Related papers: Euler calculus on spaces homeomorphic to definable…
In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic…
The Euler-Poincare characteristic, or Euler characteristic in short, is a fundamental topological invariant of compact manifolds that plays a crucial role in a variety of geometric and topological situations. From this point of view, we…
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated…
The index theorem of Euler-Poincar\'e characteristic of manifold with boundary is given by making use of the general decomposition theory of spin connection. We shows the sum of the total index of a vector field $\phi $ and half the total…
We make calculations in graph homology which further understanding of the topology of spaces of string links, in particular calculating the Euler characteristics of finite-dimensional summands in their homology and homotopy. In doing so, we…
A note connecting arguments scattered in the extant literature proving that, in any o-minimal expansion of the real field, a definable family of sets has the property that the set of parameters corresponding to finite-volume fibers is…
Combining the MPS degeneration formula for the Poincar\'e polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a…
We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through…
The Euler calculus -- an integral calculus based on Euler characteristic as a valuation on constructible functions -- is shown to be an incisive tool for answering questions about injectivity and invertibility of recent transforms based on…
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 study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set…
We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…
We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a…
For a hermitian line bundle over an arithmetic variety, we construct a convex continuous function on the Okounkov body associated to the generic fibre of the line bundle. The integration of the continuous function gives the growth of the…
The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to…
We introduce the universal Euler characteristic of orbit space definable groupoids, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids associated to proper definable group actions. We show that…
We study geometrical properties of an Ulrich vector bundle $E$ of rank $r$ on a smooth $n$-dimensional variety $X \subseteq \mathbb P^N$. We characterize ampleness of $E$ and of $\det E$ in terms of the restriction to lines contained in…
We study extension properties for morphisms of stacks of bundles for group algebraic spaces. Applications are a short proof of the classification of bundles on the projective line for smooth geometrically reductive groups and the existence…
We extend calculus from smooth manifolds to topological manifolds making use of a theory of generalized functions developed for this aim. Actually such extension fits into a boarder context: the universal construction of a site containing…
Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed…