arXiv research

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

Rank-one measurements limit feasible sets for low-rank PSD matrices.

Develops Hilbert geometries and characterizes their isometries.

New findings on geometric flows and equidistribution in Hilbert geometry.

We classify isometric actions of compact Lie groups on quaternionic-Kähler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.

Unique entropy measure found for convex projective manifolds.

Killing tensors on complex projective space are identified and generated by Killing fields.

We consider the problem of estimation of a low-rank matrix from a limited number of noisy rank-one projections. In particular, we propose two fast, non-convex \emph{proper} algorithms for matrix recovery and support them with rigorous theoretical analysis. We show that the proposed algorithms enjoy linear convergence a…

We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.

Study shows non-symmetric convex sets have full boundary limits.

Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…

The Blaschke conjecture claims that every compact Riemannian manifold whose injectivity radius equals its diameter is, up to constant rescaling, a compact rank one symmetric space. We summarize the intuition behind this problem, the proof that such manifolds have the cohomology of compact rank one symmetric spaces, and…

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic…

Let $\A$ be a line arrangement in the complex projective plane $\PP^2$. Denote by $M$ its complement and by $\M$ the set of points in $\A$ with multiplicity at least 3. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the dimension of the cohomology groups $H^m(M,\mathcal{L})$ can be compu…

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the…

The study classifies stable submanifolds in product spaces of projective spaces.

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…

We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good pr…

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,...,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and…

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compa…

New algebraic fundamental groups identified for fake projective planes.

We consider worker skill estimation for the single-coin Dawid-Skene crowdsourcing model. In practice, skill-estimation is challenging because worker assignments are sparse and irregular due to the arbitrary and uncontrolled availability of workers. We formulate skill estimation as a rank-one correlation-matrix completi…

Extended Rank-One Theorem to special metric spaces.

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is ana…

Constructs explicit p-harmonic functions on specific Lie groups.

The paper describes correlations of spectra for higher rank Anosov representations.

The study establishes uncertainty principles on harmonic manifolds of rank one.

We study the Poincaré disk ${\cal D}=\{a\in {\cal A}: \|a\|<1\}$ of a C$^*$-algebra ${\cal A}$ from a projective point of view: ${\cal D}$ is regarded as an open subset of the projective line $\mathbb{P}_1{\cal A}$, the space of complemented rank one submodules of ${\cal A}^2$. We introduce the concept of cross ratio o…

Classifies foliations on specific symmetric spaces.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

New method improves on existing algorithms for rank-one bandits.

Improved stability for matrix recovery from rank-one measurements.

Classifies rank-one submanifolds in Euclidean space.

No Einstein hypersurfaces found in Damek-Ricci spaces.

New insights into compact rank-one ECS manifolds, proving they are bundles over circles.

Study closed manifolds with rank one ray structures, proving completeness or covering properties.

Compact rank one symmetric spaces are rigid under certain curvature conditions.

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case…

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $Ω_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (α^*L)$, where $α$ is the addition map on $A\times…

Estimates joint probability distribution from 1-way marginals using low-rank tensors and random projections.

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

Researchers describe the metric structure of compact ECS manifolds.

The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.