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7
votes
The ten most fundamental topics in geometric group theory
Stallings's theorem on ends of groups, together with some applications (my favorite would be the fact that groups of cohomological dimension one are free, but that might be too algebro-topological for …
-1
votes
Proofs without words
Countable union of countable sets is one of my all-time favorites (and surely one of the all-time best):
here
It's a picture of Cantor's pairing function.
39
votes
Algebraic theorems with no known algebraic proofs
Here is my favorite one (though not so elementary).
Theorem (Grothendieck). Let $X$ be a smooth projective variety over an algebraically closed field $k$. …
3
votes
Examples of combinatorial problems where the only known solutions, or most "natural" solutio...
My favorite example is the paper of Dubuc, which proves the conjecture is true if one of the transformations is an $n$-cycle. …
12
votes
Compactly generated and paracompact $\Rightarrow$ Hausdorff?
In fact, this is a general construction: take your favorite paracompact Hausdorff locally compact space, then double a non-isolated point. …
- 1,422
1
vote
Accepted
Weak Archimedean property instead of Archimedean property
To see this, consider any real number $ x $ (in your favorite system of real numbers $ \mathbb R $). Let $ b _ n = \frac { | x | } { 2 ^ { n - 1 } } $ for all natural numbers $ n $. …
- 346
0
votes
Do continuous martingales satisfy this nice terminal inequality?
Let $X_t=Z$ for all $t$ where $Z=W^2+1$ for your favorite random variable $W$. Then $X_t$ is a continuous martingale and $G(t)=t\mathbb P(X_1\geq t)=t$ for $t\in [0,1]$. …
- 1,904
5
votes
0
answers
201
views
How do you go about making ranges (for integer variables) independent?
(I've edited this question: I originally asked "what is your favorite way to go about it?", but was told that was too subjective. …
- 20.2k
1
vote
Non-complete space verifying uniform boundedness
My favorite example is the space of scalar-valued simple functions on a $\sigma$-algebra $\mathcal{A}$ (or a measurable space $(X,\mathcal{A})$ if you want). …
- 419
1
vote
Exponential sums over a linear subspace
My favorite reference about such things is Cai, Chen, Lipton and Lu's paper "On Tractable Exponential Sums". …
- 731
19
votes
$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O)|}{|K_{2n+1}(O)|} R_K$ and replace $K$-theory with $\mathb...
specifically about the kind of idea you describe, using complex-analytic methods to handcraft an "ad hoc" zeta-function to have desired special values which would recover orders of homotopy groups of my favorite …
- 1,335
4
votes
Accepted
Holomorphic manifolds with an Einstein structure and non constant holomorphic sectional curv...
To give a non-trivial example with positive curvature, take your favorite Fano complex surface (a del Pezzo surface). …
- 6,001
5
votes
1
answer
573
views
Was homology influenced by Euler's polyhedron formula?
And also if you read Lamb - A few of my favorite proofs: The Poincaré homology sphere,
Early topologists wanted to try to find ways of distinguishing spaces by finding invariants: numbers or other mathematical …
- 245
9
votes
0
answers
179
views
How should we picture the set of monomial orders (= positive monoid orders on $\mathbb{N}^k$)?
The two seemingly favorite ways of counting medals are lexicographic (“a gold medal is worth more than any number of silver”) and graded lexicographic (= degree lexicographic: “the total number of medals …
- 32.5k
4
votes
Accepted
Unital nonalternative real division algebras of dimension 8
If anyone who comes across this question is interested in the 2-dimensional case, my favorite reference for that case is
Hübner, Marion; Petersson, Holger P., Two-dimensional real division algebras revisited …
- 2,178