arXiv research

Studies projective geometry and partial differential equations prolongation.

Projective geometry aids in analyzing fields near compact manifolds.

New framework uses elliptic operators to study projective maps.

Study cylindrical symmetric Finsler metrics that are projectively flat.

Paper links set derivatives to its orthogonal projections.

Proof confirms preservation of projective limits in synthetic differential geometry.

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …

Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.

New method detects projective equivalences and symmetries in rational 3D curves.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.

Optimizes differentially private kernel learning with random projection.

This paper extends Markovian projections to semimartingales with jumps.

Two projective structures on Riemann surfaces are described and shown not to be identical.

Linearized Einstein equations simplified via Calabi operator.

The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.

PNDEs project neural dynamics onto constraint manifolds, improving accuracy and stability.

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this …

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan con…

Develops optimal low-dimensional approximations to high-dimensional SDEs.

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our ma…

A method to visualize multidimensional local subspaces using implicit differentiation.

The paper introduces DP algorithms using random projections and sign random projections for improved privacy in machine learning.

We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invariants for analytic hypersurfaces is given. The self-dual strongly C-linearly convex hypersurfaces are characterized.

Paper develops a new method for differential privacy sampling using Wasserstein distance.

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space c…

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion o…

We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan-Shalen compactification of the character variety. We show that the dual R-trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan-Shal…

Bounds projective structure norms by bending lamination lengths.

Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.

Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the real-projective structure as the conformal structure is fixed and the cubic diffe…

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with pres…

New Finsler metrics describe trace function growth rates in convex projective surfaces.

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology.

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformiza…

We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, sho…

Let $M$ be either a projective manifold $(M,Pi)$ or a pseudo-Riemannian manifold $(M,g).$ We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on $M.$ As operators,…

Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the puncture…

This article is a survey of recent work of the authors developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, …

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a min…

We show that the group of smooth homotopy $7$-spheres acts freely on the set of smooth manifold structures on a topological manifold $M$ which is homotopy equivalent to the real projective $7$-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective $7$-space. We also show that…

Maps complex varieties into buildings with harmonic properties.

Classifies and constructs intertwining differential operators between vector bundles over real projective space.

An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.

Study on parabolic points and cylindrical surfaces in Euclidean 3-space.

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.