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This question is inspired by this Tim Gowers blogpost. I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key …

There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations …

Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually interestin …

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathematic …

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 …

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 $n<5$ ( …