arXiv research

Develops Hilbert geometries and characterizes their isometries.

Characterizes higher rank model geometries using antipodal sets.

Study on potential functions for tensors of rank four in differential geometry.

Fried's theorem proven for symmetric space boundaries.

Constructs geometries with nonvanishing curvature and essential automorphisms.

Paper explores geometry of covariance matrices using associated bundles.

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

Study shows nonexistence of certain geometric structures in complex geometries.

Proof of cohomogeneity two actions on positively curved manifolds.

Paper studies polynomial decomposition in system identification and machine learning.

New concept of coarse medians for higher rank symmetric spaces.

Study on manifolds with higher-rank lattice actions in projective and conformal classes.

Estimates covariance matrices for matrix-variate data via core covariance geometry.

We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\times n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on $\mathcal{M}(p,m\times n)$ by viewing it as the base manifold of the submersion $π:(M,N)\mapsto MN^T$, selecting an adequate Riemannian metric on the total space, and …

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

Researchers prove unique bisectors in a complex hyperbolic space.

If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.

CoreFlow models matrix-valued distributions efficiently, preserving shared low-rank structure.

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivar…

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…

New Ricci flat Kähler metrics found on complex symmetric spaces.

Proves effective Chen ranks conjecture for Koszul modules.

Researchers show mixtures of ranking models are generally identifiable.

Introduces NMF for hyperspectral imaging and discusses its geometry and complexity.

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…

Derives smooth homogeneous structures for low-rank tensors.

Develops a method to interpolate data for efficient inversion in nonuniform geometries.

New findings on geometric flows and equidistribution in Hilbert geometry.

We construct a sequence of primitive-stable representations of free groups into PSL(2,C) whose ranks go to infinity, but whose images are discrete with quotient manifolds that converge geometrically to a knot complement. In particular this implies that the rank and geometry of the image of a primitive-stable representa…

Higher rank rigidity theorem extended to Berwald spaces.

Study the distribution for low-rank matrix learning, improving inference methods.

Study of graphs interpolating curve and pants graphs, providing formulae and geometry classifications.

The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.

Study of a specific geometry in 8D with rank 3 distributions.

New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.

New method proves asymptotic normality for matrix sensing problems.

This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.

We develop a new method for filling in missing data in tensors using Riemannian geometry.

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that t…

Geometric structures on surfaces relate to 2-plane distributions in 5D.

Study of geodesic branching in 2D sub-Riemannian manifolds.

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.

Study on Lie groups acting conformally on pseudo-Riemannian manifolds.

Solves Kobayashi's conjecture on homogeneous spaces.

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with addit…

The study examines closed manifolds with ray nil-affine structures and their completeness.

We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and…