Related papers: Electrical Networks, Lagrangian Grassmannians and …

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the…

We study regularization for the deep linear network (DLN) using the entropy formula introduced in arXiv:2509.09088. The equilibria and gradient flow of the free energy on the Riemannian manifold of end-to-end maps of the DLN are…

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the…

We study moduli of ``self-associated'' sets of points in ${\bf P}^n$ for small $n$. In particular, we show that for $n=5$ a general such set arises as a hyperplane section of the Lagrangean Grassmanian $LG(5,10) \subset {\bf P}^{15}$ (this…

Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps, which constructs a graph embedding…

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative $U(1)$ gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we…

Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces…

In this paper we generalize single-relation pseudo-Riemannian graph embedding models to multi-relational networks, and show that the typical approach of encoding relations as manifold transformations translates from the Riemannian to the…

In this {paper}, we consider holomorphic embeddings of $Gr(2,m)$ into $Gr(2,n)$. We study such embeddings by finding all possible total Chern classes of the pull-back of the universal bundles under these embeddings. Using the relations…

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the…

Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit…

Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to…

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\textrm{edeg} G(k,n)$ of the real Grassmannian $G(k,n)$ as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random…

A linkage $\mathcal{L}$ consists of a graph $G=(V,E)$ and an edge-length function $\ell$. Deciding whether $\mathcal{L}$ can be realized as a planar straight-line embedding in $\mathbb{R}^2$ with edge length $\ell(e)$ for all $e \in E$ is…

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick,…

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study…

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma(N,L)$ whose vertices are $N$ random points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges are the…

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group $G$, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded…

We provide a new branching rule from the general linear group $GL_{2n}(\mathbb{C})$ to the symplectic group $Sp_{2n}(\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a…