Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi :P\to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N\left(\xi \right)$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N\left(\xi \right)$ onto a certain representation space ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty }\left(N\right(\xi \left)\right)$ and $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$, where $\Gamma$ denotes the universal central extension of...

We generalize the Malgrange preparation theorem to matrix valued functions $F(t,x)\in C^{\infty }(\mathbf{R}\times {\mathbf{R}}^n)$ satisfying the condition that $t\mapsto \det F(t,0)$ vanishes to finite order at $t=0$. Then we can factor $F(t,x)=C(t,x)P(t,x)$ near (0,0), where $C(t,x)\in C^{\infty }$ is inversible and $P(t,x)$ is polynomial function of $t$ depending $C^{\infty }$ on $x$. The preparation is (essentially) unique, up to functions vanishing to infinite order at $x=0$, if we impose some additional conditions on $P(t,x)$. We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...