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With several nice collaborators, we finished the first stage of a project that came out of last summer’s AMS MRC on Algebraic Combinatorics. Building off of our group leader Rosa Orellana‘s experience in the area, we studied the chromatic symmetric function of unicyclic graphs. We found several nice formulas including some that generalize recent work of her and her students on trees, as well as some interesting phenomena pertaining only to graphs with cycles. Like most good projects, we came out with a lot more questions than those we were able to answer, and some of us will likely continue working in this area.

One of the fun things about this project was learning about various graph algorithms and methods of computing the CSF. I will post some of the Jupyter notebooks with code we found helpful under the Code area soon. Also keep an eye out for forthcoming work on principal specialization of CSFs with another cohort from the MRC!

I’m also excited that my article with Fernando Díaz on shellability and posets of rook placements (sects) has been accepted for publication in the Electronic Journal of Combinatorics. It should appear at some point this summer. Finally, an article on starting and running a math circle at a county jail will appear in another fine publication (Journal of Humanistic Mathematics), probably this July.

Phew! It’s been a while since any updates happened here so some work has gone out in the meantime.

First, a few mastery/standards-based testing (MBT) pieces got printed in the past year or so. I started working on designing and implementing these grading systems with Becky Swanson when I started at Mines in 2022, and our first write-ups about our experiences went out in Fall 2023. We created an extensive web resource with the help of Carter Moulton at the Trefny Center detailing our decision-making process and giving a preview of the results. Later that fall, we wrote a piece for IMAGE, the bulletin of the International Linear Algebra Society, to encourage other Linear Algebra instructors to try out MBT. Last summer, a results article was accepted for publication and recently appeared (with a companion blog post)in Teaching & Learning Inquiry, a journal of the International Society for the Scholarship of Teaching and Learning (ISSOTL). A final practitioner article on instructor collaboration with campus teaching and learning centers is underway!

Meanwhile, Becky and I also wrote a blog post for the MAA Math Values blog on teaching postdoc positions and our collaborations. This then turned into an article in the January/February 2025 issue of MAA Focus magazine.

Finally, in January we finished a pre-print on Kohnert polynomials from a project started at GRWC in Milwaukee last summer. It is currently in submission, and hopefully will be published soon! See more on my Research page.

Another piece on math circles at the county jail has been accepted for publication, and will appear soon. Also, we are working on finishing one project on chromatic symmetric functions of unicyclic graphs very soon! I will try to post when these things get put out.

In preparation for a piece of writing I did about starting a math circle in county jails in Boulder and New Orleans, I collected the activities that we’ve developed (based on many other math circle resources available online and in print) to share broadly via Google Drive. The piece got long and may turn into a journal article rather than the original intention of blog post, but when it is available I will link to it here. It describes the issues, obstacles, and joys of engaging in this kind of outreach which I’ve been organizing on a weekly basis since April 2023. Look out for more coming soon!

My friend and academic brother Néstor Díaz and I started talking about working on a project together probably a bit over a year ago. I pitched this idea on proving shellability of Bruhat order for some of the symmetric varieties I was familiar with, figuring that it would be a fun way to learn more about poset topology, and that it wouldn´t be terribly difficult to get some results building off of past work of Incitti, Can, Cherniavsky, Twelbeck, Wyser and others.

The results we set out for turned out to have some fight in them! We thought we had something several times and then found problems and counterexamples in the course of writing things more carefully. Eventually, we shifted gears slightly to working with the sects of the (p,q)-clans. The idea here was that (1) the sects are slightly simpler and smaller than arbitrary intervals of clans, as they group “like” clans together, and (2) we know a nice bijection of sects and collections of rook placements. This problem was also not without challenges and subtleties, but after working out a way to associate a partial permutation to a clan in a way so that covering moves could be coherently labelled, the argument came together.

We had pretty much worked this out by the time we met up at the Schubert Summer School at UIUC in June, but for various reasons it took us a while to write up the details. We’re excited that we were finally able to post a pre-print to arXiv.org last week just in time for the holidays. Enjoy!

Recently I gave a personal record of three (3!) talks in one day in two different languages. I was honored to be invited to speak in the online colloquium of the Universidad de El Salvador, but the date available happened to be on the same day that I had already agreed to speak in the Rocky Mountain Algebraic Combinatorics Seminar (RMACS) at Colorado State in nearby Fort Collins. The topics were similar so fortunately it wasn’t too too much work to prepare, but on top of two Calculus classes in the morning, seldom have I spoken so much in one day.

I enjoyed the format of RMACS which gave the opportunity for an introductory talk before a regular research talk to try create an open environment for graduate students and folks from other fields. It reminded me of a concept I’m still eager to try to make happen in math of talks-as-dialogues. That is, I think it would be interesting to have math talks operate as conversations between two people where one is trying to explain something to the other and the “explainee” can interrupt, question, elaborate or make connections as much as they like.

I think the most enjoyable math talks I’ve attended have more-or-less functioned in this way already, as conversations between the attendees and the presenter. This is often to the expositor’s credit, having selected and introduced their topic (or themselves) in a way that invites this sort of interaction. But building a conversational flow into the structure of our talks I think would frequently force the rendering of ideas in ways that are, if not more clear, at least more diverse and therefore accessible to an audience.

Good discussion is part of good exposition. One can observe this in the way radio programs, podcasts or videos media that interview experts (like, say, Numberphile) are structured. Radiolab also comes to mind as a program exploits of this technique. An expert isn’t just invited on the show to explain their theory and known results for an hour. There are constant interruptions for clarifications, “what ifs”, auditory “illustrations,” philosophical tangents, and expressions of wonder and amusement. You could also see parallels in the idea of masterclasses from music performance. Though the tone and politic is more authoritarian than what I am imagining, a masterclass (in which musician performs, receives coaching from a “master,” and responds and adjusts before an audience) appeals to our interest in the dialogic element of engaging with art.

For a math talk, this would put some pressure on the “explainee,” but it would also alleviate the pressure on other audience members to feel guilty for interrupting the speaker or asking questions at the wrong time etc. Interaction from the rest of the audience would hopefully encouraged in this set-up, and we could democratize research talks in the ways we are beginning to do with our classrooms. The audience would be freer to dictate what it is they want to get out of the talk. Multiple perspectives could be heard. Knowledge would be shared, and interpreted.

I’m not saying all traditional math talks should go away. There’s always a time and place for a well-delivered lecture. But I think every mathematician’s experience with research talks is uneven enough for us to suspect that maybe there should be other ways for us to communicate our research.