Benjamin Grant: Topologizing infinite quivers and their mutations https://arxiv.org/abs/2604.16660 https://arxiv.org/pdf/2604.16660 https://arxiv.org/html/2604.16660

Never mind we’re good. Theorem almost fully proved, this was way trickier than I initially thought it would be. Ended up having to prove a bunch of stuff about infinite sequences of natural numbers and a (to my knowledge) new result about certain c-vectors in certain cluster algebras

I think I’ve already teared up maybe 3 or 4 times just this morning about the Carroll crater. What a special and unique way to memorialize a loved one

Absolutely heartbreaking. So many mathematicians I know studied there, men AND women.

Maryam Mirzakhani studied there.

If this isn't enough to cancel/relocate from the US the International Congress of Mathematicians #ICM2026, nothing will be.

There is a universe in which it doesn’t, and unfortunately it is now more likely to be the one we live in

Alberto Marcone, Andrea Volpi: Reverse Mathematics and Dimension of Posets https://arxiv.org/abs/2603.18759 https://arxiv.org/pdf/2603.18759 https://arxiv.org/html/2603.18759

Ultimately I am rapidly approaching the point where I have to put a hard cap on the content of the paper for myself and save everything else for a follow-up project. Though there is one little nugget I want to add before I’m done with this one, and it wouldn’t really fit well in a follow-up…

I will yield that one of the reasons this project has gotten larger than I initially expected is that I just keep thinking of other things to try to work out about it that I find interesting. But also the work is kind of just a proof of concept; I’ve been building on it in order to Prove the Concept

There is a nonzero chance this ends up just being my thesis, lol. I would definitely be happy with that. My committee may have a different point of view; there is still time to show them The Way of Infinite Quivers

Magically, after some edits and some much-needed elaborating, 40 pages became 50. I am nowhere near close to making all the edits and changes I need to, and I still have a pesky theorem I need to finish proving (it will go through, there is no universe in which it doesn’t). See you at 60 pages?

Definitely a rough, rough, rough draft. But I am giving a presentation about the work on Tuesday morning with a group of faculty who have agreed to give feedback about the content and form of the paper; I am excited. This has been quite a fun project and I am happy to move on to the next stage!

I hit the 40 page mark for the solo paper I’ve been working on, which feels very nice. I chunked it up into sticky-note-sized tasks and have mostly been blindly drawing from a pile of those to dictate what part of the paper to write next. Only one sticky note to go (though it’s a longer proof)!

Also, there is a very nice connection between mutations and representation theory. If one understands the representations of a quiver, then one can (typically) understand the representations of the quiver’s mutations as well (cf. “cluster categories”). The simplest case of this is when Q is acyclic

3. Delete all directed 2-cycles that may have been created.

Quiver mutations are the main “dynamical” ingredient in the definition of a cluster algebra (though one needs to extend the definition of mutations to some algebraic data to fully realize this). There’s a lot still open about them!

Mutations are certain local transformations defined for quivers with no loops or directed 2-cycles. Given such a quiver Q and a vertex v of Q, one defines a new quiver mu_v(Q) by the following steps:
1. For every pair of arrows i->v->j through v, add a new arrow i->j.
2. Reverse all arrows at v.

categories). Here are some good slides I found on the topic: www.math.uni-bielefeld.de/birep/meetin...

I can’t speak for everyone of course, but I prefer thinking about quivers as raw combinatorial objects instead of categories because it is easier (for me) to think about their mutations this way

Some people do consider path algebras and their representations this way. This is what I’ve heard referred to as the “functorial approach.” You can do interesting things with this perspective that are not as obvious to try without it (e.g. iteratively taking representation categories of module

Vishal Bhatoy, Colin Ingalls
Discrete Invariants of Koszul Artin-Schelter Regular Algebras of Dimension four
https://arxiv.org/abs/2602.13178

'measles outbreak at the child prison' seems entirely avoidable, it's really the kind of thing that only happens if you do several unthinkably evil things on purpose all at once

Breaking: ICE is now reportedly following white people on grocery runs in Minnesota, suspecting they're delivering food to neighbors too afraid to leave their homes.

Neighborhood instructions:

Don't put the address in your phone.

Don't use GPS.

Write it on paper.

And if you get stopped, eat it.

Benjamin Grant, Zhongyang Li: Self-avoiding walk, connective constant, cubic graph, Fisher transformation, quasi-transitive graph https://arxiv.org/abs/2601.12571 https://arxiv.org/pdf/2601.12571 https://arxiv.org/html/2601.12571

If you’re wondering why your friends in academia are a little on edge right now, it’s because an eighteen-year-old who hasn’t done the reading, doesn’t look at the assignment, and has does no critical thinking skills more complex than “because I think it’s in the Bible” can literally end your career

Jonah Berggren, Khrystyna Serhiyenko: Classical tilting and $\tau$-tilting theory via duplicated algebras https://arxiv.org/abs/2512.13893 https://arxiv.org/pdf/2512.13893 https://arxiv.org/html/2512.13893

For the longest time, I tried to convince myself that I didn’t find combinatorics all that interesting, and that if I found anything interesting about it, it was probably just because it was similar to something I liked about abstract algebra. Turns out it was the other way around this whole time

Scott Carter, Benjamin Cooper, Mikhail Khovanov, Vyacheslav Krushkal: An Extension of Khovanov Homology to Immersed Surface Cobordisms https://arxiv.org/abs/2510.14760 https://arxiv.org/pdf/2510.14760 https://arxiv.org/html/2510.14760

let alone one of the form we are interested in, x_n=f^n(x). So f cannot exist.

Thanks for reading!

/end/

contradiction with the assumption that (x_n) is an integer sequence, since L=1+1/k requires our (integer!!) differences between consecutive terms to eventually be arbitrarily close to 1+1/k, which no integer is. So there doesn’t even exist an integer sequence (x_n) with x_{n+k}-x_n=k+1 for all n,

We can rewrite the LHS as a telescoping sum with k terms:

x_{n+k}-x_{n+k-1}+…+x_{n+1}-x_n=k+1.

Taking the limit on both sides as n—>infty (and letting L be the same limit as in the previous problem), we see that kL=k+1.

But then L=1+1/k, which is not an integer (since k is at least 2). This is a

use this technique for: suppose we are told to show that for any fixed natural number k at least 2, there does not exist a function f:Z—>Z such that f^k(x)=x+(k+1) for all x in Z. Run the same “replace f^i(x) with x_{n+i}” bit we did before and rearrange to get x_{n+k}-x_n=k+1 for all n \geq 0.

f(x)=x-4 does indeed satisfy this functional equation, but this is easy to verify).

I think this is a pretty neat method. Like I said, I wasn’t familiar with it until yesterday, so forgive me if this is a common trick you have seen before, since I certainly hadn’t.

Another quick application we can