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

Proving a topology theorem by using category theory to translate it into a group theory problem.

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

An animated explainer on homotopy groups. Discusses geometric intuition for π₁/π₂/π₃ using loopspaces of metric spaces. Includes an original, nonstandard visualization of the Hopf map.

Beautiful typesetting and animation. Homotopy is really weird; surprised that people turned this stuff into type theory.

Ranked #1 among over 400 entries in the 2025 Summer of Math Exposition (SoME4) competition

!!

I think this is the first highly technical Topos Institute blog post where I walked away and felt like I understood everything I was meant to understand. Heck yeah.

Great introduction to sheaf theory.

a community database of topological counterexamples

Knot theory is far more complicated than I initially gave it credit. Would love to learn how they found the counter example via computer search.

The inscribed square/rectangle problem, solved using Möbius strips and Klein bottles.

Excellent intuition-building for topology. I jumped out of my chair when I recognized the Möbius strip construction :P.

via: ~azurylite

This video singlehandedly helped me understand coverings and Lie algebras way better than any Wikipedia article I’ve ever read. :P

This video singlehandedly helped me understand coverings and Lie algebras way better than any Wikipedia article I’ve ever read. :P