Let $G$ be a finite group and we take for the field the complex numbers. Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are irreducible for $k=1,\dotsc,m$. Question 1: Can perfect irreducible characters be characterised in an...

The weakly group-theoretical conjecture, which suggests a negative response to [ENO11, Question 2], is formulated as follows: Statement 1 (open): Every integral fusion category is weakly group-theoretical. We are exploring whether Statement 1 can be inferred from its simple version: Statement 2 (...

In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (oriented) curve for a primitive element of $H_{1} (S_{g} ; \mathbb{Z})$, where $S_{g}$ is a surface of genus $g$. Thi...

Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ in $M$ intersects a compact leaf of $\mathcal{F}$. Must $L$ itself be compact? Note that the exist...

With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$. In contrast, given $n \in \mathbb{Z}^+$, even though there are to find the $k \in \mathbb{Z}^+$ such that $\phi(k) = n$, I'm not aware of any method to deter...

Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same dimension. Is it possible to build a word map $$\mathrm{Cl}(g)^{n+1} \to G^n$$ $$(g_1,\dotsc,g_{n+1})\m...

Given a simplicial commutative ring $R$, you have the usual K-theory spectrum $K(R)$ defined as the group completion of the symmetric monoidal category of finitely-generated projective modules over $R$. A more naive construction would be to just define a simplicial spectrum $[n] \mapsto K(R_n)$ a...

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$, one has $$f_1(x_1)^2 f_2(x_2)^3 \leq g_1(x_1)^2g_2(x_2)^3 $$ then $$ \left(\int_\mathbb{R} f_1...