arXiv research

Proves rigidity for higher rank three-manifolds without curvature constraints.

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…

Study degenerate Bianchi transformations for pseudo-spherical submanifolds in 5D space.

Artin groups of spherical type are commensurable if they have the same irreducible components and rank.

The paper studies K-stability of spherical varieties and their degenerations.

The paper studies pseudo-isotopies of spherical 3-manifolds and computes ranks of certain groups.

Study spherical cap packing with probabilistic methods for detecting low-rank structures.

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for …

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation t…

Study on the limit set of spherical CR uniformization of cusped hyperbolic manifolds.

Given a 3-manifold M with no spherical boundary components, and a primitive class φin H^1(M;Z), we show that the following are equivalent: (1) φis a fibered class, (2) the rank gradient of (M,φ) is zero, (3) the Heegaard gradient of (M,φ) is zero.

SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.

Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a pre…

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities. These inequalities are given in terms of the combinatorics of the spherical Coxeter…

Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.

We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.

New rigidity theorem for manifolds with specific negative curvature.

Study spherical doubly warped spacetimes for stellar collapse and cosmology.

The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.

Extended dual Coxeter and Artin groups theory to rank-three systems.

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical bui…

Convex optimization method recovers low-rank matrices from rank-one projections efficiently.

We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explic…

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compacti…

Let $X=G/K$ be a symmetric space of noncompact type and rank $k\ge 2$. We prove that horospheres in $X$ are Lipschitz $(k-2)$--connected if their centers are not contained in a proper join factor of the spherical building of $X$ at infinity. As a consequence, the distortion dimension of an irreducible $\mathbb{Q}$--ran…

New random walk results on rank one symmetric spaces.

Researchers prove unique bisectors in a complex hyperbolic space.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

We give a construction of $G_2$ and $Spin(7)$ instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the $G_2$ case, we show that, in the asymptotically conical model, the conn…

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid b…

New graph convolution captures local features on non-Euclidean grids.

Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation o…

The study classifies singularities of spherical orthotomic curves.

Paper introduces spherical knot mosaics for knot and link invariants.

Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.

Minimal products of spherical immersions are studied with geometric and dynamical properties.

Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.

Study geodesics on spherical polyhedra, estimating their number.

Paper finds unavoidable set for spherical curves with reductivity four.

Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.

The Z-loss improves efficiency and flexibility in neural network training.

The abstract proves spherical surface decompositions with conical singularities.