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 …

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 …

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 …

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: …

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 …

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_ …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …