” other people.

I think because mathematics discovers itself as very objective, we think we can just say, “Well, logically, this seems to make sense that we’re doing everything correctly.” I think sometimes we’re a little bit oblivious as to what is the culture of a place, or who feels greet, or exactly what we we doing to attain them seem welcome?

So when I try to create mathematical spaces, I try to be very mindful of letting people be their full human egoes. And I hope that will give people more access to tools and opportunities.

What are some of the ways you do that in your teaching?

In a classroom I’m the prof, and so in some sense I’m the culture keeper. And one thing that I try to do–and it’s a little bit scary and it’s not easy–is to really try to shift the power dynamic and make sure that students feel like equally powerful contributors to the place. I try to create spaces where we’re kind of together erecting a mathematical reality.

So, for example, I taught a combinatorics class, and in every single class every single student did something active and communicated their mathematical ideas to somebody else. The arrangement of the class was such that they couldn’t merely sit here and be passive.

I believe in the power of music, and so I got each one of them to play a anthem for the rest for us at the beginning of each class. At the beginning it felt like this wild experiment where I didn’t know what was going to happen, but I was truly moved by their responses.

Some of them would dedicate the song to their mommy and talk about how whenever they’re studying math, they’re very is conscious that their mama operated unbelievably hard to give them the possibility of being the first ones in their family to go to college. Another student played this song in Arabic called “Freedom.” And she was talking about how in this day and age it’s very difficult for her to seem at home and welcome and free in this country, and how mathematics for her is a place where nobody can take her freedom away.

That classroom felt like no other classroom that I’ve ever taught in. It was a very human experience, and it was one of the richest math classrooms that I’ve had. I envision one worries when you do that, “Are you covering enough maths? ” But when students are engaged so actively and when you really listen to their ideas, then magical happens that you couldn’t have done by preparing a class and only delivering it.

Mathematics has this stereotype of being an emotionless topic, but you describe it in very emotional terms–for instance, in course curricula you promise your students a “joyful” experience.

I think doing mathematics is tremendously emotional, and I think that anybody who does maths knows this. I just don’t think that we have the emotional awareness or vocabulary to talk about this as a community. But you walk around this building and people are inducing these breakthroughs, and there are so many feelings going on–a lot of frustration and a lot of joy.

I think one thing that happens is we don’t acknowledge this as a culture–because maths is emotional in sometimes very difficult behaviors. It can really make you feel very bad about yourself sometimes. You can be pushing on something for six months and then have it breakdown, and that hurts. I don’t think we talk about that hurt enough. And the exultation of detecting something after six months of working on it is really deep.

Your own research is in combinatorics. And the paper you’ll be presenting at the Joint Math Meetings connects two different ways of understanding combinatorial structures, through the lenses of geometry and algebra. How do those two approaches operate?

When you look at the geometric side of things, suppose, for example, you want to study the permutations( the ways of rearranging a collection of objects ). It’s pretty well known that if you have n objects, the number of ways of putting them in a row is n factorial( the product n( n-1 )( n-2 )… 1 ). So it’s not a very interesting problem to count how many routes there are. But what is their inherent structure?

The three-dimensional permutahedron, a geometric depiction of the ways and means of rearrange the numbers 1, 2, 3 and 4. Two permutations are connected by an margin if one can be transformed into the other by swapping two consecutive numbers.

Tilman Piesk

If you look at when two permutations are related to each other by just swapping two elements, then you start understanding not only how many there are but how are they related to each other. And then, when “theyre saying”, “OK, let’s take all the permutations, and threw an rim between two of them if they’re a barter away, ” then you find that you get this beautiful shape that’s a polytope( a geometric object with flat sides ). I think it’s totally surprising initially that the inherent closer relations between permutations are captured in this beautiful polytope called a permutahedron. So all of a sudden you have this geometric model, and you can use tools from polytope theory to try to say new things about permutations. And that polytope has existed for a very long time and is very well understood.

And then you can also think of permutations algebraically–there’s a natural sort of “multiplication” on permutations, in which the product of two permutations is the permutation you get by doing one permutation after the other.

This is one of the most important point objects in algebra, this group of permutations.

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