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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.

138 votes

Most harmful heuristic?

Along the same lines as Qiaochu's and Zach's responses, the commonly taught heuristics pertaining to functions, differentiability and integration are a pet hate of mine. I certainly left school think …
55 votes
17 answers
16k views

Computer science for mathematicians

This is a big-list community question, so I'm sorry in advance if it is deemed too soft but I haven't seen anything similar yet. I've seen computer scientists post questions looking to learn things …
34 votes
Accepted

Introductory text on Riemannian geometry

Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it …
11 votes
5 answers
4k views

Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in p …
8 votes

What is the definition of "canonical"?

I would say that "canonical" ought to be used to describe when no choices have been made. A nice example of a non-canonical identification: A principal bundle is made up of principal homogeneous spac …
6 votes

Ingenuity in mathematics

Mathematicians are used to seeing these sorts of problems and are generally more intrigued by their solutions, but I came across this recently: Colour every point in the plane either black or white. …
6 votes

Which book would you like to see "texified"?

Leon Simon - Lectures on Geometric Measure Theory
2 votes

Proving theorems by using functions with fixed points.

I am familiar with a good example from the theory of 2nd order elliptic PDE. Technicalities omitted... A special case of the Leray-Schauder Theorem says the following: Let $T$ be a compact mapping o …
0 votes

Applications of connectedness

You can get the maximum principle for subharmonic functions: Write $F$ for the supremum of $f$ in $\Omega$, write $A$ for the set where $f = F$ and $B$ for the set where $f < F$. Then $\Omega = A \cu …