This summer at the Topos Institute, under the supervision of Dr. Sophie Libkind, I studied the composition of attractors. The project itself started earlier with my advisor, Dr. William Kalies, who asked me the following question: how do attractor lattices behave when we combine dynamical systems? In this post, I explain how attractor lattices in decoupled product systems can be characterized algebraically in terms of the lattices of their component systems.

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

Very nice philosophy paper by one of the progenitors of category theory on structure. The idea to show a correspondence between Bourbaki and category theory seems like a nice grad school project.

In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.

via: https://doi.org/10.1093/philmat/4.2.174

We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.

We give a quick review of group theory then discuss homotopy of paths building up to the definition of the fundamental group.

A nice walk from homotopy to Lie theory to category theory to combinatorics.

I’m a fifth-year Ph.D. student in the Department of Mathematics at The Ohio State University. I’m interested in algebraic topology, semigroup theory, and computer science.

via: https://www.youtube.com/watch?v=CxGtAuJdjYI

Can’t fully understand because there are no words, but the author looks to be making a very neat connection between coalgebras and the executions (here “traces”) of transition systems. Became aware of her work through Adjoint School 2026.

via: https://www.cs.uni-salzburg.at/~anas/talks.html

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

Part of Prof. Cohn’s larger matmul program. Don’t completely have the chops for it yet, but definitely something to hold onto.

This is the kind of mathematics I was put on this earth to do. The equation explorer is also just a genuinely useful tool for looking up equations; did so earlier this year as part of preparations for a lecture I gave.

We’ve all known about integer multiplication since grade school… but when did you learn about the second multiplication?

Exploration of generalization of commutativity through category theory.

When you hear that someone is “studying algebra”. What comes to mind? Are they drilling through thousands of factorisation problems? Are they an undergraduate student of mathematics, pursuing studies you can’t think of any real-world applications for? Well, you’re all wrong (or maybe you’re not).

This blog post provides an overview of the work I had done with José Siqueira this summer. Inspired by the free Boolean/Heyting algebra of a given set, we develop a free-forgetful adjunction between posets and PLTL temporal algebras, where PLTL denotes propositional linear temporal logic. We provide a description of their induced Eilenberg-Moore categories. We describe how this could be used to temporalise systems and logics through hyperdoctrines and connect this to the stream comonad. We end with future research directions, connecting this topic with the cofree comonad of polynomial functors and temporalising doxastic logic.

The logic algebra stuff was neat. Got lost in the cat sauce as per usual LOL.

Great pair of talks.

via: https://golem.ph.utexas.edu/category/2025/10/a_complex_qutrit_inside_an_oct.html

My research focuses on “homotopical physics”: the interplay between mathematical physics (particularly quantum field theory) and higher (aka categorical) algebra. Other buzzwords that describe my work: higher symmetries, topological field theory, phases of condensed matter, “moonshine” phenomena, perturbative quantization, categories, representation theory, and algebraic topology.

via: https://golem.ph.utexas.edu/category/2025/10/a_complex_qutrit_inside_an_oct.html#c069081

Zvezdelina Stankova discusses the raffle function - and her epic proof ends with an interesting connection.

Beautiful, beautiful problem. Abstract algebra, calculus, number theory, and combinatorics all wrapped up into a bow. :)

The database currently contains 544,831 groups from many different sources

Is there one ring to rule them all? Kevin Tucker goes deep into the world of mathematical rings, among other things.

Beautiful weaving of geometry, group theory, and visual art.