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.

Machines for Math

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.

Knuth on P v NP, god(s)

I don’t remember exactly how I got there, but reading through a short profile on Don Knuth I was glad to learn that my worldview roughly coincides.

Taking the interview for this article as a mini-instance of “All Questions Answered,” this reporter asked about the question of P versus NP. “It’s probably true that P equals NP, but we will never know why,” Knuth answered. The question has two aspects, he explained. The first: Given a computational problem, does there exist a polynomial-time algorithm for its solution? And the second: Is that algorithm knowable—that is, can we actually write it down? “What I suspect is that there is some algorithm, it’s out there, but it’s so complicated that for practical purposes, it makes no difference because nobody will ever know what it is,” he said. A suggestive example comes from the Robertson-Seymour theorem, which says that for any minor-closed family of graphs, there exists a polynomial-time algorithm to recognize whether a given graph belongs to the family. But “almost never do we know what the algorithm is.”

But just because something is hard doesn’t mean we should give up!

Also worth noting that this appears in the middle of a piece which characterizes the early 21st century mainstream North American academic position on diversity pretty well. Knuth is being praised for his contributions to diversity for giving money that allowed MathSciNet to add authors’ names typeset in their native languages instead of just transliterating to English. I am for this, but the tenor hews a little too close to the willful ignorance of claiming mathematics is a diverse and inclusive field because it is international. How eager we were to “celebrate diversity.” Oops!

Also, if your read further, it talks about Knuth’s lectures on religion and coming out as a Christian. I think all I can say about that is that I appreciate the acknowledgement that “God” is really a just a catch-all for mystery. It makes more sense to me as a metaphor though.

Renaming the math building at T. U. of Louisiana

**The following letter was delivered (Sep. 2020) to the board and senior administrators of the university on behalf of a group of graduate students and faculty in the math department, and other concerned community members.**

We, the undersigned members of the Mathematics Department and Tulane University community, call upon the senior university administration, President Mike Fitts, and his appointed Naming Review Task Force to undertake the renaming of

(1) Gibson Hall,

(2) the online university platform called Gibson, and

(3) any other monuments to Randall Lee Gibson within the university’s purview.

We make this petition with the facts in mind that Randall Lee Gibson

(1) was a slaver and sugar plantation owner in Terrebonne Parish,1 

(2) adhered to hardline beliefs in racial inequality, authored and published pro-slavery essays, ran for political office as a secessionist before the Civil War to preserve slavery in an independent South,1

(3) enlisted as a Confederate soldier upon outbreak of the Civil War, eventually rising to the rank of Brigadier General,3

(4) helped to restore former Confederates to political power in the backlash to Reconstruction and benefited from violent white terror campaigns meant to suppress black voters,4 and

(5) convinced Paul Tulane to “confine his bequest [to the university] to white persons.”4

We support the efforts of the Naming Review Task Force and President Fitts’ message that “racism has no place” on our campus, and so we insist upon the removal of all monuments to individuals aligned with slavery, racial segregation, and other forms of oppression. It is unacceptable that Tulane University continues to honor the name of a person that profited by and fought to protect chattel slavery. 

Our purpose is not to deny history, but rather to recognize it and connect its meaning to our present so that we may move beyond the moral deficiencies of our forebears.  Gibson Hall was named in honor of Randall Lee Gibson’s role as the first president of the Administrators of the Tulane Educational Fund, in which he oversaw the transformation of the public University of Louisiana into the private, exclusively white Tulane University of Louisiana. This conversion was made with explicit racialized intent through Paul Tulane’s act of donation.2

The university is much different now than it was in Gibson’s time, but its entanglement with white supremacy remains. The removal of monuments to oppressors is essential to our university’s project to become a more inclusive and equitable institution. With this aim in mind, we assert the necessity of renaming Gibson Hall.



[1] Allardice, Bruce S., and Lawrence Lee Hewitt, eds. Kentuckians in Gray: Confederate Generals and Field Officers of the Bluegrass State. University Press of Kentucky, 2015.

[2] Dyer, John Percy. Tulane: The biography of a university, 1834-1965. Harper & Row, 1966.

[3] United States Congress (1893-1894). Memorial address, and 2d session, 52d Cong. Memorial Addresses On the Life And Character of Randall Lee Gibson, (a Senator From Louisiana,): Delivered In the Senate And House of Representatives, March 1, 1893, And April 21, 1894. Washington: Govt. print off., 1894. 

[4] Sharfstein, Daniel J. The invisible line: A secret history of race in America. Penguin, 2011.