Patty Vest:
Welcome to Sagecast, the podcast of Pomona College. I'm Patty Vest.

Mark Wood:
And I'm Mark Wood.

Patty Vest:
In these extraordinary times, we're coming to you from our various homes as we all shelter in place.

Mark Wood:
This season on Sagecast, we're talking to Pomona faculty and alumni about the personal, professional, and intellectual journeys that have brought them to where they are today.

Patty Vest:
Today we're talking with professor of mathematics Edray Goins who specializes in number theory and algebraic geometry.

Mark Wood:
Welcome, Edray. Glad to have you with us here in cyberspace. And how are you adjusting to life here in the time of quarantine?

Edray Goins:
The best I can. I can't really say that things are horrible. Probably if anything, I really miss talking with the students one on one, being able to see their faces. But I think I'm surviving with the new adjustment to classes.

Patty Vest:
Edray, you grew up in L. A. Tell us a little bit about your early years. And did you always gravitate toward math, or how did you fond your way toward mathematics?

Edray Goins:
No, I wasn't always into mathematics. I was actually really big into science. Actually, being a scientist was always one of my dreams. I grew up in South Central Las Angeles, not too far away from Englewood. And actually, every day I would see the airplanes flying over going to LAX. So I wasn't so much interested in the whole thing of traveling to other countries or wondering where these planes were going. I was more interested in how the planes worked. So literally, every day seeing the planes fly over I would always wonder how are they still up there in the sky. How does flight work? What's really happening with these engines?

Edray Goins:
There was one time when my mom, actually my father, was really worried about me being too much of a nerd, and maybe he wanted to get out and be more social. So they put me on a flag football team. And I remember one day being out there on the field and an airplane flew over, and it was very loud. I was completely fascinated by this plane flying over. I didn't know during all of this time that there was a play in motion, people running on the field. The coach was yelling at me I had absolutely no idea until I looked around. I thought nobody was there. That's a typical thing that would happen.

Edray Goins:
So I did grow up there in South L.A. almost all of my schools were, I'd say, 99% black. All of the students were black. All of the teachers were black. And I would say that I really grew up with a very strong sense of community. A lot of the people that I was around were really big into helping out with others. It was always about learning about black history in the classroom, figuring out what you could do to help out with the community. Maxine Waters, who I think is very big now, she was my congresswoman growing up. So just that whole thing of really doing for community, helping out others, was always something that I was really big into. It was something that was instilled in me from a very young age.

Edray Goins:
Still, I wanted to be a scientist, and I really wanted to learn about how the world around me worked. I can't say that I went to any fancy private schools or that I had any fancy tutors. I definitely was a product of the public school system there out of Las Angeles Unified School District, and I'm still very proud of that. But I didn't really go to the best schools in the city, so I feel very fortunate that I had a lot of friends and also a lot of teachers who really saw how interested I was in science. And even though the classes maybe weren't the best classes in the school district, I still had a lot of teachers who were very, very encouraging of me.

Edray Goins:
I remember at one point there at the high school that I went to, it was certainly one of the worst in the city. And, of course, we had a lot of problems with gangs and shootings that happened either on campus or just outside of campus. There was one gang member, in particular, who came up to me one day. First I thought, "Here it is. I'm probably going to get harassed by this guy." He said, "You know, you are one of the best and brightest of our community and of our generation, so if anybody ever messes with you, let me know. I'll take care of it."

Mark Wood:
Wow!

Edray Goins:
That whole thing of just knowing that I'm always going to be protected, that people are very proud of me, I just really can't say enough. I really felt like growing up in South Central L.A. in the 80s was, for me, one of the best experiences.

Mark Wood:
No, you were the first student from your high school to attend Caltech, is that right?

Mark Wood:
[crosstalk 00:04:31] Why did you pick Caltech, and what was your experience like there?

Edray Goins:
Well, I came to Caltech from a slightly roundabout direction. Growing up in L.A, of course, you know about earthquakes and if you know about earthquakes, yeah going to hear about it from Caltech. And the evening news would always say things like, "The scientists at Caltech said that the recent earthquake was a 6.5." If you're growing up in L.A. hearing about earthquakes, everybody knows about Caltech because of the earthquakes. But not many people know about Caltech for either its academics or for the science that they do, so I remember hearing about Caltech from a very early age.

Edray Goins:
In addition to that, I was fortunate in that I had my parents' godparents who essentially were my godparents, we would make it a point that after church on Sundays, we would all get together in the car and we would drive to Pasadena. And Pasadena was where Caltech is located. So really almost every Sunday it's the way that the family would get together. We would drive to Pasadena. We would go have dinner at a cafeteria there called Beadles Cafeteria. And I would always ask then, "Can we just drive past this place called Caltech?"

Edray Goins:
So I knew where it was, I knew from the news, in a sense, what it was, but I didn't really know what happened on campus. So I knew from a very early age, I would say probably from about maybe age 15 or so, that Caltech was my number one choice. It was where I really wanted to go. I didn't really know much about the Ivy League colleges or even about the University of California system, but I knew Caltech was where I wanted to go.

Edray Goins:
So my senior year, I was fortunate enough to be one of the best students in all of Las Angeles Unified School District, and I knew I wanted to go into science. So at the time, I had applied, at least wanted to apply to one of two schools. It was either going to be Caltech or MIT. I actually sent off a request for an application from MIT. This was before the internet, so you couldn't really go online and do all of this stuff like the common application. You really had to write a letter or send a postcard to get the application. So I sent a postcard to MIT, never heard back. Never heard back, so I never got an application. So I never applied to MIT.

Mark Wood:
Wow! Their loss.

Edray Goins:
Yeah, definitely it was their loss. But I got an application in the mail from Caltech. I didn't request one. It just appeared one day. Come to find out later, there actually was a back staff member who worked at Caltech named Lee Brown. And he made it a point to search out in the country the top black students. And once he found out who they were, he would look them up, he would get their address, he would send them an application in the mail.

Edray Goins:
So I got an application because there was a black staff member at Caltech who actually knew who I was. I didn't even know anything about this guy, but he knew who I was, and he sent me an application. So that's one of the reasons why I applied to Caltech. Of course, I definitely knew about it. It was certainly my number one school. But just the fact that I got the application, that put it even more over the top. And, of course, it probably didn't hurt that I didn't even get an application from MIT at all.

Edray Goins:
On the flip side, and I wonder if a lot of students listening to this podcast know about this too. You mentioned at my high school that I was the first person to get admitted to Caltech. I'm pretty sure I was the first person, probably even to this day the only person to even apply to Caltech.

Mark Wood:
Wow.

Edray Goins:
So I was a senior starting in 1989. I graduated in 1990. Of course, everyone on campus knew I was big into science, wanted to go to Caltech, was completely fascinated by this place. And so, of course, I told my high school counselor that I was going to apply to two places. It was going to be Caltech and MIT. Well, my high school counselor looked at me and said, "Caltech is a really difficult place to get into." And I said, "Well, yeah. I understand this." So she said, "Well, maybe you should think about applying to some other places when you don't get in."

Edray Goins:
It wasn't really a matter of if, it was more a matter of well, don't set your hopes up too high, probably won't happen. I wanted to be sympathetic to her because I understood there were something like 650 people who graduated from my high school every year. I would say only maybe the top 100 even applied to college, which meant another 500 people didn't even think about going to college. And of those 100 that she saw going to college every year, the vast majority would end up going to either a school there in the UC system or the Cal State system. There might even be a handful of others that would go to some of the HBCUs, these historically black colleges somewhere in the South.

Edray Goins:
So for her to even know somebody who's even thinking of applying to Caltech, let aloe was cocky enough to say that he was going to go there, I know that for her it was not even something that was even in her worldview.

Mark Wood:
It was beyond her imagination I guess.

Edray Goins:
Yeah, I think it was. I think it was beyond her imagination. So she said, "Fine, go ahead and apply to Caltech, but please apply to maybe a safe school." So she forced me to apply to USC. So I said, "Fine, I'll apply to USC. I did get in, but I still joke to people that really there were only, I'd say, three places that I applied to, and I got into. USC was one, Caltech was the second one, and ironically, Harvey Mudd was the third.

Edray Goins:
Of course, I then have to debate when I got into these three places where I was going to go. USC I crossed off the list right away. And I joke this with a lot of people in my family because my mother actually went to USC. She actually got a master's degree from there. But I was just never impressed by the campus. I had done a couple of summer programs there. I could see that the professors there weren't really that interested in working with kids from the neighborhood. Of course, I grew up like five, 10 minutes away from USC. And seeing that, unfortunately, with a couple of professors, that made a difference. That really hit home.

Edray Goins:
Caltech, on the other hand, I still have my acceptance letter to this day. The fact that I got into Caltech, that was one of the happiest days of my life. When I got into Harvey Mudd, I really had to debate which school should I go to. And I think it is ironic that I'm back here at the Claremont Colleges because I remember having a serious debate which school should I go to. Should it be Caltech? Should it be Harvey Mudd?

Edray Goins:
So at the time, I looked up in a brochure how many black students are there at Harvey Mudd and how many black students are there at Caltech. That, for me, was a real serious issue, and I could see that the numbers were pretty much zero. I can't remember at the time how many students there were at either school, but I can tell you that we're talking probably 20 at most. It might have even been five to 10 that were listed in the brochure.

Edray Goins:
So I decided if I'm really going to have a lonely experience, screw it, it might as well be at Caltech. So that's really the reason why I ended up in-

Edray Goins:
[crosstalk 00:12:04]

Mark Wood:
So what was it like there for you?

Edray Goins:
I have to say it was a mixed experience. In terms of academics, I don't know if I could have found a better fit than Caltech. When I got there, of course, I wanted to major in physics. It's a running joke in Caltech that almost everybody that starts at Caltech says that they want to major in physics, but you have a long history if Caltech really being one of the top schools in the world when it comes to physics. I didn't know about a lot of these names at the time, but Richard Feynman, this very famous physicist, he was there at Caltech for a very long time. He had passed away maybe two years before I got there. He was well known for teaching freshmen physics every single year.

Edray Goins:
On of the people who invented the term quartz, Murray Gell-Mann, he was actually a visiting professor there for several years when I was there. Caltech was just very much in the psyche of everyone who really wanted to do physics. And so to be at Caltech really meant that you wanted to be a physicist. I'm also really happy with the classes that I had taken because even now the way that I teach my math classes is very heavily influenced by what I saw as an undergraduate. And I just remember seeing a lot of crazy crossovers.

Edray Goins:
In math classes, they would talk about physics problems and how you could solve all these physics problems by using fancy math. I had a calculus instructor named Tom Apostol who actually as a number theorist, and I'll get back to that connection in just a second. But Apostol actually worked very closely with the physics department when it came to math education. He did this series of videos called Project Mathematics, and at the time, it was groundbreaking. There were computer animations he would have to showcase different ideas in math and how they applied to physics. And he started working on these videos in like 1985, on something very, very early.

Edray Goins:
Now, one of the things that he spent a couple of lectures talking about in the honors calculus class was the whole thing about the invention of calculus. And a lot of this came down to the motions of the planets. So for example, he would say that Johannes Kepler really wanted to know about the motions of the planets. There wasn't a lot of politics that, I guess, Kepler was interested in, in the sense of is the universe really centered around the Earth. That is, do the planets and stars orbit around the Earth? He was more interested in this idea that the planets and stars orbited around the sun. But what he really wanted to know was what's happening in terms of the numbers he was seeing.

Edray Goins:
So he realized things such as the lengths of the orbits. For example, we orbit the sun every roughly 365 days. The lengths of the orbits somehow matched up with how far away the planets were away from the sun. But it was actually more than that. The time of the seasons changed depending upon the different times of the year. We might say that in some sense the Earth is actually passing faster when it's closer to the sun, and it's moving slower when it's further away from the sun. And these are things that no one really talks about in popular culture much these days, but apparently, this was a big deal in the days of Kepler.

Edray Goins:
This was perplexing mathematicians for a long time, and apparently, at some point, there was a young kid by the name of Isaac Newton who saw all of this data. And he said, "I'm going to figure it all out." So he went away for a little bit. He invented something called calculus, but he did it so he could explain what Kepler was seeing with his data. So once he was able to explain it, he then showed there's this cool application of this thing called calculus that explains all of it.

Edray Goins:
So nowadays, we call these Kepler's Laws of Motion, but the reality is, Kepler could only explain it using data. He really couldn't explain why it was true. It was really Newton and the invention of calculus that could explain all of this. Well, I saw all of this for the first time in my calculus class, and I remember being fascinated hearing the story of the history and the physics behind it. This was in calculus. My instructor was explaining all of the physics, but at the end of the day, he said, "It's math that does it. It's math that explains it really."

Edray Goins:
That's where I changed my mindset from being a physics major to being a math major. It was really in this calculus class seeing how mathematics can explain the physics is what totally did it for me. And remember, up to this point, I wanted to understand the world around me. I wasn't so much interested in math, or physics, or chemistry. It was what can I do to understand how the world around me goes. And it all happened right there in my calculus class. So I would say that Caltech really influenced me in how I saw things, how I teach things. Even now when I talk to students, I really like to blend the math, and the physics and the chemistry all of it together because to me, it's just explaining how the world around you works.

Edray Goins:
But I say that on one hand Caltech really influenced me when it came to the academics, but on the other hand, when it came to the social aspects of life it was a very frustrating and in some ways depressing place. At the time, there were around 200 freshmen that came in, so there were just under 800 undergraduates there on campus. There were about maybe another 1,000 graduate students, but not more than 2,000 students total on the whole campus. So it feels very much like Pomona.

Edray Goins:
Of the 200 freshmen, my year there were 14 black students. They told me that this was the largest number of black students Caltech have ever had coming in, in its freshmen class. I think the year before that there were maybe eight. And I think the year before that there were like two. So we were a very large group. But all of this is kind of relative. When you take a look, in general, there were really no other black students on campus, which meant that it was really hard to find friends, to find community, to do much of anything.

Edray Goins:
Even for women, I believe that we had, well what we called at the time, the ratio. There were I want to say maybe eight men for every one woman that was on campus. So there really were no women at all. I hear that nowadays, Caltech is much, much better. I think that it's sitting at about 50% women that are there coming in the freshmen class. And I know that his year they have the largest freshmen class of black students ever, which is sitting at something like 20 or so. I think that things have gotten a lot better. But at the time, in 1990 when I started, it was a very, very depressing place.

Edray Goins:
I'll just tell one little story just to say in some sense how frustrating it was. At the time there was this brand new form of music that was hitting the scenes known as rap music. I guess nowadays the kids call it hip-hop. And it was a very controversial art form. Not a lot of people played it. And I remember, in particular, MTV refused to play rap videos. In the early 90s, you could not find rap anywhere. A lot of stations wouldn't play it. The radio stations wouldn't play it. The TV stations definitely would not play it.

Edray Goins:
If you wanted to hear any rap at all, it all had to be underground. You had to hope you could get some radio station out of Santa Monica that was playing something at 2:00 Saturday morning. But otherwise, it was nearly impossible to find anything. There were a couple of times when my friends and I would go to parties on campus and with would ask the DJ would they play rap music. I remember this one time, in particular, I went in with a tape that I had, that I'd spent a long time trying to find it at the local Tower Records. And I asked the guy to play it, and the guy said, "Well, what kind of music is it." I said, "It's rap." He said, "No, I'm not going to play this."

Edray Goins:
And I remember my friends and I were really depressed that we couldn't get anything played, so we actually decided we were going to throw our own parties. So I remember that we had this one summer program where it was mostly minority kids that were there, black and Latino kids that were there for about six weeks or so. And at sone point, I just got so frustrated at finding DJs who were willing to play rap music, I myself DJed the whole party. It was a totally crazy experience that I had to have all of my tapes together and had to figure out between two cassette players. How am I going to play this music and mix from one song to the next song? It's nothing like nowadays where you can pull up everything on a laptop, you click on an MP3 and it plays.

Edray Goins:
I didn't even have two turntables, I had two cassette players trying to go back and forth playing all of this. And it was for a party of like 60 students in the room. That was a surreal experience. But still, this is things that we did to help establish that sense of community at Caltech. It was good at some times, it was frustrating at other times, but I still feel, in general, really fortunate for that experience.

Patty Vest:
Edray, you said you found your love for mathematics through your undergraduate experience. You, later on, went to Stanford for your PhD.

Edray Goins:
Right.

Patty Vest:
At what point did you find your interest or love for academics and for teaching?

Edray Goins:
I think I always had a love for academics, at least I know freshmen year is when I really had found a love for academics. I think of the same way I also found a real disdain for academia. And unfortunately, there's a big difference. My love for academics comes from the fact that I really love just to learn. And I learned that I could talk to other students. I could talk to other faculty. I could just learn about a lot of things.

Edray Goins:
And so from a very early age, I'll say roughly probably 12 years old, I knew that I wanted to get this thing called a PhD. I had no idea what it meant to get a PhD, but I knew that it was something that I wanted to do because I always had this love for learning. And I remember asking around when I was at that age what careers could I have where I could just spend the rest of my life learning, and people told me, "Well, you have to be a professor and to be a professor, you have to get a PhD." So I said, "Fine, in order for me to be a professor I'm going to get my PhD, and that's what I'll do." So I knew from a very early age that just learning was something that I wanted to do.

Edray Goins:
I think when I was in grad school, I was very fortunate in that I had a lot of friends from many, many different departments on campus. I mean literally, you name the department, I had a friend in that department. For a while, I was president of the Black Graduate Student's Association, which probably had about 400 members or so. There were almost exactly 100 black doctoral students at Stanford, which mean almost exactly 50 women and 50 men. And so there were people from every department people from the med school, from the business school, from the law school. I had friends in geology and chemistry, women studies, the history department. I mean really, you name the department, I had friends in that department.

Edray Goins:
And that's when I think I gained an appreciation of the whole college experience. Up to that point, I think I had been more focused in maybe science or learning physics and mathematics, but that's when I really gained this appreciation for no, it's all connected. It's not just about doing STEM or about doing science, it's really just, in general, about the whole college experience and learning what it's like to be I\in academia.

Edray Goins:
I think, in the same way, I also saw a lot of the negative sides as well. So I did see a little bit of how some professors can be very negative, how they can make student's lives miserable. I definitely had run-ins with different professors that did not take me seriously. When I was in grad school, certainly I was not told about any of the studies groups that were going on. I didn't even know that they were happening until one of my friends asked me about four months into my first year at Stanford, "How come you're not coming to the study groups?" I said, "What study groups?" So he just happened to tell me that they were going on, so then that's when I eventually learned that I wasn't being invited, so I tried to go a little bit more.

Edray Goins:
So I did see a lot of the negative things that could happen, which is why I'm always sympathetic when I see that there are good parts and bad parts of being in college. And it's not going to be this complete great experience for everybody, that it could be different things for different people. And even now, I try to keep that in mind. But I'll say that this idea of being at a liberal arts, I think that always stuck with me from when I was in grad school. I had so much fun having friends in the humanities. It was something I never saw at Caltech other than the work I did in the history department. Most of what I saw as an undergraduate were entirely in STEM, but now seeing this brand new experience of the possibility of being a professor when you don't necessarily have to do science all day. It was something I never thought about, but it was something that really fascinated me.

Mark Wood:
Let's move into your work as a mathematician. It's one of those things that's kind of hard for us to do much with on a general podcast because I know you're at a level where there's a lot of preparation that has to go into understanding the things that you're working with. But can you talk to us a little bit about the areas you specialize in? Number theory and algebraic geometry are the two that we mentioned. I know there are others. Can you tell us a little bit of what attracted you to those areas, in particular?

Edray Goins:
Yeah. Yeah, I'll try my best here. Well, the quick story that I'll give is when I was a freshman, I mentioned that I was taking honors calculus with Tom Apostol. Tom Apostol himself was a number theorist. Well, he never really talked about number theory when he taught calculus, and he was mostly talking about physics, and history, and what have you. But one day he mentioned that there was going to be a Caltech alum who was going to be giving a talk on campus, and he thought it would be really great if the undergraduates went to go hear his talk. He didn't really say much about what it was or about who this guy was. He just said, "Well, just try to go to the talk."

Edray Goins:
At the time, because I was trying to learn more about this whole thing of mathematics, if you're growing up in South Central L.A., you're not going to know anything about what a number theorist is or algebraic geometer was. I would go to the bookstore all the time just to look at titles of books, and there were two books, in particular, whose titles caught my attention. One was one that said, I believe it was An Introduction to Number Theory. I had never heard the phrase number theory. It was just bizarre to me. How can you have a whole theory of numbers? But here it was.

Edray Goins:
But the author of the book looked familiar. The author was Harold Stark who was the same person that Tom Apostol told me, "Go see his talk." So I had seen this book, I figured it must be fate or something. I see this book here in the book store and decide to go to the guy's talk. I probably understood the first 15 minutes and didn't understand any of the rest of it after that. But he was talking about some really cool ideas.

Edray Goins:
Let me see if I can give you a flavor of things that he was mentioning. A lot of us hear about these so-called Pythagorean triples, so things like three, four, five triangle. And what that really means is three square plus four square equals five square. Now, one thing I like to tell the students that I work with over the summers is when you hear this kind of equation, there should be two questions that come to mind. Number one, are there more examples like this one? And some people in high school might hear of things like a 5, 12, 13 triangle, five squared plus 12 squared is 13 squared. Maybe you might hear of an 8, 15, 17 triangle. So you start to write down more examples.

Edray Goins:
But then the second question should come in, which is, can you write down the formula that will generate all of these? So I was more interested in the latter because I realized that you could probably write a computer program that could help you find a few examples. But how do you know with a mathematical proof that you've actually gotten all examples? I became completely obsessed with this. And one thing I'm almost embarrassed to say is in reading Harold Stark's book, he showed a method of how you can actually write down all possible solutions, how you can actually write down a formula for these. And I just fell in love with this whole concept that you can prove that you have all solutions.

Edray Goins:
So what I used to do every single week is I would post on my dorm room door the equations that I could find that week where I found all solutions. So the equations of the week, I would write down 10 equations where I could find all solutions. Even at Caltech, people would scribble back these notes saying, "Oh this guy is a nerd." But I totally fell in love with it.

Mark Wood:
So you were a nerd even at Caltech.

Edray Goins:
Even at Caltech. Every single week, that was my thing freshmen year. I would put up on the board outside of my door, equation that I've solved this week. I was a little bit embarrassed to tell people that I was really in love with number theory because I was a physicist. I was at this school basically that Richard Feynman built. And I wasn't going to be one of these lowly mathematicians that was just talking about these silly equations. I was supposed to talk about how the universe worked, and about black holes, and quartz and all the rest of that.

Edray Goins:
I was a closeted mathematician for the first few years I was there because I would tell people I was a physics major, but then I would quietly say that I was still really in love with math. But more specifically, I was really in love with number theory. I just loved this whole concept that you could say with proof that now you found all true solutions to this very specific equation.

Mark Wood:
When do you come out of the closet?

Edray Goins:
Second year of graduate school. I'm embarrassed to say that yes, I went to grad school in mathematics, but I did it at Stanford where I knew I could jump over to the physics department, and I was still taking classes in the physics department my first couple of years.

Mark Wood:
You were just keeping your foot in the door in case you came to your senses and went back to-

Edray Goins:
That was exactly right. That's exactly right. Yes.

Patty Vest:
You had mentioned earlier that ... I can't remember if it was your mom or dad who were worried that you were "too nerdy" and involved you in flag football. What were they thinking now that you were nerdy at Caltech and nerdy at Stanford?

Edray Goins:
Oh, I don't really know that they really knew the whole story. I would always come home, and there would always be discussion about the classes I was doing or maybe me saying that I had to go to do research on campus that summer. But I never really talked about what my interests were outside of the classes and the other things. And I think even now it's really hard for me to, almost in a sense, justify doing mathematics because I really feel that there's a lot of societal issues that are out there. And at the end of the day, me being self-indulgent dealing with numbers and this kind of thing isn't really going to solve the world's problems.

Edray Goins:
I think that there are mathematicians out there who are doing things to solve the world's problems, but I can't say that the research that I'm doing will be one of them. So I do go back and forth. Yes, there are things that I try to do to help build the community and help make the world a better place, but I'm one of the first ones to say that I think there is a to of mathematics, including the stuff that I do, that's just very self-indulgent. So it's things that I enjoy doing, but really, at the end of the day, it's something that I personally enjoy.

Mark Wood:
Well, it's for the beauty of it, right?

Edray Goins:
It is, it is. It's the beauty, it's the simplicity. I was going to mention that I definitely got into algebraic geometry because it involves two words that I think a lot of people know, algebra and geometry. And what I like about it is it combines a lot of ideas, so I'm not necessarily going to say that it's a lot of mathematics, but there's a lot of philosophy that goes behind all of this.

Edray Goins:
What I work on these days really combines a lot of different areas, so yes, I do like geometry, which means that I deal with a lot of really weird geometry. This is even something that you see in physics. So you might say that the world that we live in is flat. I mean literally, you might say that we live on the planet Earth. And I know that there's something people who believe that the Earth is flat, but there's others of us who believe that the Earth is round. And you can even argue well, it's not a perfectly spherical shape. It's a little bit like an ellipsoidal. But the point is that these are all different kinds of shapes.

Edray Goins:
Well, in physics, you actually would argue that a lot of modern physics says that the universe that we live in is some kind of weird curved, warped type of universe, that the space that we live in isn't necessarily a flat space. Think of it as if you're moving your hand through air, it's not saying that it's just uniform air, but it's almost like if you put your hand in a bucket of water, swim around in a pool, that as you transition from the air to the liquid, things change. You can actually feel the viscosity in how things change. The same thing happens in the universe, that the type of universe that we live in, it's going to move and curve, and move around in different kinds of ways.

Edray Goins:
So even in geometry, it's not as simple as what you see on a daily basis, that things are going to be a little bit weird and a little bit strange. So you can ask yourself, if you're taking the typical geometry class back in high school, this might be draw lines, and look at triangles, and ask yourself with a ruler and compass what kind of things can you come up with. Now, what happens if the sheet of paper that you're working with is curved? Maybe you've tried to mat it out onto the surface of a bowling ball. Now, what if you tried to draw lines or tried to take a look at angles? Now, everything that you learned in high school is going to be completely different. Things are going to be very, very strange. That's one of the things that I like to work in. So trying your best to understand geometry, but now what if things are curved in a very bizarre way.

Edray Goins:
In the same sense, there's algebra. And in algebra, some of it starts with what we are used to in high school, so it could be the idea first of arithmetic, so adding numbers together, multiplying numbers together. But then eventually, we go from arithmetic to algebra. So now let's say trying to solve for X, looking at solutions to quadratic equations. Well, when you have algebra, there are different kinds of algebra. A lot of our undergraduate students have to take linear algebra. So this is a kind of algebra where now you're not just solving for X as a number, you might be solving for X as a matrix or as a vector. So now you're generalizing different types of algebra.

Edray Goins:
And we hope students will be interested enough to take the next sequence in algebra, which is abstract algebra. And so now you're completely abstracting all of these concepts. Well, this is again something that I like to work with. So now abstracting algebra to the point that things become really, really crazy. So if you're dealing with the integers, what's an abstraction of the integers? If you're dealing with the rational numbers, what's an abstraction of that?

Edray Goins:
And then there's a way to combine all of this together, which is really hard to explain, but this is the concept of algebraic geometry. So there should be a way that you're looking at geometry and looking at algebra almost one and the same. And I will admit that when I was a freshman, I saw these two words on a book when I was there at the Caltech bookstore. And I thought, "Well, this will be easy reading because I know algebra. I know geometry. So algebraic geometry can't be that bad." And I remember opening the book and I saw a lot of really pretty pictures, and I thought this was great. I tried to read it, made no sense, no sense at all.

Mark Wood:
And that was a challenge?

Edray Goins:
That was a challenge. I'm almost embarrassed to say this. I purchased a copy of that book freshmen year, and I said, "One day, I will understand this book." I didn't actually understand it until after I received tenure after being a professor for about seven years.

Mark Wood:
Yeah, I can imagine. Yeah. Looking at your resume, there aren't many words that jump out at me as a non-mathematician, but one of them did and that's origami.

Edray Goins:
Right.

Mark Wood:
And I was curious about whether there's something in that that could help laypeople like us understand a little better what you're dealing with.

Edray Goins:
Well, let's try. Unfortunately, mathematicians like to use the same words over and over again. So there are actually two types of origami that are out there. One is the one that most people are used to, which is you take a sheet of paper, you try your best to fold it like this Japanese paper folding idea. And there are a lot of mathematicians out there that are working on the concept of origami. One of the first things that comes to mind is a professor named Eric Demaine who is at MIT, and he's actually well known even in the physics world these days because you have questions such as how do you actually do origami in real life.

Edray Goins:
So, for example, I know that NASA is very interested in origami because what they like to do is a lot of their spacecraft needs to have solar panels in order to be powered. And, of course, to have a solar panel, you can't need a lot of surface area. Think about the roof on your house, the larger roof you have, the more solar panels and the more electricity you can get coming into your home. Well, as you might imagine, you don't want a spacecraft that has this huge solar panel as it's lifting off from the Earth.

Edray Goins:
So what you'd like for it to do is maybe be in a very small enclosed area, and then once it gets into outer space, or maybe once it goes to Mars, let say, then somehow it can unfurl and you'll have a nice big, big solar panel where it could take in a lot of energy, a lot of electricity, and then it'll work just fine. Well, one of the questions is, how do you get a solar panel that will have a really big surface area, but fit that into a really, really small space? That means you need paper folding, you need origami in order to do this.

Mark Wood:
Yeah.

Edray Goins:
This is a fascinating question some mathematicians are working on, this whole concept of paper folding. But now you have a very specific application to space travel. Unfortunately, what I work on isn't as lofty. It's a little bit more down to Earth. Here, I'm working on a different kind of origami that actually relates to various branches of mathematics. So yes, it is part of algebraic geometry, but maybe some of the listeners will know about this. It also relates in topology, differential geometry some group theory. There's a lot of different branches of math where all of this comes into play.

Edray Goins:
The origami that I'm thinking of, you can almost in a sense go like this. Start with, let's say a square sheet of paper. And now, I'm going to do the following: take the square sheet of paper, and let's say that you make a whole bunch of copies of it. So let's say maybe you make 10 copies. Well, once I have these 10 copies, I'm going to glue together various edges of these different 10 copies of that sheet of paper based on a certain rule. And the rule depends upon what I might feel like at the start of the day. So it could be for a very specific rule I'm going to say, "Alright, now try your best to glue these edges in this specific fashion. And once you do that, at the end, you're going to have some shape is what I'm calling an origami."

Mark Wood:
I see.

Edray Goins:
Now, one of the shapes that you could get could be this idea of what's called the Taurus, or you can think of it as a doughnut. Maybe I ought to start with one sheet of paper, and now I'm going to say, "Well, why don't you glue the top edge and the bottom edge together, and why don't you glue the left edge and the right edge together?" Well, if you glue the top and bottom edges, that'll form a cylinder. It'll form a tube. And then as you glue the left edge and the right edge together, you're gluing the circular ends of the tube together. And now you have a Taurus, you have one of these doughnuts.

Edray Goins:
Well, that would be one type of origami that now what I have is the sheet of paper I'm trying to glue. There are different types of origami that you'll get, and this is where the whole thing gets a little bit crazy. I'm not going to remember exactly, but there's one type of object that you get that I think is called ... Oh, I'm going to say I'm not going to remember the name here. Something like the three-headed platypus, I'm not going to remember exactly. It's some really bizarre name.

Edray Goins:
But the point is that this thing, you glue it together in such a bizarre way that you get this really weird looking object, but mathematically, you can study this object. I am very interested in these types of things because again, this is the geometry so this is where you have the shapes. But also, there's the algebra. It turns out that in gluing all of this together, there is a certain set of automorphisms. There's a certain set of symmetries that I have with this that gives me these very beautiful ideas.

Edray Goins:
So I actually plan to work with a group of students this summer where we're actually going to focus on these types of origami. So what I want them to do is let's say we're going to have sheets of paper, we're going to talk about these various gluing rules. We'll try to glue all of this together. I want to know if I can hold this in my hand, what will this look like? I don't know at all how this is going to go, but I've been thinking about maybe using 3D printers or 3D mathematic software, some kind of things where I can actually see what these things are going to look like. So it isn't so much origami in the sense of paper folding, it's more origami in the sense of now you're paper gluing.

Mark Wood:
That makes sense.

Patty Vest:
Can you tell us about some of the classes you're teaching this spring or some of the classes you've been teaching at Pomona?

Edray Goins:
Let's see. Mostly, I've been trying to teach linear algebra, so math 60. And that's actually been a really fun class. I definitely have made it a very complicated, difficult class, but what I've tried to do is emphasize that math isn't just about arithmetic. It's not about playing around with numbers, being able to do a computation. I'm totally convinced that Pomona students are brilliant enough that they can do computations. Once I tell you what to do, it's just a matter of sitting down with a calculator or computer, you can do it.

Edray Goins:
I like to use that class to talk about mathematical logic. So one of the first questions I typically ask in this class is, say that you make a claim of something. Let's say that you fundamentally believe that it's true. Go ahead and prove it to me. What does that mean? How do you convince someone of a certain mathematical concept? We spend a lot of time in that class going over a proof, going over a regular proof.

Edray Goins:
What I like about that class is that the students who have taken it with me rise up to the challenge. And we don't have math majors that are in this class. Some, I think are students that have placed out of calculus and so they want to try something new. Others, maybe the computer science majors. Some of them I know are economics majors. But what I love about this class is that students are very interested and really trying to say mathematically how do you prove something. How do you convince someone of a certain concept? So for me, linear algebra, math 60, is one of the most fun classes that I've had to teach.

Edray Goins:
Other classes that I've taught would be abstract algebra. I believe that's 171. I've taught a sequence of classes after that, so this would be Galois theory. And this semester, I'm teaching an even further sequence from there, algebraic geometry, the area that I work in. Next semester, I'm hoping to branch out a little bit and then teach what's called real analysis. So this is actually building on one of the classes that I'm teaching this semester, which is introduction to analysis, math 101, which is one of these speaking intensive classes. So the hope next semester is to build on that. I'm really hoping that some students that are in the speaking intensive class will continue along with that class.

Edray Goins:
Next semester, oh sorry, I should say at the spring a year from now, so the spring of 2021, I'm really looking forward to teaching differential equations. So I did teach it before at my previous job, but what I'm hoping to do now is really combine all of my knowledge of math and physics all together into one class. I'm hoping to go back to what inspired me to go into academia but now do it with the students here. So to really say, "Yes, you could talk about the numbers and how to do all of this mathematically, but what does this really mean for the world around you?

Edray Goins:
For example, a lot of what's happening nowadays in the age of COVID-19 comes down to looking at these mathematical models, so trying your best to figure out can you predict the spread of a disease. Can you predict how many people are going to eventually get infected? This all comes down to something that we would do in this differential equations class. So the question would be, how do you come up with these models? How do you actually know that some of these things are correct? Can you actually use this to even influence policy on when we should have shelter-in-home, when we should have safer at home? And it all comes down to things that mathematicians are doing.

Edray Goins:
I myself have actually been really talking with a lot of my friends who are in some of these modeling communities to see what types of mathematical equations are they using. What are some of the differential equations are they using? What I would love to do is use a class like this differential equations class to actually say, "Here's how you can do this stuff in real life in real-time." It all comes down to just these mathematical formulas, but these are things you can actually do in a class. You don't have to be some hardcore researcher sitting in a lab somewhere. You can be an undergraduate really taking this math class, but actually seeing what's happening with the numbers, even doing your best to predict what government should do.

Mark Wood:
I'm sorry to say we're out of time. On that note, we're going to wrap this up. We've been talking with Edray Goins, professor of mathematics. Thanks, Edray. This was fun.

Edray Goins:
Yeah, thank you. Thank you for having me.

Patty Vest:
Thank you. That was great. And to all who stuck with us this far thanks for listening to Sagecast, the podcast of Pomona College. Stay safe, and until next time.