0
votes
1answer
114 views
Does there exist a polar decomposition of matrices over finite fields?
There exists a polar decomposition for matrices over the reals.
What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would …
1
vote
1answer
153 views
Finding Decision Boundary from empirical distribution
Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't …
71
votes
54answers
14k views
Which math paper maximizes the ratio (importance)/(length)?
My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7 …
33
votes
11answers
7k views
Most striking applications of category theory?
What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
…
2
votes
0answers
88 views
Existence of particular embeddings in euclidean spaces for non compact manifolds
Let $M$ be a $n$-dimensional smooth non-compact manifold such that the singular cohomology groups $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a sufficie …
0
votes
1answer
25 views
Transitive subgroup of $S_p$ containing a $p$-cycle and a double transposition
Let $p$ be a prime other than 5 or 7. Are $A_p$ and $S_p$ the only subgroups of $S_p$ that contains a $p$-cycle and a double transposition?
As for $p = 5$, the dihedral group $D_ …
9
votes
1answer
393 views
The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance a …
2
votes
1answer
237 views
About Sectional Curvature
In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel tra …
1
vote
1answer
73 views
non-convex Polytope definition.
I have a simple question.
I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex Polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu …
4
votes
1answer
189 views
On Geometric Langlands Correspondence
The Geometric Langlands correspondence introduced by Drinfeld and Laumon conjectures a 1 to 1 correspondence between
(A) local systems on a projective smooth curve over a field
a …
10
votes
2answers
455 views
Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?
QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to …
1
vote
1answer
313 views
“thematic” algebras
I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a l …
4
votes
2answers
223 views
Cofibrant replacements of a given object in a combinatorial model category
In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-fil …
33
votes
12answers
2k views
Why don’t more mathematicians improve Wikipedia articles?
Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. Here is a list of the 500 mo …
2
votes
1answer
528 views
Pole data of meromorphic matrix function
Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a rig …

