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1500 questions
50
votes
10 answers
How is the physical meaning of an irreducible representation justified?
This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO.
I've been reading Singer's Linearity, Symmetry, and Prediction in the Hydrogen…
Qiaochu Yuan
- 114,941
50
votes
1 answer
Rearrangements of a power series at the boundary of convergence
Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = \sum_{n=0}^{\infty} a_{\sigma(n)} z^{\sigma(n)}$$
where…
echinodermata
- 1,203
50
votes
37 answers
Structures that turn out to exhibit a symmetry even though their definition doesn't
Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry.
As an example, I'm thinking of the…
Wolfgang
- 13,193
50
votes
7 answers
Good lattice theory books?
A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - lattices seem to show up everywhere, the author…
Zev Chonoles
- 6,722
50
votes
8 answers
Puzzle on deleting k bits from binary vectors of length 3k
Consider all $2^n$ different binary vectors of length $n$ and assume $n$ is an integer multiple of $3$. You are allowed to delete exactly $n/3$ bits from each of the binary vectors, leaving vectors of length $2n/3$ remaining. The number of distinct…
Simd
- 3,195
50
votes
4 answers
Motivation for concepts in Algebraic Geometry
I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
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Question
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Are there any well-motivated introductions to scheme theory?
My idea of what "well-motivated" means…
Steven Gubkin
- 11,945
50
votes
13 answers
Erratum for Cassels-Froehlich
Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has past, I've spent days going through the results,…
Kevin Buzzard
- 40,559
49
votes
1 answer
Exploding primes
Suppose every prime $n$ could "explode" once.
An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being
uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n \rfloor$.
If a particle hits a composite or a…
Joseph O'Rourke
- 149,182
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49
votes
7 answers
Is there an algebraic approach to metric spaces?
It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there is a complete dictionary between topological…
Mark
- 4,804
49
votes
1 answer
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up while working on a paper. We need to impose that…
javier
- 2,911
- 19
- 20
49
votes
20 answers
What are good non-English languages for mathematicians to know?
It seems that knowing French is useful if you're an algebraic geometer. More generally, I've sometimes wished I could read German and Russian so I could read papers by great German and Russian mathematicians, but I don't know how useful this would…
Qiaochu Yuan
- 114,941
49
votes
2 answers
Intuition for coends
Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end
$$
\int^{c\in C} F(c,c)
$$
as the co-equalizer of
$$
\coprod_{c\to…
K Shao
- 603
49
votes
7 answers
What is a coalgebra intuitively?
How to think about coalgebras? Are there geometric interpretations of coalgebras?
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic examples of coalgebras should one keep in mind?
Anything…
Jan Weidner
- 12,846
49
votes
95 answers
Undergraduate Level Math Books
What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much…
person
- 13
49
votes
3 answers
how to use arxiv?
This is a soft question. How do people usually use arxiv to put their papers? At which stage does one usually put his/her paper/report there? Someone suggests me to submit a paper while putting it on arxiv. Is that the convention that people…
Anand
- 1,619