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Topics: LGBTQ+ Idenity in MATH, Math Research as an Undergraduate, Homotopy Theory

Natalie Downey (She/They) is a transwoman who does research in Homotopy Theory. During her undergraduate degree, she was a part of four REU's (Research Experience for Undergraduates). Her reserach has led to two papers, one of which has been cited. She graduted in Math from CU Boulder in 2020 and will go to the University of Illinois Urbanna-Champaign for her gradute work. 

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Transcript for Part 1 & Part 2. 

(The Following Transcript has been lightly edit for clarity.) 

ARPI: Hi, my name is Arpi.

N: Hi, I’m Natalie.

A: So, I am speaking to Natalie Downey right now. She just graduated from CU Boulder with a degree in mathematics and will be going on to the University of Illinois-Urabana Champaign as a graduate student in math. And today, we are going to discuss research in mathematics, what it’s like to get interested in mathematics, and LGBTQ identity in math. So, what started your interest in mathematics?

N: So, from a very young age, like, math always was neat, and I was pretty good at it. It helps that I have synesthesia - which I cannot say [the word], which means that, for me, mathematics produces songs and beautiful pictures in my head. And later, in junior year of high school, I still found math neat, though, it wasn’t like a diehard passion. I had a very great teacher at Boulder High School, Ms. Speckle, who made my Calc B/C course very fun and engaging to the point where I asked so many questions that I got limited to two a day. From there, my senior year I did Calc 3 and then my school had me go up to the University for my next class which was Linear Algebra. And, the class was
taught professor Dr. Carla Farsi who is a mathematician and professional artist. And seeing how she spoke about mathematics — it wasn’t a bunch of formulas to memorize. It wasn’t, “Oh, follow these exact logical steps to get an answer.” It was more intuitive. It was listening to your emotions and dealing with it like art. Sure, in art there are definitely good, standard techniques, and brushes and tools you have to use to make art, but you don’t always have to follow it and you can go in leading with your emotions and intuition and get very beautiful results. The year I took her class, I also read a book on the history of chaos theory which mentioned the Poincaré Conjecture. This conjecture is the only solved Millennium Prize problem of the seven that got posted at the start of the 21st century. But it was like nothing else I’d ever seen. It was talking about connectedness, and the loops and just all these different mathematical objects. And it hooked me, and it led my college math career to studying homotopy theory.

I: So, you got interested in research fairly early as a student. I was wondering if you could give us a little bit more background on how you got started.

R: Because of that book, I was talking to my professor Dr. Farsi about learning more about the Poincaré conjecture. And it turns out, she does differential geometry, which is the area that the conjecture lies in, roughly. It’s a homotopy theory conjecture, proved with differential geometry. So, she let me know about a grant at CU — the undergrad research opportunity program as well as the math department’s own funding for undergraduate summer research. We applied for both. We got it, and I was able to have my first REU. But, that research really came from me going out, talking to my professor about it, um, and learning how to find or learning the ways I could go get funding to do what I
wanted. Um, and from there, it led to three other projects which led to some papers, but it was really that first step of talking to my professor, having an interest, and really just being interested inmathematics, which she was really receptive to. And most professors are receptive if you go to talk to them.

A: And we just want to emphasize too, that, being assertive, and going to talk to professors is really the key aspect to getting research opportunities. We are going to be talking to, we’re going to be talking to a lot of students and graduate students who got started this way. And, it can seem a little intimidating at first, but we also want to emphasize, and I think that it’s been a lot of our experiences that professors are fairly open to speaking with students about this stuff. Especially in mathematics. Like, they are interested, and they want to speak to students. And not all professors are going to say, “Yes,” but that’s why asking around a little bit. Getting a little bit of feedback is really key, into getting those
opportunities. And it can also seem a little intimidating in addition to doing your classes well, but we like to emphasize that research is this exciting opportunity to really create something new. To really learn something really deeply that you are not going to see in your classes. And it’s also really helpful to think about when you want to go into graduate school because really, what you’re doing part-time as a researcher, as an undergraduate, you’re going to be doing almost full time as a graduate student or a post-doc or going further into that. So, it’s just a really good opportunity to stretch that limb and to see if it’s something that you’d like to continue doing. Is there anything that you’d want to add to that?

N: So research, and specifically math research, if you look online at the archives of something and you look at those papers, it’s extremely scary! The good part is that those papers are…to help contribute to a paper post online, even if it’s a scary paper, you can do that as an undergraduate after taking your first proof-based math course. Because the first step in any research is learning the background. It doesn’t matter if you’re 18/19 fresh out of high school or you're a 50-year-old tenured professor, you
still have to learn the background. And then, the next step is doing computations and looking for patterns. This is the step [at which] you will know that you like math research or you’ll hate it. Because you’re going to be banging your head against the wall for a long time and you have to be okay with being wrong. For me, I found that being open to making really bad, dumb, simple mistakes let me learn exactly why they didn’t work. But would still give me good intuition and I could fix them. And this led to two papers which were cited. And it’s because I was not afraid to make the simple mistakes, the dumb mistakes, or even the super technical mistakes, and learn why those or what went wrong, and how I can avoid them.

A: Okay, so now we want to switch gears a little bit, and we wanted to talk about being LGBTQ in a math field. So, what was it like being an LGBTQ person on campus while you are doing your math [degree]?

N: So, on campus, my experience sort of varied from department to department. None of it was really bad, um, but some departments like CU’s math department, most of the people are amazing! And it actually, in the past few years, it’s one of, I’ve noticed it turn into one of the queerest departments on campus with a large number of LGBTQ people as grad students and working in the help lab.  As well as just in my classes, like, my last years, I wasn’t the only queer person. And it wasn’t just like [one other] …there would be two to four in most of them. But, it’s still, I still receive some looks. They went away about… That was about the worst thing I got on campus. The occasional glare.

A: And what kind of thinking got you through those microagressions or aggressions?

N: So, when being on campus I knew that I was there as a math student and I knew that I was a math student and I knew that I had a right to be there. Those looks and whatnot  they can hurt, but it’s a stranger who doesn’t know you giving a quick judgement on you. Which is wrong. And so that, remembering that and having allies on campus which, there are plenty of, luckily, helped me though.

A: And, also, just share a little bit about your experiences going to symposiums and the interactions you had there, and interactions you’ve had with other LGBTQ people in mathematics beyond CU Boulder.

N: So, this January, I went to the Joint Mathematicians Meeting in Denver which is one of the largest international general math conferences. I was there presenting a poster, and I noticed that I got a lot of looks when I was there. They weren’t glares, there were just….people around the world might not have seen a transperson before. And if they had negative thoughts or other things, they knew enough not to say it at the conference. When I did the poster session, I did notice that a lot of people would walk right past me, look right past me, and give me like glances but for the most part ignore me. But the people who did pay attention were very fine with it. Treated me like a cis woman. And even in the math world though, I went to a reception with the group, Spectra, which is the LGBTQ mathematicians’ organization. And at that part, it was great. And it is growing. We had a room that was…it wasn’t the biggest room because they didn’t expect a lot of people, and it was packed full. So, while, like support and acceptance of trans people in the larger match community isn’t the best, it has a lot of …career mathematicians have a strong base of support and Spectra is supported by the two largest math organizations in America. And we’re growing. We’re becoming more visible. We’re getting more noticed. Fewer people are hiding.

A: I think we talked a little bit about, how, I’m in cosmology and you’re doing mathematics, and how a lot of people in cosmology and mathematics and these more abstract sciences and how people in those areas think that because they are more abstract that they are free of bias, free of the world, whatever they think themselves. But that’s not really true. Right? They do have a bias. Almost that ignorance is what’s fueling that bias, and continuing that bias, rather than them being ahead of it in any kind of way.

N: So, it’s a big issue in mathematics and I think that it stems from the fact that, at the most formal level, mathematics is inherently non-biased. But, we’re not emotionless machines performing computations without any idea of the real world. It’s a bunch of humans performing mathematics and there a lot of implicit biases that the mathematical community, or large parts of the community, will rally against them, [saying] that, it’s impossible to have them. We only care about the work that mathematicians produce.  And it’s not just an issue for LGBTQ mathematicians, but for mathematicians of color and women mathematicians. They are all in this struggle together knowing that their voices are ignored because of appearances or name, or gender—things that they can’t really control. And so there’s support there, making sure that it’s not, that the bias of the mainstream isn’t allowed to continue and that, they can help support those less heard voices and underrepresented voices.

A: and do you have any advice for LGBTQ students in high school now, because there’s a lot of uncertainty going on right now with the COVID-19 virus and what colleges are going to look like. How can students approach this while making steps towards their STEM careers?

N: So, like, this is definitely a rough time where it’s hard to know, “What’s next week going to look like? What’s next month going to look like? What’s the fall going to look like?” And, in that uncertainty, especially an an LGBTQ person, it’s very scary. But you can’t think about the long-term. It will happen but thinking about it will not affect it. So, take a look at where you are right now and focus on taking the next step. It can be a small step. Like waking up and putting on clean clothes for the day. It can be a big step, like applying to a tier 1 research university. But it’s going to be, as long as you are taking a step forward, just focus on that. I’ve had a lot of things happen in the past four years that have put me out of school for like a day, a week, months. Six months was the longest time. And each time, I just made sure to keep taking steps, whether it was mental health, research, fitness, and re-evaluating, I am here today - what is success? Is success not crying three times today? Or is success running a marathon?  But making sure you have those steps and knowing that even though we are in a scary world, you’re there and to ….sorry, I’m being a little rambly but… something in a quote from Captain Holt from Brooklyn 99 where he says, well I’m going to paraphrase it. Like, every time, someone like you steps up and announces to the world who they are, the world becomes a better place. Doing that is an insanely courageous act and just you existing day to day sends a message to people, “I am here; I am beautiful; and I exist.” We exist and I’m not like …we can’t be silenced. And you can go, and just doing that is a great thing, but you are also you’re going to be improving the world. You’re going to be working on a new force(?) engineering stuff, understanding that the world isn’t sunshine and rainbows and that we have things that we have to get worked on. And you’re going to do it, and you’re going to stand there, and you're going to be proud. And you are going to help people just by existing. Because they’ll see you and they’ll see it’s okay. And that one day they could even if they are not in a safe situation, just watching someone be out is inspiring.

A: Well thanks for talking with us!

Transcribed by Robyn Sandekian, June 7, 2020

Transcript for Homotopy Theory Interview

Arpi: Cool. So, you do homotopy theory research. And I wanted you to tell us, what is this? What is this field of mathematics that you research?

Natalie: So, to start with, I’m going to talk a little bit for two things to be equal in mathematics. Everyone knows in math that 7 = 7, or X^2 is equal to X^2 +0. Things are equal to themselves and themselves exactly. Even if you label it something different. Then, if you take high school geometry, you learn a second type of equivalence called congruence. Where let’s say, you have two triangles, and you can take one triangle and move it on top of the other by just shifting it around, rotating it, or flipping it, they’re congruent. But you can’t change the shape at all. In homotopy theory, we are interested in, instead of a rigid triangle, what if it were made out of a rubber band and it’s malleable and you can change it? You can shrink it. You can grow it. Size doesn’t matter. The rigid shape doesn’t matter. You can’t cut it and you can’t poke holes in it, but you can do just about anything else.  And it turns out, in homotopy theory, we do this with circles. I’ll take, let’s say, a flat piece of paper, draw a circle on it, and see what it changes into and study that. But a flat piece of paper is a good model of the Euclidian plane, or the XY-plane that you guys work with in your high school math courses and I can shrink it to a point, I can grow it forever, but I can’t really do much else with a circle. It’s not very interesting there. Though let’s say instead of that, I were to do it on a water bottle which has holes and {loops? time stamp 0:10:05.3} or a car, or a donut. These are much more complicated things where you can’t always tie the loops to a close or shift any loop into another. On a donut, if you draw one around the donut, you can’t turn it into a point without cutting or….if you put a rubber band around a donut, you can’t shrink it to a point without lifting it off the donut, which is not allowed. Or cutting it, also not allowed. So, all of a sudden, once you get to shapes, instead of the xy-plane, things become interesting. And it might not seem very useful, but it actually has applications in a lot of pure mathematics, theoretical physics, and data science and statistics. So, it is a very useful tool in studying the physical and abstract world.

A: And those data sets that you are talking about, those can also be datasets related to epidemiology and issues like the COVID-19 virus, right?

N: Yes, you could look at the COVID data, turn it, there are ways of turning that data into a shape naturally, and then using homotopy theory you can study it and see things which other techniques just cannot see. It is very good at handling “bad data” (laugh!).