On setting boundaries and managing the Manitoba eXperimental Mathematics Laboratory (MXML): Meet Meredith Sargent, PIMS PDF at UManitoba.

Ruth A. Situma, Program and Communications Manager.

It’s clear from Meredith’s responses that she is waiting for the Pandemic to be over. She was at the University of Arkansas (UArk) when COVID hit. Once it became clear that all programming would move online, she moved in with her parents in Montana to help with the isolation, as well as get some more space. “I really dislike working from home, so it was really nice to be able to have a room I could use as an “office” that wasn’t the same space as the rest of my life,” she says. Many of us share the same sentiment no doubt!

Over the summer, after accepting her position at the University of Manitoba, Meredith packed her bags and moved up to Winnipeg in her dad’s little white pickup truck (UHaul wasn’t allowing cross border moves at the time- she says). But she had to endure much the same inconvenience when she got to Canada. “It was definitely hard adjusting back to working in my one bedroom apartment. My main method of work/life balance is to have separate spaces for math and for my life, so sitting in the same couch dent for 14 hours a day was less than ideal”. Since the Province of Manitoba eased restrictions, Meredith is glad that she can venture out to the office and eat lunch with friends and talk about math, teaching, and whatever else she pleases. She notes “I’ve been so much more productive this semester now that I have boundaries again”. I connected with Meredith at the end of a very busy week, and term. Our communication has been edited for brevity.

Tell us about your academic journey: What field are you in and how did you get to work with your current PDF supervisor, Prof. Robert Martin?

I’ve taken a long way around to get to my current focus of research: it’s meant a lot of starting from scratch! My thesis was about Dirichlet series (considering them from a functional analysis perspective), which is somewhat popular in Scandinavia, but there’s not a huge amount of interest here in North America. I have some vague plans to work with them again, but once I finished my thesis, I felt like I needed to branch out. That’s actually right around the time I saw Robert give a talk at the Joint Math Meetings. I think the non-commutative analysis is very cool, and I really enjoyed the way he was presenting it, but I ended up not following up on that yet.

I worked on a couple of different projects with friends of mine, and I’ve gotten somewhat into the question of optimal polynomial approximants. The idea is that you want to find an approximation for 1/(a nice function) that is close in the norm of your space. This is connected to the idea of cyclicity of the function with respect to the shift operator that I’m discussing in my talk.

These are the zero sets for a counterexample Meredith and her co-author Alan Sola found. The goal was to have the function have no zeros on the closed bidisk, but have an optimal approximant with zeros on the interior of the bidisk. Here, we look at the “facial zero sets” of the function; notice that the curves are both above the line at 1, so the zeros are all outside the bidisk.

Last year, some collaborators and I started looking at these optimal approximants in the non-commutative setting. Learning all the NC theory has been a huge hurdle, but it’s been nice too since I’m also looking at that with Robert, but from a different direction. It’s got a really different flavor than the single variable case or the commutative multivariable cases that I’ve worked with; some things actually work better in the NC realm than in the commutative several variable setting!

You are involved in the UManitoba MXML program. Tell us about it in a few short sentences.

Part of my job at UM is to run the Manitoba eXperimental Mathematics Laboratory based on the University of Washington program with a similar name. The idea of this is to give students who maybe wouldn’t otherwise consider doing math a chance to work in groups on undergraduate research projects. UM already has students who do undergraduate research (over the summer for example), but these are usually students who are already planning on graduate school for math. For me, the goal of this project is to expose students to material outside of the standard curriculum and to work on problems that don’t have an answer, even if the students aren’t necessarily math majors or planning on graduate school.

This is also a good way to give graduate students and postdocs experience mentoring students in a different context than teaching a class. (For example, Josh Males is running a project this semester!) We have graduate student mentors who meet with the undergraduates and provide another point of contact while they all learn together.

“This is work used as an example in a recent paper. We were considering monomials arranged in a lattice and asking which ones had non-zero inner product with the one circled in teal. The undergrad researchers I’m working with just did this for a similar example!”-Meredith.

I’m leading a group this semester that’s looking at a particular multivariable example of optimal approximants. I think these make for good undergraduate research projects because, for analysis, they actually don’t require a TON of mathematics background. A student who knows about Taylor series and orthogonal projection/Gram-Schmidt is equipped to learn some of the basic ideas about the Hardy space and finding these “closest points.” I’ve loved working with the graduate mentor and the undergrad researchers this semester, and I’m looking forward to working on another project in the fall (hopefully being able to meet in person!)

Aside from the MXML program Were you teaching this semester? Do you think your time at the University of Arkansas prepared you for this?

I’ve been teaching a multivariable calculus class this semester, which I think is a fun class. I taught a similar class when I was at UArk, but that was in person and I had various activities and projects that involved the students standing around in the classroom, and that doesn’t translate well to zoom. Instead, I’ve been really leaning into using the GeoGebra 3d graphing calculator. It’s nice because it’s easy for students to use for basic graphing, but I’m also able to make some neat demonstrations and move surfaces around to view from different directions. It’s been a good class; I get a lot of chatter in the zoom chat, which I appreciate, but I’m very much looking forward to teaching in person again hopefully soon.

An image that Meredith made in Geogebra’s 3d graphing calculator — which she uses a lot in her teaching, especially over zoom. It shows an orthogonal projection onto a subspace. “Basically what we end up doing in the optimal approximants project, except there the vectors are functions”, she notes.

What do you do to balance your research and life, and what does a typical Sunday look like for you?

I try to keep my math and teaching work at work (now that I can go to the office on a regular basis). Having different spaces for things helps me stay focused on what I want/need to be doing! When I’m at home (and on the bus!) I like knitting and listening to podcasts and audiobooks. I have a sweater in progress currently, and will probably finish it just in time for summer… I like dyeing yarn and knitting it up into socks as well: I have a collection of colorful handknit socks that I appreciated during the very cold winter.

Pairs of hand-knit socks that Meredith made. “I didn’t dye all of the yarn, but some of these are my own dye-work”!-Meredith Sargent.

I also really like vegetable gardening. I had a very lush balcony garden when I was living down south, and when I was staying with my folks, I helped my dad build a raised bed for his garden, as well as planting the tomatoes, beans, corn, squash, peppers, etc etc etc. It was also nice because the garden has a really tall fence to keep the deer out, so I took my cats out to have some nice outdoor time among all the plants. I’d pull weeds and they would chase bugs. I missed the growing season when I moved up here, and my apartment isn’t quite big enough to have an indoor tomato plant, but I’m thinking of getting a plot in a community garden or something.

Meredith’s cat, Beans, being frightened by a dog, who was on the other side of not one, but two fences. Image credit Meredith Sargent.

My Sundays are usually not very exciting: I like to sleep in and take things easy and then do some of the basic life maintenance that gets neglected during the week. I’ll sometimes play some video games or work on my knitting, but mostly I try to take the day to take care of my body and my habitat.

What is your best discovery since arriving at the University of Manitoba ?

I know I should probably say something mathy or exciting, but it’s got to be Honey Dill sauce. Apparently, this is a classic Winnipeg condiment and it is delightful.

Meredith Sargent received her PhD from Washington University in St. Louis in 2018. Her thesis work was about Dirichlet series (generalizations of the Riemann zeta function), their integrals on vertical lines, and how these connect to integrals on the infinite polytorus. She then worked as a Visiting Assistant Professor at the University of Arkansas from 2018–2021 where she focused on optimal polynomial approximants in several variable contexts. Since Fall 2021, she has been a PIMS postdoctoral fellow at the University of Manitoba working with Robert T. W. Martin on problems about analytic functions of non commuting variables. She also organizes the Manitoba eXperimental Mathematics Laboratory, modeled on the program at the University of Washington for undergraduate research and is supervising students this semester.

Meredith will be speaking at the PIMS Emergent Research Seminar Series, on April 27, 2022, at 9:30 AM Pacific. Details on her talk, Shift operators and their adjoints in several contexts can be found here