**My Life through Mathematics**

The opportunity to write this chapter has been a breath of fresh air amid very tumultuous times. It is my sincere hope that when we look back to the year 2020, it will be as a watershed moment where a large majority agreed that racial injustice, xenophobic sentiments and less than competent leaders cannot be tolerated in society. Something that has come up again and again during this reflection is the idea that every perceived personal success has had many people behind it. Mentors who, with small acts of kindness, have a huge impact. I will try to describe a bit of my journey as a vehicle to articulate these thoughts.

**Early life.** I was born in 1984 in Santiago de Cali, the third most populous city in Colombia, to Adiela Benítez and José Lúcio Perea. Geographically speaking, Cali is nestled between the central and western Andean mountain ranges; it has a beautiful year-round tropical climate, and it is a rich melting pot of indigenous and Afro-Latin cultures. People refer to it affectionately as *La Sucursal del Cielo* (Heaven’s branch office) or *La Capital Mundial de la Salsa* (The world’s salsa capital). The Afro part of Cali can be traced back to several migrations, particularly from the Colombian Pacific. This predominantly Black part of the country emerged from early slave settlements in territories like Chocó—a region historically plagued by poverty and government neglect. My dad and my maternal grandma, Rosa Lia Mosquera, were born in the neighboring river towns of Santa Rita and Santa Ana, Chocó. I don’t think they knew each other while living there (age difference), but they both emigrated to Cali in search of a better life. They came with nothing; not even a high school diploma, but they both built a life for themselves as immigrants. My grandma worked very hard to singlehandedly raise my mom, and my dad put himself through school—almost finishing a university degree in accounting—while building a small tailoring business to support my family.

I credit my dad with breaking a very resilient cycle of poverty in his side of the family. When I was around fifteen years old, he took me on a trip to his childhood home in Santa Rita. I remember the two of us traveling by bus from Cali to Chocó, and then helping paddle a small boat for the better part of a day—a four-person canoe—along the Iro river. We finally made it to Santa Rita, where I found a beautiful town filled with warm and welcoming people, though with few paved roads and even fewer homes with electricity or running water. My dad’s home was not one of those few. Going back to those memories always makes me think of the effect that initial conditions can have, the importance of pushing for equitable policies, and of our responsibility as mentors to not let said conditions get in the way of someone’s work ethic and talents.

Growing up we were not middle class, but my parents always made sure we had everything we needed. My mom and my grandma would repeat mantras like “being poor is not an excuse for having dirty shoes or wrinkled shirts,” which were part of a bigger theme. For example, in middle school I had horrible handwriting; my mom would sit with me to go through my notebooks, she would take out all the messy jumbled pages and make me rewrite them neatly. Today I have better handwriting thanks to her, and an appreciation for the benefits of putting effort into getting the small things right. I believe these were, in their way, an attempt to lessen our chances of being subjected to discrimination later on. My grandma had a line that went something like this: you’ll be judged as a Black man first, and your actions will reflect on Black people in general; it will not be José did this or that. As unfair as it sounds, this was in all likelihood a result of her upbringing, interactions with other people, and personal journey. I’ve had similar feelings as an immigrant in the U.S.; that I would be judged as Colombian first (I’ve heard all the cocaine-related “jokes” and they are still annoying) and that my failures could reflect poorly on an entire country. While these feelings are legitimate and, I imagine, shared by others, it is crazy to think that minorities—in addition to everything else—have to deal with the constant pressure of being “the” representative. I hope we can dispel these notions through honest conversations with our students, while we help build more diverse environments where people don’t have to feel this way.

**High school years.** My love for math came almost at the end of my high school years at *Centro Educativo Industrial Luis Madina*. I had always been a good student with top grades, and learning was something I enjoyed, but I wouldn’t say that math was something I was passionate about. Physics, on the other hand, blew my mind. When we learned Newtonian mechanics, it was incredible to me that one could understand and formalize the world with math. I was also very lucky to have an unconventional physics teacher, Prof. Jesús Rivera. Once he noticed I was doing well on the tests and reading ahead in the book, he told me to go to the school’s library instead of coming to class; that I should study the material at my own pace and periodically talk to him as I made progress. This freedom to learn was empowering, and I was convinced that physics would be my major in college.

In Colombia, students declare their major when they apply to a university; they submit their materials to a specific program and are admitted according to a ranking of test results.

The test in question is called the ICFES exam. [1] It is a standardized test like the SAT, administered nationally to all high school seniors, and used by Colombian universities to determine admission. Needless to say, it is a big deal. The large public universities in Colombia are quite good, though there are few of them, and the ICFES scores are essentially their only admission criterion. Thus, for students who cannot afford private education, it is a high-stakes exam. This was certainly my situation, and my mom embarked us on a mission to make sure I got into *Universidad del Valle (Univalle)*—the third-largest public university in the country, located in Cali, and ranked among the top five nationally. She signed me up for a pre-ICFES course that met every Saturday. I was of course very reluctant—I thought I could study on my own—but I was pleasantly surprised with the unforeseen benefits.

The preparation course was structured to review all relevant material for the test by splitting the content into classes. Among them was obviously mathematics, but the course was entirely subverted by the teacher in charge. On the first day of classes, he said, It is very unlikely we’ll be able to cram in a few Saturdays what you haven’t learned in six years; let’s learn something interesting instead.” The teacher was an undergraduate student in mathematics at Univalle, and he proceeded to review several “math facts” from one viewpoint—Why is this true? We covered several topics in arithmetic, algebra, and geometry from a discovery/proof-based perspective, which was entirely alien to me. Until then, I had seen math as a formulaic and rote memorization exercise, which is unfortunately still very common. From then on, I was hooked.

The course ended, and the test came and went. In the end, I got the top ICFES score in my school, and the question became whether I would go for physics or mathematics in my application to *Univalle*. I didn’t have a good framework to make this decision, so I went to Prof. Rivera for advice. He asked me, “What it is that you find interesting?” I responded, “Figuring out why things work, why they are true.” Without missing a beat he said that I’d probably be happy doing math, and several years later I can report that he was 100% correct.

I went to *Univalle* for my bachelor’s in mathematics between 2001 and 2006. As I mentioned before, the Colombian higher education system is geared towards students focusing on their major almost exclusively from day one, though other complementary courses are also included. By design, I ended up taking a lot of math classes: from number theory and set theory to abstract algebra, real and complex analysis, measure theory, functional analysis, differential geometry, and point-set/algebraic topology. Everything seemed super interesting. Along the way, I met several professors who inspired me to pursue mathematics, and selflessly helped me succeed. Among them, I want to highlight Dr. Doris Hinestroza, Dr. Gonzalo García, and Dr. José Raul Quintero. They all graduated from *Univalle* and went on to earn PhDs in mathematics from universities in the U.S.: Doris from the University of Cincinnati, Gonzalo from Cornell University, and Raul from the University of Maryland. Doris passed away in February of 2019; she was kind, warm, generous, a force to be reckoned with, and genuinely excited about doing and teaching mathematics. She is sorely missed.

During my third year in college, I took Real Analysis with Gonzalo. My friends and I used to study for tests by attempting to solve all problems in the book (in this case Rudin’s *Principles of Mathematical Analysis*) for the units covered in the exam. Of course, there were problems we couldn’t do, so we would go to office hours often, and talk to Gonzalo about everything math-related. At the end of the semester, I told him that I wanted to learn algebraic topology, and asked if he would be willing to advise me. He agreed and the very next semester we set up a reading course (Massey’s *A Basic Course in Algebraic Topology*) and with his guidance, I was able to complete a laureate thesis titled *The Borsuk-Ulam theorem and its applications*. I thank him not only for his guidance but for the many opportunities he facilitated so that I could be successful. One of the first things he told me when we started working together was that he thought I was capable of getting a PhD overseas, but for that to happen I needed to speak English. At the time I could read it, but I certainly couldn’t write/speak or understand it without subtitles. Gonzalo, through his contacts, found me a job as a substitute math teacher in a bilingual (English-Spanish) school, and with the salary, I was able to pay for a one-year intensive English course. When the time came for me to take the TOEFL, GRE and GRE-math exams, Gonzalo let me use his credit card (my family didn’t have one) to pay the online registration fees. Of course, I paid him back later, but these were all things he didn’t have to do, which were huge for me. I cannot thank him enough.

Looking back, I now think of these logistical challenges as examples of structural roadblocks where the socioeconomic background can limit access: did you go to a school where learning to speak a second language was possible/encouraged? Did your parents have resources available for extra-curricular/academic activities? Did you have access to people who knew the system and could give you timely and accurate advice? I was lucky to have people in my corner who helped me circumvent these roadblocks. The goal should be for all our students to feel that luck is also on their side.

I graduated from *Univalle* in April of 2006 (Valedictorian) and started my PhD in mathematics at Stanford University in September of the same year. The application process for grad school started two years earlier, from the time Gonzalo suggested I learned English, to one afternoon in Doris’ home with her, Raul and Gonzalo helping me prepare the application packages—at the time these documents (translated and notarized grades, recommendation letters, CV, essays, etc.) needed to be physically mailed. The three of them wrote recommendation letters on my behalf and helped make sure that the right documents were in the right envelopes for each university. Due to costs, I could only apply to five schools; all in the U.S. and across a wide range of academic rankings. I was denied admission in two, short-listed for one, and admitted into the other two.

**Stanford Years**

The time I spent in California as a graduate student was filled with good experiences, both academic and social. I made a lot of good friends and learned a lot of mathematics. And even though I worked very hard, I vividly remember feeling like an impostor, that the admissions office had surely made a mistake when they let me in. There is actually a name for that, the impostor syndrome, and it must have been widespread enough at Stanford because there were several institutional initiatives to help combat it. For instance, I attended a university-sponsored workshop for coping mechanisms, in a packed auditorium, and there was also this very cool series called The Resilience Project, with videos [2] from top faculty—heroes of many—telling their stories of academic and professional failure. I had my share of those in grad school. I failed my first midterm ever within two months of starting (measure theory) and also failed the algebra qualifying exam. As painful as they were, those failures were useful. All the algebra I learned during the summer of that year, while studying to retake (and pass!) the qualifying exam, has helped me in my research to this day.

Speaking of research, I started graduate school convinced that I wanted to focus on algebraic topology. This is the branch of mathematics concerned with spatial properties that are invariant under continuous deformations. For most of its history, algebraic topology has been regarded as a purely theoretical subject. Imagine then my surprise when, during one of my reading courses, I found out that topological ideas were actually being used to solve real problems in data analysis. Today, computational and applied topology, and in particular topological data analysis (TDA), are rapidly developing disciplines at the intersection of statistics, mathematics, and computation, with many students, vibrant conferences, and faculty in universities worldwide. In its inception, the main advances in TDA were spearheaded by Dr. John Harer and Dr. Herbert Edeslbrunner at Duke University, and by Dr. Gunnar Carlsson at Stanford.

I met Gunnar in a reading course I signed up for during my second year of grad school.These are directed studies where, typically, a faculty member guides a student through advanced/seminal research papers in their area. Gunnar is a widely respected mathematician with deep contributions in pure algebraic topology. During one of said meetings—we were reading Wall’s Finiteness Conditions for CW-Complexes—I asked Gunnar about what he was actually working on those days. He proceeded to tell me all about this new field, topological data analysis, showing that it was possible to leverage machinery developed by pure algebraic topologists, but now in algorithms to solve real problems in data science. It made total sense, and I was blown away. We stopped reading Wall’s paper, Gunnar became my thesis advisor, and from then on I’ve been working in TDA as my main research focus. [3]

I graduated in 2011, and we were able to arrange for my parents to make the trip from Cali to California. I believe that was the first time they had flown on an airplane. We had to get passports and visas, I flew back to Colombia and then came back with them for the commencement; having them there was awesome. After the ceremony in the math department, where we actually got our diplomas, Gunnar came up to me with congratulations and said he wanted to meet my parents. I panicked for a second and said: Gunnar … they don’t … speak English—to which he responded: it’s ok, I just want to say hi. He goes up to my parents and tells them, slowly, in Spanish, *José es un muy buen estudiante*. [4] He then said bye and walked away. At the beginning of this chapter, I mentioned the idea of small acts of kindness from mentors, that can have a huge impact. The interaction I just described is a prime example of one.

**Post-Stanford Years—Making the World a Smaller Place**

After Stanford, I went to Duke University as a postdoc to work with John Harer. I was very excited to learn from and work with him—John has made pioneering contributions in algebraic topology, especially in the homology and cohomology of mapping class groups, and he has translated this expertise to establish foundational results in TDA. What gave me pause, however, was moving to North Carolina without any other context than the painful history of the overt racial violence that has plagued the South. As an outsider, the history books were my only reference. My experience there was positive, though. I always felt welcomed, and the four years I spent at Duke were fruitful both academically and personally. John was a great mentor to have; he helped me grow into the next stage of my professional career, and he and his family welcomed me into their home for every celebration.

After four years at Duke, I moved to Michigan State University (MSU) as an Assistant Professor with joint appointments in the Department of Computational Mathematics, Science & Engineering (CMSE), and the Department of Mathematics (MTH). At MSU I’ve worked to build a vibrant research group with undergrads, grad students, postdocs, and the support of several sources (e.g., MSU, Defense Advanced Research Projects Agency (DARPA), National Science Foundation (NSF) and its CAREER program), advancing applied topology in ways that I personally find deeply fascinating. Over the last few years, I’ve been fortunate to also mentor students (e.g., by co-directing their thesis work) from Mexico, Colombia, and Brazil. One of them is now completing a dual PhD in CMSE and MTH at MSU. With other Latinx faculty at U.S. institutions, we have also run summer schools, workshops, and programs both in the U.S. and South America, which we hope to leverage in order to reach students that may not be aware of the many existing opportunities.

The emergence of new areas in computational mathematics—like applied topology, applied algebraic geometry, computational harmonic analysis, etc.—provides a valuable opportunity to increase the representation of historically underserved communities in the mathematical sciences. Indeed, these areas blend mathematics with modern science in ways that students tend to find compelling, and with mostly democratized points of entry. Nowadays, academic software is open and widely available, online conferences and open courses can be followed anywhere in the world, and online social communities provide valuable information and mentoring that not too long ago were available to only a select few.5 If you know of someone that you think may benefit from any of these resources, please reach out to them; a seemingly small act of kindness can go a long way.

**Advice**

I would like to end with a few words of advice:

(1) Advice to my younger self: Trust in your own ideas and pursue them with unrelenting passion; even if it doesn’t seem like it at times, things will work out in the end. Also, the tenure track is a gruelling process and busyness is a reality, but you must make time to take care of yourself both physically and mentally. Learning how to say “no” early on, and abandoning perfectionism, are good ways to create that space.

(2) How to encourage and inspire younger people in math: At least when I was younger, there were two things that motivated me to do mathematics. One was the deep personal satisfaction I would get from solving a difficult problem, and the other was hearing the stories and learning about the human side of being a mathematician (Simon Singh’s Fermat’s Last Theorem, is one of those great stories). With this in mind, one of the best ways I have found to inspire students in mathematics is to facilitate their own discovery, with ample guidance, but avoiding easy/lazy answers that would deprive them of the satisfaction of figuring something out on their own. Something that I also think is valuable is connecting the act of doing mathematics to the social aspect of being a mathematician. By this, I mean the stories, the struggles, the joy of sharing with people at conferences and other social/cultural events. Also, math is just cool, with intrinsic beauty and many connections to science broadly; try (as tempting as it may be) not to make it uncool with unnecessary jargon and over-formalization.

[1] The ICFES stands for Instituto Colombiano para la Evaluación de la Educación (ICFES).

[2] *The Resilience Project* series is still available here: youtube.com/channel/UCWImwojhbas29JwQlslsLGg

[3] Here is a recent talk (in Spanish) from the Cibercoloquio Latinoamericano de Matemáticas:

youtube.com/watch?v=cBJOo0NX6So.

[4] José is a very good student.

[5] See, for instance, my lectures on TDA at youtube.com/watch?v=APgR3avai30 and the Network of Minorities in Mathematical Sciences group on Facebook.

Previous *Testimonios:*

- Dr. James A. M. Álvarez
- Dr. Federico Ardila Mantilla
- Dr. Selenne Bañuelos
- Dr. Erika Tatiana Camacho
- Dr. Anastasia Chavez
- Dr. Minerva Cordero
- Dr. Ricardo Cortez
- Dr. Jesús A. De Loera Herrera
- Dr. Jessica M. Deshler
- Dr. Carrie Diaz Eaton
- Dr. Alexander Díaz-López
- Dr. Stephan Ramon Garcia
- Dr. Ralph R. Gomez
- Dr. Victor H. Moll
- Dr. Ryan R. Mouruzzi, Jr.
- Dr. Cynthia Oropesa Anhalt
- Dr. Omayra Ortega