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 …

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: …

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 …

Hi, Is it possible to have topologically different Markov Random Fields (few different edges) and yet yielding the same inference results ? Thanks!

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_ …

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/ …

Consider the following function repeatedly applied to a rational $r = a/b$ in lowest terms: $f(a/b) = (a b) / (a + b - 1)$. So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$. I am wo …

Suppose that $\lambda_1,\lambda_2,\lambda_3$ are partititions of $n$. When do there exist permutations $\sigma_1,\sigma_2,\sigma_3 \in S_n$ such that (1) $\sigma_1\sigma_2\sigma_ …

Suppose that $X$ and $Y$ are smooth complex algebraic varieties, and that $f:X\rightarrow Y$ is an etale morphism in the sense that $d_xf:T_xX\rightarrow T_{f(x)}Y$ is an isomorphi …

Here is the criteria for a "perfect" graph editor: it should be able to perform an automated, but controllable layout one is able to make "manual" enforcements to nodes and edges …

This is something I found in trying to work on Vince Vatter's excellent question. I have no solution, but a much more precise conjecture. Recall that a rooted planar tree is a roo …

In algebraic geometry, the very useful semicontinuity theorem tells you the following: Let $X \to Y$ be a projective morphism of schemes, and $F$ a coherent sheaf on $X$ which is …