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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.

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I came across this article today while looking for information on how Jensen and Williamson implemented an algorithm for computing the $p$-canonical basis. Not recognizing any of the co-authors I assumed it was a group of students until coming across this paragraph.

In this work we prove a new formula Kazhdan-Lusztig polynomials for symmetric groups. Our formula was discovered whilst trying to understand certain machine learning models trained to predict Kazhdan-Lusztig polynomials from Bruhat graphs. The new formula suggests an approach to the combinatorial invariance conjecture for symmetric groups.

Turns out the co-authors are Google I mean DeepMind people. This seemed kind of wild to me — to throw machine learning at something like computing KL polynomials, and I guess it is sort of radical. At least enough that they published another paper in Nature describing this and other efforts at using machine learning to formulate or solve conjectures. And U. Sydney is going on a PR push about it.

Reading the Nature article, it is clear that this isn’t really the doomsday scenario we worry about where machines can prove all the theorems and we mortals are useless to them. It sounds like there’s still a high degree of specialized knowledge and intuition that goes into training the supervised learning model. At least that’s my sense from this paragraph:

We took the conjecture as our initial hypothesis, and found that a supervised learning model was able to predict the KL polynomial from the Bruhat interval with reasonably high accuracy. By experimenting on the way in which we input the Bruhat interval to the network, it became apparent that some choices of graphs and features were particularly conducive to accurate predictions. In particular, we found that a subgraph inspired by prior work may be sufficient to calculate the KL polynomial, and this was supported by a much more accurate estimated function.

I know not much about machine learning, and the second sentence doesn’t give a clear sense of how the “experimenting on” inputting went, but I would be surprised if the machine was able to actually pull out hypercube decompositions and diamond completeness as the features it needed to really predict the KL polynomials. That is, it sounds like ML could be a useful interactive tool for discovering and verifying ideas, like interactive theorem proving seeks to be, but they’re still not doing the math for us. Not to mention the proof of the main formula of the paper involves some layers of categorification and pretty heavy geometry.

Still, all in all, it looks like we’re gonna all have to learn how machines learn sooner or later.

My poor imagination can only think of making a long camel trip across the desert and getting close to a destination, but you can really feel the bobbing in these songs. This version appears to be recorded from a cassette which somehow makes it sound even better. Maybe its an old jeep trip instead.