We introduce the notion of fundamental heap for compact orientable surfaces with boundary embedded in 3-space, which is an isotopy invariant of the embedding. It is a group, endowed with a ternary heap operation, defined using diagrams of surfaces in a form of thickened trivalent graphs called surface ribbons. We prove that the fundamental heap has a free part whose rank is given by the number of connected components of the surface. We study the behavior of the invariant under a boundary connected sum, as well as addition/deletion of twisted bands and provide formulas relating the number of generators of the fundamental heap to the Euler characteristics. We describe in detail the effect of stabilization on the fundamental heap and determine that for each given finitely presented group, there exists a surface ribbon whose fundamental heap is isomorphic to it, up to extra free factors. A relation between the fundamental heap and the Wirtinger presentation is also described. Moreover, we introduce cocycle invariants for surface ribbons using the notion of mutually distributive cohomology and heap colorings. Explicit computations of fundamental heap and cocycle invariants are presented.

Let $\mathrm{\Omega }=P_{n,m}(\mathbb{Z}[C_p])$ and $A=P_{n,m}(\mathbb{Z}),$ where p is a positive integer and $P_{n,m}(R)$ is the R-algebra $P_{n,m}(R)=R[t_1,t_1^{-1},\dots ,t_n,t_n^{-1}]{\otimes }_{R}R[x_1,\dots ,x_m]$. A Swan module is an extension module of the form $0\to I^{(k)}\to X\to A^{(k)}\to 0$, where I is the kernel of the augmentation homomorphism $\mathit{ϵ}:\mathrm{\Omega }\to A$. We show that, when p is prime, every such projective Swan module is free; this is false if p is not prime and $n+m>0$. The proof relies on the fact that when R is the ring of algebraic integers in $\mathbb{Q}({\mathit{\zeta }}_p)$ and ${\mathbb{F}}_p$ is the field with p elements, then the canonical homomorphism ${\mathit{GL}}_k(P_{n,m}(R))\twoheadrightarrow {\mathit{GL}}_k(P_{n,m}({\mathbb{F}}_p))$ is surjective for all $k\ge 1$.

We study the question of when a ring can be realized as a direct summand of a regular ring by examining the case of homogeneous coordinate rings. We present very strong obstacles to expressing a graded ring with isolated singularity as a finite graded direct summand. For several classes of examples (del Pezzo surfaces, hypersurfaces), we give a complete classification of which coordinate rings can be expressed as direct summands (not necessarily finite), and in doing so answer a question of Hara about the finite F-representation type (FFRT) property of the quintic del Pezzo. We also examine what happens in the case where the ring does not have isolated singularities, through topological arguments: as an example, we give a classification of which coordinate rings of singular cubic surfaces can be written as finite direct summands of regular rings.

Let $X\subset {ℝ}^d$ be a point set in general position. For a subset of X, let the degree be the number of d-dimensional simplices formed by this subset and further points of X which are empty—that is, which contain no other points of X. The k-degree of the set X is defined as the largest degree of all k-element subsets of X. We show that if X is a random point set consisting of n independently and uniformly chosen points from a convex set, then the d-degree is of order n, improving previously obtained results and giving the correct order of magnitude with a significantly simpler proof. We also prove that the 1-degree is of order $n^{d-1}$ for $d\ge 3$.

In this paper, we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic.

We prove that every (geometrically) quasiconformal homeomorphism between metric measure spaces induces an isomorphism between the cotangent modules constructed by Gigli. We obtain this by first showing that every continuous mapping φ with bounded outer dilatation induces a pullback map ${\mathit{\phi }}^{\ast }$ between the cotangent modules of Gigli, and then proving the functorial nature of the resulting pullback operator. Such pullback is consistent with the differential for metric-valued locally Sobolev maps introduced by Gigli–Pasqualetto–Soultanis. Using the consistency between Gigli’s and Cheeger’s cotangent modules for PI spaces, we prove that quasiconformal homeomorphisms between PI spaces preserve the dimension of Cheeger charts, thereby generalizing earlier work by Heinonen–Koskela–Shanmugalingam–Tyson. Finally, we show that if φ is a given homeomorphism with bounded outer dilatation, then ${\mathit{\phi }}^{-1}$ has bounded outer dilatation if and only if ${\mathit{\phi }}^{\ast }$ is invertible and ${\mathit{\phi }}^{-1}$ is Sobolev. In contrast to the setting of Euclidean spaces, Carnot groups, or more generally, Ahlfors regular PI spaces, the Sobolev regularity of ${\mathit{\phi }}^{-1}$ needs to be assumed separately.

The standard arithmetic measures of center, the mean, and the median, have natural topological counterparts that have been widely used in continuum theory. In the context of metric spaces, it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.

We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti to epsilon multiplicity of filtrations and show that under mild assumptions, this multiplicity exists as a limit. We show that in rather general rings, the epsilon multiplicity of a $\mathbb{Q}$-divisorial filtration is positive if and only if the analytic spread of the filtration is maximal (equal to the dimension of the ring). The condition that filtrations $\mathcal{J}\subset \mathcal{I}$ have the same epsilon multiplicity is considered, and we find conditions ensuring that the filtrations have the same integral closure.