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In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels…
This paper examines the properties of real symmetric square matrices with a constant value for the main diagonal elements and another constant value for all off-diagonal elements. This matrix form is a simple subclass of circulant matrices,…
The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry. It has direct applications in geometric modeling, computer vision, and statistics. We use non-proper Morse theory to give a…
The Hilbert manifold $\Sigma$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $\Omega\subset \Sigma$ is studied from the topological and metric…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…
The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is…
We propose a novel type of duality that connects a sequence of well-known theories with even-multiplicity scalar amplitudes: it relates the Yang-Mills theory coupled to a specific scalar matter sector to the nonlinear sigma model on a…
We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-negative. Several generalizations are…
The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times…
The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry…
In this paper a generalization of the Gram-Schmidt Algorithm is presented. Actually we provide an algorithm to construct a set of equiangular vectors with a given angle $\theta\in(0,\arccos(\frac{-1}{n-1}))$ using a set of input independent…
We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries
This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the…
We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the…
Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical…
We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different…
In this thesis, we study value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces in n-dimensional Euclidean space. After reviewing basic facts, we give estimates for the number of exceptional values…