Related papers: An algorithm for computing the topological Euler c…
We introduce a notion of `proChow group' of varieties, agreeing with the notion of Chow group for complete varieties and covariantly functorial with respect to arbitrary morphisms. We construct a natural transformation from the functor of…
Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive…
Using a fully connected feedforward neural network we study topological invariants of a class of Calabi--Yau manifolds constructed as hypersurfaces in toric varieties associated with reflexive polytopes from the Kreuzer--Skarke database. In…
Given a projective algebraic variety $X$, let $\Pi_p(X)$ denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on $X$. The $p$-th Euler-Chow series of $X$ is an element in the formal monoid-ring…
This note studies the behavior of Euler characteristics and of intersection homology Euler characterstics under proper morphisms of algebraic (or analytic) varieties. The methods also yield, for algebraic (or analytic) varieties, formulae…
In this short note we present a generating function computing the compactly supported Euler characteristic $\chi_c(F(X, n), K^{\boxtimes n})$ of the configuration spaces on a topologically stratified space $X$, with $K$ a constructible…
This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the…
Let M be the blow--up of a manifold M along a submanifold X. In this paper we present closed formulae for the integral cohomology and the total Chern class of M. As applications we compute the cohomology of the varieties of complete conics…
We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series…
We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our formulas can be applied also to the case where the $A$-discriminant varieties are higher-codimensional…
Computing topological invariants in two-dimensional quasicrystals and super-moire matter is a remarkable open challenge, due to the absence of translational symmetry and the colossal number of sites inherent to these systems. Here, we…
We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which…
We introduce new algebro-topological invariants of directed networks, based on the topological construction of the directed clique complex. The shape of the underlying directed graph is encoded in a way that can be studied mathematically to…
We study $\mathbb{S}_n$-equivariant weight-graded and topological Euler characteristics of the universal Picard stack $\mathrm{Pic}_{g, n}^d \to \mathcal{M}_{g, n}$ of degree-$d$ line bundles over $\mathcal{M}_{g, n}$. We prove that in the…
We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of…
I will present an explicit formula for the intersection indices of the Chern classes of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler…
Persistent homology (PH) -- the conventional method in topological data analysis -- is computationally expensive, requires further vectorization of its signatures before machine learning (ML) can be applied, and captures information along…
Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides…
We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with…
Dwyer, Weiss, and Williams have recently defined the notions of parametrized topological Euler characteristic and parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that…