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1500 questions
67
votes
28 answers
Examples of seemingly elementary problems that are hard to solve?
I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here is that they can verify a few small cases by…
67
votes
3 answers
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex Functions-Jones, Three Dimensional Geometry and…
Martin David
- 1,216
67
votes
6 answers
Good ways to engage in mathematics outreach?
Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose children and adults to all the fun we are…
Frank Thorne
- 7,199
67
votes
16 answers
What do named "tricks" share?
There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 11 such tricks (the last of which I learned at MO):
the Whitney…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
67
votes
11 answers
How should one think about non-Hausdorff topologies?
In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/nets have unique limits, compact sets are closed,…
Mark
- 4,804
67
votes
6 answers
How to recognise that the polynomial method might work
A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(a_n,b_n)$ be a sequence of points in…
gowers
- 28,729
67
votes
2 answers
Function that produces primes
For any $n\geq 2$ consider the recursion
\begin{align*}
a(0,n)&=n;\\
a(m,n)&=a(m-1,n)+\operatorname{gcd}(a(m-1,n),n-m),\qquad m\geq 1.
\end{align*}
I conjecture that $a(n-1,n)$ is always prime.
To verify it one may use this simple PARI…
Notamathematician
- 3,358
67
votes
3 answers
Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0
Peter Scholze
- 19,800
- 3
- 98
- 118
67
votes
3 answers
Should water at the scale of a cell feel more like tar?
The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary manipulations show that if you zoom in by a factor…
vmist
- 889
67
votes
9 answers
Is all ordinary mathematics contained in high school mathematics?
By high school mathematics I mean Elementary Function Arithmetic (EFA), where one is allowed +, ×, xy, and a weak form of induction for formulas with bounded quantifiers. This is much weaker than primitive recursive arithmetic, which is in turn much…
Richard Borcherds
- 20,442
67
votes
14 answers
A reading list for topological quantum field theory?
Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic algebraic topology, and have some basic background…
user142
- 1,173
67
votes
9 answers
When are probability distributions completely determined by their moments?
If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. Does anyone know an example of two different…
Steve Flammia
- 2,569
67
votes
10 answers
Non-homeomorphic spaces that have continuous bijections between them
What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
Henno Brandsma
- 5,297
67
votes
6 answers
What is a chess piece mathematically?
Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained open for more than one millennium before getting solved by…
67
votes
5 answers
Decidability of chess on an infinite board
The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an infinite board decidable? In other words, given a…
Richard Stanley
- 49,238