Related papers: Topological Applications of p-Adic Divergence and …
I show that one can explicitly construct topologically/geometrically distinguishable data which provide isomorphic copies (i.e. \emph{isomorphs}) of the tempered fundamental group of a geometrically connected, smooth, quasi-projective…
We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von…
In this paper, we introduce translation operators on graphs. Contrary to spectrally-defined translations in the framework of graph signal processing, our operators mimic neighborhood-preserving properties of translation operators defined in…
The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to…
The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of…
This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual,…
We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with…
In this paper, we define a Laplacian operator on a statistical manifold, called the vector Laplacian. This vector Laplacian incorporates information from the Amari-Chentsov tensor. We derive a formula for the vector Laplacian. We also give…
Topological Data Analysis (TDA) is a modern approach to Data Analysis focusing on the topological features of data; it has been widely studied in recent years and used extensively in Biology, Physics, and many other areas. However,…
Topological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of Topological Machine Learning, great attention has been devoted to signal processing of such topological signals.…
There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree…
Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of…
These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main…
In many areas such as computational biology, finance or social sciences, knowledge of an underlying graph explaining the interactions between agents is of paramount importance but still challenging. Considering that these interactions may…
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…
We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the…
In this paper we study some problems related with the theory of multidimensional $p$-adic wavelets in connection with the theory of multidimensional $p$-adic pseudo-differential operators (in the $p$-adic Lizorkin space). We introduce a new…
We construct the deformation functor associated to a couple of morphisms of differential graded Lie algebras, and use it to study the infinitesimal deformations of a holomorphic map of compact complex manifolds. In particular, in the case…
We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of…
In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of…