Analytic lattice cohomology of isolated curve singularities

We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq 0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb Z}$-graded ${\mathbb Z}[U]$-module. The Euler characteristic of ${\mathbb H}^*(C,o)$ is the delta-invariant of $(C,o)$. The construction is based on the multivariable Hilbert series of the multifiltration provided by valuations of the normalization. Several examples are discussed, e.g. Gorenstein curves (where an additional symmetry is established), plane curves (in particular, Newton non-degenerate ones), ordinary $r$-tuples. We also prove that a flat deformation $(C_t,o)_{t\in ({\mathbb C},0)}$ of isolated curve singularities induces an explicit degree zero graded ${\mathbb Z}[U]$-module morphism ${\mathbb H}^0(C_{t=0},o)\to {\mathbb H}^0(C_{t\not=0},o)$, and a graded (graph) map of degree zero at the level of graded roots ${\mathfrak R}(C_{t\not=0},o)\to{\mathfrak R}(C_{t=0},o)$. In the treatment of the deformation functor we need a second construction of the lattice cohomology in terms of the system of linear subspace arrangements associated with the above filtration

Aspects of Metric Spaces in Computation

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Lê Numbers of Arrangements and Matroid Identities

We present several new polynomial identities associated with matroids and geometric lattices, and relate them to formulas for the characteristic polynomial and the Tutte polynomial. The identities imply a formula for the Lê numbers of complex hyperplane arrangements