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There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the tw …

Related MO questions: What-are-your-favorite-instructional-counterexamples, Cannonical examples of algebraic structures, Counterexamples-in-algebra, individual-mathematical-objects-whose-study-amounts-to-a-subdiscipline … most-intricate-and-most-beautiful-structures-in-mathematics, counterexamples-in-algebraic-topology, algebraic-geometry-examples, what-could-be-some-potentially-useful-mathematical-databases, what-is-your-favorite-strange-function …

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. …

The only branch where I think this is explicitly recognized in the literature is topology, where for example Munkres is careful to point out and discuss his favorite counterexamples in his book, and Counterexamples … So: what are your favorite examples of counterexamples that really illuminate something about some aspect of a subject? …

Here's one of my favorite axiomatizations of topology! To make a set $X$ into a topological space, you introduce a relation, "touches," between the elements of $X$ and the subsets of $X$. …

My favorite example of bad notation is using $\textrm{sin}^2(x)$ for $(\textrm{sin}(x))^2$ and $\textrm{sin}^{-1}(x)$ for $\textrm{arcsin}(x)$, since this is basically the same notation used for two different …

This is elementary as well, but one of my favorite ones :) $1^2 + 2^2 + \dots + n^2 = \frac13n(n+1)(n+\frac12)$ (Author: Man-Keung Siu) …

(eds.)(2016): Graph Theory—favorite conjectures and open problems. 1.) …

Of this type is the sin-example in the opening post or a favorite of mine (also occurred to me): The underlying additive group of the field with $p^n$ elements is $\mathbb{Z}/p^n\mathbb{Z}$. …

Probably my favorite way to state this fact: the minimal polynomial is not irreducible or square-free. The same thing in a fancier language: the Jordan canonical form is not diagonal. …

event where no solution is near after this effort, the resulting compilation would itself help illustrate the depth of RH. 3) It would take very little effort for an expert in a given area to post a favorite …

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. …

So: What are your favorite dinner conversation math puzzles? I don't want to provide hard guidelines. But I'm generally interested in problems that are mathematical and not just logic puzzles. …

My favorite example is the following proof of the Cayley-Hamilton theorem, which caused me some disconcertion when I was a student. …

So, feel free to mention your favorite strange spaces and maps. …