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Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples particul …

I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck. It is better to have a good category with bad objects than a bad category …

Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology: Let f be a $C^\infty$ function on a closed manifold …

The Generalized Riemann Hypothesis (GRH) influences Complexity Theory. In particular, Pascal Koiran proved that the truth of the GRH implies that the problem of "whether a set of polynomial equations …

I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a …

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, a …

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued …

I'm not sure how K-theory enters in this context, but modular forms with half-integral weight are closely related to the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds. The story starts with …

I believe all is answered by Page 196 (and the subsequent discussion) of Curves and Singularities by J.W. Bruce and P.J. Giblin, who coined the name.

I don't think that there has been a tremendous amount of progress in understanding the Kontsevich Invariant of a knot in the last decade or so. It appears that essential new ideas may be needed in ord …

I think my paper with A. Carmi Daniel Moskovich, Avishy Y. Carmi, Tales told by coloured tangles, Int. J. Unconv. Comput. 12(1) 71-105 (2016), journal version, arXiv:1511.04919 is largely just this. …

A week or so ago, I was saddened to read Jim Stasheff's post on the AlgTop mailing list, announcing the passing of Colin Day, after a long bout with cancer. I was thinking of reading Colin Day's PhD …

In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply- …

It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for $n<5$ ( …

I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Algèbres d'E …