Most Popular
1500 questions
87
votes
19 answers
Injectivity implies surjectivity
In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
Set theory
An injective map between two finite sets with the same cardinality is surjective.
Linear algebra
An injective linear map between two…
coudy
- 18,537
- 5
- 74
- 134
87
votes
8 answers
Why is Lebesgue integration taught using positive and negative parts of functions?
Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only then for real-valued functions using the crutch…
KConrad
- 49,546
87
votes
5 answers
Why higher category theory?
This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I have some background in commutative algebra and…
Alex
- 685
87
votes
1 answer
Non-amenable groups with arbitrarily large Tarski number?
Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ admits a paradoxical decomposition with $n$ pieces:…
Narutaka OZAWA
- 9,621
87
votes
4 answers
Is there an accepted definition of $(\infty,\infty)$ category?
For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with some weak forms of composition and associativity…
Theo Johnson-Freyd
- 52,873
86
votes
61 answers
Favorite popular math book
Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend?
It would be nice if you could answer in the following way:
Title: The Poincaré Conjecture: In…
Spinorbundle
- 1,909
86
votes
13 answers
How has modern algebraic geometry affected other areas of math?
I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, scheme theory, and especially the relative approach,…
Dori Bejleri
- 2,890
86
votes
9 answers
Why should I believe the Mordell Conjecture?
It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and others believed this statement in the first place.…
Barinder Banwait
- 2,797
86
votes
2 answers
Light reflecting off Christmas-tree balls
'Twas the night before Christmas and under the tree
Was a heap of new balls, stacked tight as can be.
The balls so gleaming, they reflect all light rays,
Which bounce in the stack every which way.
When, what to my wondering mind does occur:
A…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
86
votes
4 answers
Nonexistence of boundary between convergent and divergent series?
The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following Theorem 3.29, he writes:
One might thus be led to…
Timothy Chow
- 78,129
86
votes
5 answers
Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am investigating this with regard to finding the normalized…
86
votes
4 answers
Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?
Several ancient arguments suggest a curved Earth, such as
the observation that ships disappear mast-last over the
horizon, and
Eratosthenes'
surprisingly accurate calculation of the size of the
Earth
by measuring a difference in shadow length…
Joel David Hamkins
- 224,022
86
votes
5 answers
What is sheaf cohomology intuitively?
What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand sheaf cohomology in some "geometric" way?
For…
Jan Weidner
- 12,846
86
votes
44 answers
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are contour integrals and Laplace transform, but this…
Wadim Zudilin
- 13,404
86
votes
6 answers
What are the most elegant proofs that you have learned from MO?
One of the things that MO does best is provide clear, concise
answers to specific mathematical questions. I have picked up ideas
from areas of mathematics I normally wouldn't touch, simply because
someone posted an eye-catching answer on MO.
In…
John Stillwell
- 12,258