MathOverflow

I will start with an example. Consider the following basic inequality,
$$ 2xy \leq x^2 + y^2,$$
which holds for any $x, y \in \mathbb{R}$ and, in particular, holds for any nonnegative $x$ and $y$. We now promote $x$, resp. $y$ to hermitian positive semidefinite matrices $X$, resp. $Y$. Corresponding to the left-hand side of the previous inequality...

Let $g$ be a quasi-triangular Lie bialgebra and $U_h(g)$ be its Etingof-Kazhdan quantisation. Theorem 6.5 in their paper "Quantisation of Lie bialgebras I" tell us that there is an equivalence of braided tensor categories:
$$\mathcal{M}(g) \cong U_h(g)\textrm{-mod}$$...

Consider the spectral fractional Laplacian with $0<s<1$ on a bounded domain $\Omega\subset\mathbb R^N (N>1)$:$$(-\Delta)^s u = f \quad\text{in }\Omega,\qquad u=0 \text{ on }\partial\Omega.$$Assume $f\in L^\infty_{\mathrm{loc}}(\Omega\setminus...

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.Then every connection on $P$ gives a unique holomorphic...