arXiv research

New partial orders on positive-definite matrices derived from geometry.

This paper studies gradient flows for sampling using various metrics and their affine invariance.

Solves Minkowski problem for affine invariant convex domains.

New samplers improve MCMC efficiency in high dimensions.

We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearl…

Classifies non-arithmetic orbifolds in specific hyperbolic spaces.

Study shows dynamics of rank 1 orbifolds in flat surfaces.

Study equi-affine invariants for convex domains with asymptotes.

Study examines causal properties of Finsler spacetimes with cone Killing vectors.

We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independe…

This research studies affine invariance in continuous-domain convolutional neural networks.

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…

New metrics for SPD matrices explore affine invariance and symmetry principles.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

The paper shows measures equidistribute on affine submanifolds with a rate.

The paper studies semi-Riemannian cones and their geometric properties.

Cone structures in quantum field theory linked to information geometry.

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

The paper studies a specific centro-affine invariant hypersurface flow in R^(n+1).

Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a split…

The paper defines general-affine invariants for plane and space curves.

Proves openness of balanced HKT cone and studies hyperholomorphic vector fields.

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extre…

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

Employing a centro-affine flow on smooth convex bodies, we generate new centro-affine differential invariants. One class of the newly defined invariants is the object of a sharp isoperimetric inequality, while other new inequalities on known centro-affine invariants are obtained as a byproduct of the flow's study. Furt…

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove fini…

New examples of Calabi-Yau metrics on cones with irregular smooth links.

Smooth approximations of causal functions derived from Lyapunov.

Extends gauge conditions for superparticle to conic neighbourhood.

New calculus for invariant differential operators in parabolic geometries.

New geometric structures defined on SPD matrices for better understanding.

Unique soliton found on resolved cones.

Bayesian inference for neural networks improves uncertainty quantification.

This paper explores gradient flows for sampling distributions without normalization constants.

New estimator for covariance of heavy-tailed data with affine-invariant bound.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

The paper explores totally geodesic submanifolds in SPD matrices and their properties.

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$, where Aff(n) denotes the group of all nonsingular affine maps of $\mathbb R^n$. Fo…

Study shows instability of naked singularities in scalar field models.

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of …

The paper matches features in images using centro-affine invariants and heat flow.

New approach to convexity and monotonicity on metric spaces.

Wave propagation framework using cone structures and observers' vector fields.

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometr…

Holomorphic tensors on algebraic cones are invariant under certain group actions.

Study of cone structures and Finsler metrics linking geometric and physical aspects.