An edge flipping is a non-reversible Markov chain on a given connected graph, as defined in Chung and Graham (2012). In the same paper, edge flipping eigenvalues and stationary distributions for some classes of graphs were identified. We further study edge flipping spectral properties to show a lower bound for the rate of convergence in the case of regular graphs. Moreover, we show by a coupling argument that a cutoff occurs at $\frac{1}{4} n \log n$ for the edge flipping on the complete graph.

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.

We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

Suppose that $Y$ is a cyclic cover of projective space branched over a hyperplane arrangement $D$ and that $U$ is the complement of the ramification locus in $Y$. The first theorem in this paper implies that the Beilinson–Hodge conjecture holds for $U$ if certain multiplicities of $D$ are coprime to the degree of the cover. For instance, this applies when $D$ is reduced with normal crossings. The second theorem shows that when $D$ has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of $Y$. The last section contains some partial extensions to more general nonabelian covers.

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.

For $x\in (a_{j-1}, a_j)\ (j=1,\ldots, p+1;\ a_0\!:=-\infty, \ a_{p+1} \!:=\infty)$ the mapping $T_j\!: w=x-\sum ^p_{l=1}\lambda _l/(x-a_l)\ (\lambda _l$>$0, \ a_l\in$R) is onto R. It was shown by G. Boole in the 1850's that $\sum ^{p+1} _{j=1}[(\partial w/\partial x) ^{-1}] _{x=T^{-1}_j(w)}=1.$ We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.