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Results tagged with big-list
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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 …
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.
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 …