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The following story is a bit strange to be true, but we all believed it as students, and I think I still do believe that a somewhat weaker version of events must have indeed occurred. Michael Maschler …

Another urban legend, which I've heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following: As a young p …

Numerical analysis is of key importance in sciences and applications, including biology, economics, computing, and medicine. The idea of approximating a solution, and how that might be carried out. Th …

Calculus, particularly the ideas of derivation and integration, is surely the mathematical idea which has changed history most in the last 400 years. The ability to study and quantify change and rate …

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 …

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) …

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 …

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 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 …

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 …

A famous example is that John Conway, who fathered the concept of a Skein relation, didn't discover the Jones (or HOMFLYPT) polynomials. By just searching for knot invariants defined via Skein relatio …

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 …

A knot table, with the knots in it made out of a nice (pretty and pliable) material. It's aesthetic, and people might have fun playing with them. One might include also the Perko pair! They come with …