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1500 questions
47
votes
7 answers
Is it easy to produce hard-to-color graphs?
This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each child drew a graph as a challenge, to be colored…
Joel David Hamkins
- 224,022
47
votes
3 answers
Absolute value inequality for complex numbers
I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
hold for all complex numbers $a,b,c$ ?
For real…
Rene Schipperus
- 953
47
votes
1 answer
improving known bounds for Pierce expansions; cash prize
Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have thought about it. As I'm getting older, I now offer…
Jeffrey Shallit
- 2,918
47
votes
5 answers
Why was John Nash's 1950 Game Theory paper such a big deal?
I'm trying to understand why John Nash's 1950 2-page paper that was published in PNAS was such a big deal. Unless I'm mistaken, the 1928 paper by John von Neumann demonstrated that all n-player non-cooperative and zero-sum games possess an…
TSGM
- 583
- 1
- 5
- 8
47
votes
2 answers
Collapsible group words
What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?
For example, $f(2)=4$, with the commutator $[x_1,x_2]=x_1 x_2 x_1^{-1} x_2^{-1}$ attaining the…
Bjorn Poonen
- 23,617
47
votes
15 answers
How does the work of a pure mathematician impact society?
First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.
I am in a really bad position, because the people who are in charge of the video want me…
Joaquín Moraga
- 1,585
47
votes
0 answers
Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$. A subspace $E$ is invariant for $A$ if $A(E)…
Nik Weaver
- 42,041
47
votes
23 answers
Taking lecture notes in lectures
Do you find it a good idea to take lecture notes (even detailed lecture notes) in mathematical lectures?
Related question: Digital Pen for Math: Your Experiences?
Gil Kalai
- 24,218
47
votes
3 answers
Connected sum of topological manifolds
A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \mathring B'$ along their boundary (an…
ACL
- 12,762
47
votes
5 answers
What axioms are used to prove Gödel's Incompleteness Theorems?
I understand Gödel's Incompleteness Theorems to be statements about effectively generated formal systems, which basically makes them theorems about algorithms. This is cool, because despite being very abstract, they actually constrain my…
Andrew Critch
- 11,060
47
votes
2 answers
current status of crystalline cohomology?
The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (just compare Illusie's survey from 1975 with that…
Thomas Riepe
- 10,731
47
votes
6 answers
Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is usually functional analytic and imposes strong…
Vidit Nanda
- 15,397
47
votes
1 answer
Brown representability for non-connected spaces
In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this:
If $F$ is a contravariant functor from the (weak) homotopy…
Mike Shulman
- 65,064
46
votes
2 answers
Open problems/questions in representation theory and around?
What are open problems in representation theory?
What are the sources (books/papers/sites) discussing this?
Any kinds of problems/questions are welcome - big/small, vague/concrete.
Some estimation of difficulty and importance, as well as, small…
Alexander Chervov
- 23,944
46
votes
4 answers
Why could Mertens not prove the prime number theorem?
We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x).
$$
Thus both these divergent…
Nilotpal Kanti Sinha
- 4,638