In Episode 71, Confluence talks with Bertha Morton Award winner and native Montanan Anastasia Halfpap, a Ph.D. student in the Mathematical Sciences program at 老虎机攻略. During our conversation, Halfpap discusses finding a family in the mathematics department, her journey from math prodigy to flourishing Ph.D. student, and the value of the simply beauty of math.
STORY TRANSCRIPT
Ashby Kinch
This is Confluence, where great ideas flow together. The podcast of the Graduate School of the 老虎机攻略. I'm Ashby Kinch, Dean of Graduate School on Confluence. We travel down the tributaries of wisdom and beauty that enriched the soil of knowledge on our beautiful mountain campus.
Corey Palmer
Anna's clever, she's hilarious, she's great to talk to. There'll be this 1 hour preamble before we get to the work about what's going on in life and not just me asking us, trading stories and telling the silly things, whatever bad horror movie she watched over the weekend and things like that.
AK
You just heard the voice of Corey Palmer, professor of Mathematics, talking about his student, Anastasia Halfpap, one of the Bertha Morton Graduate Student scholarship winners for 2022- 23.
This episode is part of a series recognizing the achievements of some of our outstanding graduate students. The Bertha Morton Award, named for a great Montana who dedicated her life to public service, was endowed to support graduate students by recognizing the distinctive contributions they make in research, creative activity, and public service. As a native Montana who was raised in Missoula, Anastasia is a fitting recipient. Of the birth of Morton legacy. She has several coauthored, publications and more in the works, as she wraps up her PhD and looks to a future as a math professor. In this episode, she tells the story of her journey into math research, from. Her high school experience as a math. Prodigy to a flourishing intellectual collaboration with her advisor, Cory Palmer. We're proud to share her graduate story with listeners. Enjoy the float.
Anastasia Halfpap
Someday we'll find it the Rainbow Connection the lovers, the dreamers and me.
AK
Welcome to Confluence, Anastasia.
Ana
Hey, thanks for having me.
AK
Why are there so many mathematical equations about Rainbows? Do you like The Rainbow Connection or do you consider it cheesy?
Ana
I gave an undergraduate talk called Someday We'll Find at the Rainbow Connection about that Rainbow work.
AK
Did you perform as a singer when you were younger?
Ana
Yeah, I actually been taking classical voice lessons for eleven years now.
AK
Oh, wow. And you're still doing that?
Ana
Yes
AK
Oh, fun. And do you sing with the university?
Ana
Chorus no, I'm not like a professional performing level, but it's very related to math in some sort of way. There's lots of mathematicians who are also into music, and I think it tickles the same part of my brain that math does.
AK
Have you kind of dug into that, like the part of the brain that it tickles. That's common.
Ana
I'm not sure. There are a lot of connections between one of the professors in the department actually teaches a course for non majors about math and music and connections there. And I know that there are certain musical forms that are very structured in a similar way. So there's one professor, I think, at Harvard who can just compose fugues on the spot, like you give him two bars of a melody, and he composes a feud based off of it. And he says, well, when you have this mathematical understanding of the form, not necessarily that's great art every time, but you can just on the fly do all the processing to make that up.
AK
Yeah. Create the formula, small variation moving forward, and then the art would be in the repetition, and it would be in finding through that repetition some kind of underlying structure. And that's, I guess, very similar in terms of the research process itself. It's not just in other words that it's the part of the brain that processes, but it's kind of maybe common to mathematical research.
Ana
Yeah, I would say mathematical research is much more creative than many non mathematicians might think. It's not just, I'm going to use a bunch of formulas that I already know. It's not just, I'm going to apply a theorem that somebody else has proved. You have to do a lot of lateral thinking. You have to try different things. A lot of combinations that you try, it might be true, but it doesn't get you to the results you want. It's not necessarily useful. And so I definitely think it's a hugely creative process to try those different ideas, those different combinations, and find something that makes an argument.
AK
Yeah, I guess that would be the case with almost any discipline that at its highest levels, creativity is the thing that distinguishes people from being able to just do the root level thing from the people who are making innovation or breaking through. But with math, the root level thing is hard for a lot of people, right? In other words, it's hard. Only a certain number of people seem to be able to kind of really get those fundamental mental procedures, practices, processes, fully mastered, so they can kind of move up.
Ana
Yeah. And so I think that's maybe part of the reason why there are so many misconceptions about math is because it's not like reading, where we all learn to read when we were in grade school. And then we understand that writing a book requires more than just being able to recognize words and make grammatical sentences. With math, there's such a barrier to entry that most people, they don't ever get to see. Like the creative stuff.
AK
Yeah. And you leaped over that barrier at a very young age. You've been kind of in the math for a long time. You've been working, in fact, with your adviser Corey Palmer, since you were quite young. Tell us that story. How did that unfold and develop?
Ana
So when I was 14 and a sophomore in high school, my family was on sabbatical. Both of my parents at the time were professors here at the 老虎机攻略. And when we got back, I had been in a high school that had a lot of Advanced Placement courses. I had already sort of finished all of the math that I would have been able to take had I just gone back to a public high school. And so they decided they would try homeschooling, and I ended up in the department. And sort of on the first day I was taking a linear algebra course, I said, oh, this is going to take a little bit of time to get challenging. Why don't I go and sit in on a high level course and just see how it goes? And that's not the way that I would recommend people get into math, but I was just so taken.
AK
You don't dive into the deep end immediately, right? Typically.
Ana
Not typically, but I was just so taken. And my mom, who is a mathematician, said, okay, this is going to be a lot of hard work. You don't have the prerequisites. You're going to need to learn to write proofs, which normally you would do in a different class. You're going to have to struggle a lot more than other people to read the book and understand things. But I was just really motivated, sort of from the first day, I was looking at this graph theory. This is beautiful. This is how my mind works. I want to plug in. And so that first class wasn't actually with my advisor, but then the next semester, he was offering a special topics class, and it was on a similar subject, and I decided to register, and that's how we met.
AK
Wow, that's incredible. So I'm going to stick with that. We'll come back to Cory in a second, but I'm going to stick with that. So you're a sophomore in high school taking a college level math course, but then separately, you're homeschooling on everything else on the other subjects.
Ana
Yes.
AK
In other words, how does that work itself out for you socially, like, you're interacting with college students at age 15, 16?
Ana
It was tough. I would say it wasn't until I actually got to grad school that I started feeling like I could interact with my peers as a peer, as a friend, because, man, all the people in those classes, they were mostly graduate students. They were, like, 24, 25. And my mom tried to set me up on a date with one of them, and he was like, I'm going to get arrested if I take her to a movie.
AK
Yeah.
Ana
I would say the downside was that it was a lonely few years. I was hanging out with my parents and my cats a lot.
AK
But the path you've taken is interesting. I mean, a lot of times you'll do an undergraduate degree, which you did here at as well, and then you'll be pushed to go somewhere else for grad school, but you decided to stay. Why is that?
Ana
Yeah, so actually, a lot of people were encouraging me to go somewhere else, maybe somewhere with a bigger program, more competitive. I went somewhere for about three weeks, and then it just so happened that there was actually some without getting into sad details, some personal tragedy, and basically I was not even in the country. I'd gone up to Canada for the program and I basically thought, I can't do two years, four years, whatever. It would end up being in a foreign country. And I came back home for personal reasons. And at the time, that was such a rough experience. I didn't even know if I was going to end up doing grad school. And it was because all the people in the department here are so nice and so like family. Wonderful. It is like family.
AK
You've been with them and you know them.
Ana
Yeah. I came back on sort of the second day I was back in town and I went to see Corey and I said, look, this bad stuff happened. I don't know what I'm going to do. And he said, hey, you don't have to make a decision now, you don't have to register, but if you want to think about math, you can come in and we can talk this semester. And that's what convinced me to stay here. And it feels perverse to say that I'm glad it happened, because obviously at the time, it was deeply traumatic what happened. I'm really glad.
AK
Yes, you found a solution that works for you and your overall trajectory. And of course, your wings have not been clipped at all. You've been publishing and publishing with Corey, which is really interesting, and we love that in the graduate school. We love to highlight kind of how these mentor advisor relationships work out over time. And in this case, the two of you have now co authored two publications. You have a few more kind of in publication or in process, but tell us a little bit about how that evolved, that intellectual collaboration at the level of actually publishing a paper.
Ana
Yeah, so when we first started talking about research together, of course, as a 15 year old, I wasn't fully aware of how much work you put in before you can publish a paper. But even then, when I was just woefully naive, he always really tried to put me at ease and take my ideas seriously. And so I think that that set the stage for what I consider really a collaboration between equals. Obviously, he knows more than me. He has a lot more experience. But through the years, as I took more classes, as we talked about more problems together, and I sort of built up that foundation where we can solve open problems now. I always really felt that the most important part of our collaboration was that he made me feel comfortable talking to him. He made me feel comfortable being wrong. He made me feel like I can present ideas, which, when you do research, a lot of your ideas are going to be wrong before you find the right one. And I would not be looked down upon for being wrong. And he made me feel like I can tell him, hey, I don't think that this works.
AK
Yeah, I love the way you're talking about this as Graduate Dean. It makes my heart sing to hear you say, “I think of myself as an equal.” I mean, that is the ideal relationship that our graduate students have the confidence to grow and engage with their professors. And for sure he feels the same way. In other words, I'm sure he on his side loves to hear you say that. Part of what his work has been is to help you gain confidence. You get it from some other places, I'm sure, but he's a key part of that. Well, I don't mind saying I'm completely ignorant. How hard would it be for you to explain Rainbow Cycles versus, this is one of your publications, Rainbow Pass. Is it futile to even give it a crack?
AK
I could give it a try. We could see how it goes. Okay, so the basic object that my research is about is something called graphs, which is not the same as the graph of a function that you might have seen in algebra. It's a structure that basically you can visualize as a bunch of points put down on a piece of paper, and some of them are connected by lines. And my work is looking at very broadly, how extreme of a graph can I make while avoiding a certain property? A simple example might be how many lines can I put down between these points before three of them form a triangle? So, Rainbow Cycles versus Rainbow Paths. The setup is you put down your points and lines, and you want to figure out, how many lines can I put down and then color them.
AK
That's the rainbow connection.
Ana
That's the rainbow connection. So that I want to have lots of Rainbow Cycles. So if you trace edges around, you want to have like a pentagon, say. But I don't want to have a rainbow path of equal length. So if you trace edges and they don't connect into a cycle, they just sort of lie there in a line. I don't want five edges in a line that are all different colors.
AK
Got you.
Ana
And so we're skipping a few things. There are some coloring rules because if you color all the edges the same color, obviously, well, you don't have any Rainbow Cycles, but you definitely don't have rainbow paths. But that's sort of the base set up. What I would say to a nonmathematician, just think about a problem like that. How many edges can you have without getting these structures?
AK
In a confined space? Yeah, I think I understand that roughly right. But then, of course, the mathematics is the challenging part. Why do you solve a problem like that? What does it do in the world? Or does it not do anything? Is it just say beautiful?
Ana
There are kind of the two sides right there's. The people who want math to be applicable and the people who think math is an art. I am on the peer side where I do math primarily because I think it's beautiful. It tickles my brain in a good way. That isn't to say that there are no applications of the stuff I do. So, the field of graph theory broadly is useful in a lot of different settings. It has lots of algorithmic and computer science applications. If you ever use Google Maps or another GPS app that finds the shortest route between two points, that's a basic graph theory algorithm that then people work to sup’ up to make it efficient and really good. But I would say that the stuff I do, I can't give you an application right off the bat. I would say that many things that we now use math, for instance, encryption, the math that we use, it wasn't developed to do that. It was developed because somebody said, hey, this is a cool problem, and I want to think about it. And then later on, we have this big toolbox of things that people say, hey, this was a cool problem. I wanted to think about it. And someone comes along and they have all these tools.
AK
Yeah. So that's so great to talk about, because, again, pulling up to the broad level that across the board in higher ed, we're under a lot of pressure to constantly justify what we do. But the reality is that the history of higher education, the history of ideas, is a lot of play, is a lot of creativity, is a lot of exploration, and a lot of dead ends, quote unquote, from the standpoint of practical applied. But that the great innovations come out of that our Grad school mission, is actually innovation, imagination, and intellect. And we really believe in all three. Right. That the imagination part of it. The creativity that our best graduate students are exploring, whatever their field, whether it's actually a creative output field or math, that's a real root of talent in our society, that we want to make sure we keep cultivating that curiosity. So, I love to hear that story. Well, you had great success too. We're here because you're a Berta Morton Award winner. Does that have any particular residence for you, that award and its history?
Ana
Yeah, when I read the little synopsis of the award, and it talks about Bertha Morton's life and how she wants to reward people at this school who are doing excellent research at Harvard or something, I decided to stay here. And I think Montana is an excellent school, but the math department is smaller. It's not necessarily on the map. And I sort of had a little bit of an uphill battle to convince some people in the community that, hey, you can do excellent research here. The fact that we're a smaller school doesn't mean I'm getting a worse education. It doesn't mean my advisor isn't the best advisor I could possibly have. And so, of course, I feel a lot of personal validation, but I also feel like, hey, no, this is a good award. This is recognizing that there are people here who are putting out really good research into the world.
AK
Yeah. And as a Montana, lifelong Montana raised here in this community, it must feel validating on that level, too.
Ana
Absolutely. I am absolutely in love with this school. I'm sad that I'm going to graduate because I don't want to leave.
AK
What do you think is next to you when you graduate and decide to leave?
Ana
Well, I'm hoping to get a postdoc, so probably three-year position where I would get to just focus on research. My ultimate goal is to be a professor, either at a four-year school or maybe somewhere sort of like here. I want to be somewhere that I think I can make a difference in students' lives. I definitely want to still pursue my research, but I think one of the things that I've learned from my own journey is that having professors, especially in a smaller program where you can give individualized attention, having professors who are working hard, putting out good research, but have the time to mentor students is really crucial for growing the discipline and making the next generation. I mean, I don't think I'd be sitting here if I didn't have all these great people back in the department here who were helping me along.
AK
That's wonderful to hear. Love to hear it. Well, thank you so much for joining us on Confluence, Anastasia.
Ana
Thanks so much for having me. It was a pleasure.
AK
If you like what you've heard, you've got Kate Lloyd to thank. She's a student in our MFA program in Media Arts. Her deft ear and keen editing touch have created the sonic landscape through which you're floating. We'd like to thank 老虎机攻略's College of Arts and Media for providing studio space and talent to support this production. Confluence is brought to you by the Graduate School of the 老虎机攻略. Innovation. Imagination. Intellect to serve the state, the region, and the world. You can subscribe to our podcast on Apple, Stitcher, Spotify, and Google by searching Confluence 老虎机攻略, or click a link at the Confluence website, www.umt.edu/grad. On the Telling Our Story tab, you'll find podcasts, videos, and other media that help us share with the world the groundbreaking research and creativity happening at the 老虎机攻略. Enjoy the float.