On September 25, 2024, Gabriel Sosa Castillo delivered a talk about his research on reconstructible monomial orderings at the Amherst College Math Colloquium. Sosa Castillo previously worked at the College as an assistant professor in the Mathematics Department and now works at Colgate University.
Sosa Castillo specializes in computational and combinatorial commutative algebra, which refers to mathematical concepts called rings with certain properties and applies them to problems such as counting or finding patterns (more on this later). He is also involved in research, and sometimes he works with undergraduates to do so. In particular, he researched monomial orderings and their properties both at Amherst College in 2017 and at Colgate University in 2023, with help from undergraduates at each institution. In an interview with the Amherst STEM Network, Sosa Castillo talked about how he became a mathematician, the importance of pure mathematics, and his experience as a Hispanic student.
Interview edited for length and clarity:
LF: Could you introduce a little bit about yourself and your past research?
GSC: When I was a kid…everybody was like, “Oh, you’re so good at math, and you like reading, and you’re so smart. You should be an engineer.” I was like, “Okay, I’ll be an engineer.” But I really didn’t know what it meant to be an engineer. It was just that’s what they told me I should do. Luckily, when I was in eighth grade, we had a new math teacher, and this guy said, “We’re going to participate in the Math Olympiads.” So, I participated, and then I made it to the finals. It was a shock because the finals involved proofs. I [had] never really seen a proof, and when I arrived, they were explaining things that I had no idea what they meant…Everybody seemed to know what was happening, and I didn’t. So I didn’t do too well in the finals, but after the finals, [there was training] for six weeks for international competitions. I stayed maybe a day and a half.
And then one of the coaches called me on the phone [asking], “Why did you leave? Why didn’t you stay for the training time?” [And I answered], “I had no idea what was happening and everybody seemed to know what was happening, so I’m not cut out for this.” [The coach answered], “All of these people that are here, they have made it to the final for the last three years. They all have had the training. They all know what’s happening because they have seen it before. That’s why I kept asking if you were following because you are the only new person to make it to the final.”
I made the mistake of not speaking up because I was afraid of [seeming like] a fool. They told me, “We’re gonna send you some copies of some books. Try to read them on your own.” So I did. The next year, [I] went again, and then I made it to the team. I went to the Iberoamerican Mathematical Olympiad. Then I went to the international one. Now I was in my senior year of high school, and somebody asked me, “What is your plan? What are you going to study? Aren’t you going to study math?” And then I was like, “Wait, you can study math?” So I went back home, and I told my dad, “I don’t want to be an engineer. I want to be a mathematician.”
I moved to Costa Rica. I did my undergraduate degree there… When my sophomore year was ending, I came out [as part of the LGBTQ+ community], and that was a pretty rough experience. At that point, I was essentially homeless. One of my professors reached out to me [and said], “You’re not in a good place right now. And it sounds like a lot of your classmates have a lot of mixed feelings towards you.” This was my advisor, and he actually saved me, in that sense, because he [proposed], “I know this bilingual high school that is looking for a math teacher, and you can get a teaching certification over the summer. I can vouch for you…” So, I worked full time as a high school teacher for three years. And then [I decided to] go back [and finish my undergraduate]…After four years, I graduated and did my Ph.D. at Purdue University. I think that Purdue was the right place for me. It had a lot of camaraderie between the graduate students. I had seen other places where I applied, and the graduate students were not as friendly with each other. To me, math is a very social activity. I have had a couple of papers on my own, but I really enjoy [writing a paper when I do it] with other people, discuss ideas, and make progress together. That’s the kind of mathematician that I am. I’m a very social mathematician.
I’m a computational and combinatorial commutative algebraist. What this means is that I do commutative algebra, a field that explores rings in which the product is commutative (the order of the factors doesn’t matter). But the things that I explore are tied to combinatorial objects, which means that I try to exploit properties related to counting things, and exchanging things.
The computational part is much more like math; it is about noticing patterns [out of what] you’re seeing and trying to understand. I think one of the misconceptions that people have about work in algebra is that you actually have to create examples. Thank goodness we have computers because then you can program the computer to do a lot of examples at once. Essentially, I use computers to generate data about algebraic objects that I want to study. Then I look at that data searching for patterns. And once one is clear, I try to prove it (most of the time I succeed). The proof requires me to have some type of combinatorial fun. It can be that there’s a graph attached to a combinatorial object, and something is happening in the graph as I do things.
I enjoy talking about my research on monomial orderings because I think it is accessible to a general audience. I like to put it like this: imagine that I tell you that there’s an alien language. And I tell you what the letters are, but these are letters that you have never encountered, like weird symbols. And I tell you which letter comes first in the alphabet, which one comes second, an so on; and then I give you a bunch of dictionaries, each one missing one of the letters. So, [one] dictionary has all the words that don’t have the first letter. This [other] dictionary has all the words that don’t have the second letter, and so on. And I give you all of these dictionaries. So then the question is, are you able to find out what the full dictionary looks like if I just give you these partial dictionaries? The answer is, yeah, as long as the alphabet has more than three letters. If you have three letters or less, then you really cannot do it.
LF: Great, thank you. I see now how patterns play into your work. Given that we are in Hispanic Heritage Month also — I didn’t know this before I came here, which is kind of fun — what do you think about being a Hispanic student, professor, researcher, [and] mathematician, here in the U.S.? Has the community been inclusive?
GSC: I was one of the founders of Lathisms, which is a website [for] Latinx and Hispanics in the Mathematical Sciences. I’m no longer associated with Lathisms… but I worked with them for the first four or five years. Something that I think is important to highlight is the marked differences between Hispanic and Latinx mathematicians who were born in South and Central America versus the ones who were born in the U.S. The obstacles that the U.S.-based ones faced were incredible. Some were even insurmountable in many senses that we, coming from another country, never faced. So we didn’t have people thinking that we were not able to do the math. And by highlighting that Latinx students in the U.S. face more problems, you actually can show that there is an issue. The issue is a little lessened now, but it’s not gone. Amherst College does a really great job of bringing not only international students, but also students who [have Latinx parents but were born in the U.S. or moved to the U.S. at a young age], having a very diverse student body, finding strength in that, and promoting it. I wish that more places would do that.
LF: Thank you for sharing your experience with Lathisms. To conclude the interview, could you please share your thoughts on pure math and its applicability to other things? What do you think about this concept?
GSC: I teach a course called The Hidden History of Mathematics, and I start talking about what math is and what math has meant or been interpreted as, by different cultures and different people throughout history. I can go on for hours because I have classes where we discuss these whole ideas of how math is sometimes like philosophy, or math is sometimes like literature, or math sometimes is a cult, too…
If you read Ray Bradbury’s The Martian Chronicles, there’s a story about a guy who is on a bus, and he has a device that tracks exactly where he is at all times. His family can reach him through the device [at] any moment of the day. He has no sense of privacy because whatever he puts on the device, there are people who can look into it. People know who he is because of the things that he does with the device. And then he essentially jumps off the bus, because he’s horrified by this. And so this is science fiction. And of course, like in the 70s, you read this, and it’s like a horror story. And now we are in 2024 and everybody has that device. It’s an iPhone. Anybody can know where you are. I don’t see people having this struggle, but it’s because now it’s normalized. But back then, it was science fiction. So again, it’s something that was imagined, and now it’s reality. So that’s [how] a lot of…science fiction works, but I feel that math is the same, right?
Like planets move in elliptical orbits. [We found that out with] Johannes Kepler. [He had] all of these things that he knew about ellipses. The ancient Greeks worked on ellipses for years, just for the pleasure of understanding what an ellipse was and all its properties. There was no direct application at the time, but the application showed up eventually. It’s the same with a lot of these algebraic geometry [and] commutative algebra [concepts]…Some math has been created because there was an application question that they wanted to answer immediately, but a lot of the math was created for no particular reason. It was just for the joy of doing math. So, I feel we have to find a middle point.”