Related papers: Degree theory for orbifolds
We associate to any endomorphism of the punctured affine space over some field an element in the Witt group of the base field that we call degree. We use this degree to give a counter-example to a question on unimodular rows
We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…
This text arises from teaching advanced undergraduate courses in differential topology for the master curriculum in Mathematics at the University of Pisa. So it is mainly addressed to motivated and collaborative master undergraduate…
We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
Perhaps the very first elementary exercise one encounters in graph theory is the result that any graph on at least two vertices must have at least two vertices with the same degree. There are various ways in which this result can be…
The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the…
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…
Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…
We consider the degrees of the elements of a homogeneous system of parameters for the ring of invariants of a binary form, give a divisibility condition, and a complete classification for forms of degree at most 8.
Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the…
We study toric orbifolds of real dimension four with vanishing odd-degree cohomology and obtain a basis for its degree-two equivariant cohomology with integral coefficients by identifying it with the intersection of certain lattices. As…
Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the…
The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…
The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples.…
In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study…