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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 …

In 1993, Pat Gilmer asserted as Theorem 1 of Classical knot and link concordance, that certain Casson-Gordon invariants vanish for all slice knots, which would be true if the kernel of the inclusion $ …

Two widely believed conjectures: The proportion of hyperbolic knots amongst prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. The crossing number (the minimal number o …

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 …

Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered t …

Levi Ben Gershon (1288-1344) (see also here) is commonly known to us as the RaLBa"G. Again, this is a nickname rather than a pseudonym- RLBG = "Rabbi Levi Ben Gershon", much in the same way as Shah Ri …

You might be interested in some of the answers to this conceptually similar question: What is the high-concept explanation on why real numbers are useful in number theory? My understanding is that p …

The Selberg Trace Formula- general case Hejhal's original 1983 proof is 1322 pages long! As far as I know, the proof remains famously very hard.

The Four Colour Theorem might perhaps be a canonical example of a very hard proof of a major result which has improved, but is still very hard- no human-comprehensible proof exists, as far as I know, …

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 …

The Smale Conjecture. This was proven by Hatcher in 1983. It states that the diffeomorphism group $\mathrm{Diff}(S^3)$ of the $3$-sphere has the homotopy type of the orthogonal group $O(4)$, which …

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 …

Reading Bhaskara II's Lilavati (written in 1150) was an eye-opening experience and provided me with many gems with which to liven up a calculus course. It's quite readable, and its approach is playful …

Perhaps the canonical example is Nikolai Ivanovich Lobachevsky? His work on hyperbolic geometry was subject to severe ridicule, stemming from a negative review of Ostrogradsky, the leading Russian mat …

Novikov's proof of the topological invariance of rational Pontryangin classes, for which he was awarded the 1970 Fields Medal. Fundamentally new (complicating a fundamental group to simplify geometry) …