Given a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$, there is an embedding of $(G,d)$ (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$, where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

We give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

Let K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler ℤ-bilinear form associated to C, the Euler characteristic χC(M,N) of left C-comodules M,N, and the defect ∂C(M,N) ∊ ℤ associated to any computable Euler pair (M,N) of left C-comodules. We show that (lgthM, lgthN) = χC(M,N) + ∂C(M,N), for any computable Euler pair (M,N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect ∂C(M,N) is zero and (lgthM,lgthN) = χC(M,N), if the comodules M,N are finite-dimensional.

Let G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set XCn and the multiplicities of indecomposable summands of M considered as a kCn-module are related via a localization theorem in equivariant cohomology, where Cn is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.