In case you're looking for something to do in a bit over an hour, I'm giving a Zoom talk for the University of Nottingham's #MachineLearning seminar at 3pm BST on discrete neural nets and polymorphic learning. You can find a link to the abstract at https://kasprzyk.work/seminars/ml.html and you can find the preprint I'm discussing at https://arxiv.org/abs/2308.00677. In case you'd like to watch later, you'll be able to find a recording on the seminar page.

#AI #CategoryTheory #algebra #combinatorics

Petition to rename the pigeonhole principle the Fundamental Theorem of Combinatorics #combinatorics

I even struggled to formulate the problem, but in case somebody wants a math "riddle": https://math.stackexchange.com/questions/4759043/how-to-calculated-the-expected-value-of-the-overlap-between-groups-of-numbers Any hint would be highly appreciated. #math #combinatorics #probability

This week I posted a paper which I basically wrote during my first couple years of graduate school to the arXiv. You can find the preprint "A partition formula from idempotents" at https://arxiv.org/abs/2308.10177.

I feel a bit conflicted about this paper. Even after posting it I'm not sure how interesting it is. I showed the formula obtained therein to several people, including my advisor, while I was in grad school and people seemed generally unsure what to make of it.

Partition numbers are not something I ever studied much. My impression is that there is neither a known elementary formula nor a proof that one cannot exist. I'm not even sure if there is a rigorous, accepted definition of what would constitute a closed-form elementary formula in this context, since lots of fun can be had in summing over neat sets as in this paper.

#combinatorics #RepresentationTheory #AbstractAlgebra #algebra #NumberTheory

Anyone have good references and/or favorite examples of the use of generating functions (in the sense of combinatorics) in probability or physics? #combinatorics #physics #probability

I made my first post to the #AI section of the arXiv this week! You can find the preprint "Discrete neural nets and polymorphic learning" at https://arxiv.org/abs/2308.00677.

In this paper a learning algorithm based on polymorphisms of finite structures is described. This provides a systematic way to choose activation functions for neural nets whose neurons can only act on a fixed finite set of values. These polymorphisms preserve any specified constraints imposed by the learning task in question.

This paper is the result of a 2021 REU at the University of #Rochester. I am working with a great group of students on a follow-up project right now, so videos of talks and a sequel preprint should be out soon!

#NeuralNets #MachineLearning #math #combinatorics #UniversalAlgebra #ComputerScience #teaching

Sometimes a Numberphile post hits hard.

Yesterday, YouTube autoplayed me the latest episode of Numberphile, about the game of SET: https://youtu.be/EkFX9jUJPKk

My best friend and I used to play this a lot. So much so, that we were way too good for the game to be fun. We'd both see sets before the dealer had finished putting down all the cards, so it became impossible to play.

After she died, I played by myself for a long time. I guess until I moved in with my then boyfriend?

Haven't played it in years, but still cannot think of the game without thinking of her. I recently came across it again in Berkeley, and bought it for a research group at ASU. I hope they enjoy it.

I love the contribution of one commenter on the video, who gave the tip of buying three copies of the game, so you can add a fifth property (a border) and make the game playable again by making it hard again. Maybe I'll buy myself one or more copies.

Also love the variations that are explained in the video ❤️

Thanks to Catherine Hsu, Brady Haran and Pete McPartlan for the lovely video and the trip down memory lane!

#Numberphile #Mathematics #Games #Combinatorics

The new version of my paper with Semin Yoo, "Orientable triangulable manifolds are essentially quasigroups" is now available on arXiv! You can find the preprint at https://arxiv.org/abs/2110.05660 and you can find some videos of me talking about it on my YouTube channel (https://www.youtube.com/channel/UCT0qXiThOxzbCO36U-iXNTQ).

In addition to new images which illustrate our constructions we also have filled a gap in the proof of the main theorem. In order to show that all orientable triangulable manifolds could be created from an \(n\)-ary quasigroup by our construction, we needed to make an appropriate \(n\)-quasigroup for each manifold. What we actually did in the original paper was give a presentation of such an algebraic structure, which is not quite enough to prove the desired result. This new version contains an explicit description of such an \(n\)-quasigroup.

You can look forward to hearing more from me on connections between #quasigroups and #topology in the future!

#UniversalAlgebra #combinatorics #AlgebraicTopology

Here are two figures I just made for the last section of the paper I’m working on, and I wonder if anyone can determine what the section is about, based on these? (It’s only tangentially related to the main topic of the paper, but it was something neat I discovered along the way.)

#math #research #combinatorics

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Tech #wired #TechNews #IT #Technology via @morganeogerbc #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

Wired: A Computer-Assisted Proof Solves the ‘Packing Coloring’ Problem https://www.wired.com/story/a-computer-assisted-proof-solves-the-packing-coloring-problem/ #Science/PhysicsandMath #QuantaMagazine #combinatorics #mathematics #Computers #HardMath #Science #puzzles #math

@pieter @bones There is also Flajolet and Sedgewick's “Analytical Combinatorics”, a much bigger work, which in many aspects was much clearer to me. It is completely accessible online (https://ac.cs.princeton.edu/home).

For the real beginner, I would still recommend “Concrete Mathematics” (https://en.wikipedia.org/wiki/Concrete_Mathematics). That's at least where I learned it first.

#combinatorics #GeneratingFunctions

The LOOPS'23 conference on #loops and #quasigroups just concluded. There were lots of great talks, great Polish food, and great people. This palace in Będlewo, Poland was a neat venue at which to study #AbstractAlgebra!

#conferences #UniversalAlgebra #math #combinatorics

12 secondary school students prepared diligently this week for the International Mathematical Olympiad (IMO) taking place in Japan 🇯🇵 and for the Middle European Mathematical Olympiad (MEMO) happening in Slovakia. 🇸🇰
The training concentrated on the fields #geometry, #combinatorics, #algebra and #numbertheory.

We wished them good luck and hope to see them as students and/or researchers at
@ESIVienna and
@univienna

Sometimes when artists do #combinatorics their math gets a bit suspect--- but this Claude Closky piece "All the ways to close a cardboard box" (1989) seems legit.

https://ww.closky.info/?p=126

Each small flap can be either above or below each of the large flaps, giving 4 binary possibilities, so there should indeed be 16 cases -- not all of which are stable configurations, of course.

Since I have a fair amount of work from before this venture into social media, I'll share some older things I did sometimes.

My first single-author paper was about multiplayer versions of rock-paper-scissors. You can find a copy on arXiv (https://arxiv.org/abs/1903.07252) which is pretty similar to the version published in the journal Algebra Universalis.

I had the idea for this paper when I was stranded in #Yosemite national park for a month in 2017 after I finished my bachelor's degree. I wanted to explain to my non-mathy friends that I was into #UniversalAlgebra and this is what I came up with. There are a lot of connections with tournaments from #GraphTheory. You can find videos of me talking about this on YouTube too.

#math #combinatorics #algebra #AbstractAlgebra