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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.

13 votes

Where do root systems arise in mathematics?

They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable r …
3 votes

Suggestions for good notation

Since this one is on the front page again: In my personal notes, I have started writing sums/integrals over complicated index sets as $\sum \left( \text{summand} \mid \text{condition} \right)$, rather …
5 votes

Listing applications of the SVD

See my other answer for the caveat that I am writing about things I have learned from student presentations, so they may be flawed. Suppose we have a block of raw material which has some shape $X \sub …
15 votes

Listing applications of the SVD

I teach a course on Applied Linear Algebra, intended for Engineers, where the final project is always to give a presentation on an application of linear algebra in the student's field of study. Since …
13 votes

Lunch seminars for PhD students

When I was at UC Berkeley, there was a seminar of this sort called "Many Cheerful Facts" (see here if you want to know why). It was organized by graduate student. Every week, one student would be recr …
24 votes

PhD dissertations that solve an established open problem

June Huh's recent proof of Rota's conjecture (stated by Read in 1968 for graphical matroids and Rota in 1971 for all matroids) formed his 2014 Ph. D. thesis. For matroids over $\mathbb{C}$, this appea …
10 votes

What are some very important papers published in non-top journals?

Alastair King's "Moduli of representations of finite-dimensional algebras" (1994) is foundational for everything that has happened in quiver moduli, wall crossing formulae, and various other stability …
10 votes

Examples of common false beliefs in mathematics

Multiplication of differential forms is inherently anti-commutative. Thus, if $x$ and $y$ are coordinates on a surface, then $dx \wedge dy$ makes sense but $(dx)^2+(dy)^2$ is either nonsense or, if it …
17 votes

When has the Borel-Cantelli heuristic been wrong?

I may as well convert my comment to an answer. The probability that a randomly chosen degree $d$ polynomial in $\mathbb{F}_p[u]$ is irreducible is $d^{-1} \cdot (1+O(p^{-d/2}))$. So, if $g(T,u)$ is a …
David E Speyer's user avatar
1 vote
Accepted

Decidable theorem or result that is not weaker than Tarski's theorem

Ax and Kochen proved decidability for the ring of $p$-adic numbers, and many rings like it. That certainly doesn't follow from Tarski, and I would say it is more difficult.
David E Speyer's user avatar
41 votes

Suggestions for good notation

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG $$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^ …
4 votes

Computer algebra errors

We found some interesting bugs in Mathematica's integration software on this thread. To wit, set integral[m_,n_] = Integrate[Log[2+Cos[2Pi x]+Cos[2Pi y]] Cos[2Pi m x] Cos[2Pi n y], …
2 votes

Examples of non-abelian groups arising in nature without any natural action

Higman's group seems like a pretty good example. Of course, it acts on itself, but it has no action on any finite set or finite dimensional vector space, and the only reasonable description is by gene …
5 votes

Asymptotic Methods in Combinatorics

If you want to know about quantities which (1) have nice generating functions and (2) depend on more than one parameter, the most thorough guide will be found in the papers of Robin Pemantle. to the b …
9 votes

Asymptotic Methods in Combinatorics

At a lower level than Flajolet and Sedgewick, Chapter 5 of generatingfunctionology by Wilf is a good introduction to complex analytic methods. (Yes, my two answers look very similar. As usual in a big …

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