arXiv research

Survey of geometry developments, including complex structures on surfaces.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

Advances M-polyfolds for complex geometry applications.

The study sets limits on the complexity of Klein geometries.

Quantum complexity lowerbound proved using differential geometry.

Diffeology extends differential geometry to complex spaces.

PINNs solve differential geometry problems in complex shapes.

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential g…

New complexes refine multicomplexes for subRiemannian geometry.

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…

Study of $H$-eigenvalues for complex tensors and their applications in differential geometry.

Studies projective geometry and partial differential equations prolongation.

Characterizes conformal classes of tori using differential geometry.

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker…

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of…

Smooth complex surfaces with triple intersections using differential geometry.

Revisits Koiso's rigid metrics on complex projective spaces.

Script Geometry offers a new approach to discrete differential geometry.

Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and resu…

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.

Study on complex line fields on almost-complex manifolds, proving existence conditions.

New complex-valued maps found on complex geometries.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…

This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…

We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein…

GeoFunFlow tackles inverse problems on complex geometries with efficient learning.

Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.

New theory proves representability of PDE solutions without complex machinery.

Geometrization Theorem solves complex geometry problems.

Paper proves inequalities for forms on sub-Riemannian manifolds.

The paper explores cone structures and their connections to parabolic geometries in complex manifolds.

I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.

Paper proves polynomial equivalence of quantum complexity metrics.

Different topics on the differential geometry of the complex Grassmann manifold are surveyed in relation to the coherent states. A calculation of the tangent conjugate locus and conjugate locus in the complex Grassmann manifold is presented. The proofs use the Jordan's stationary angles. Also various formulas for the d…

Surveying connections between algebraic geometry and surface topology.

The aim of this article is to use generalized complex structures in order to extend the definition of twistor spaces given by Penrose. We will adapt the integrability result of Atiyah, Hitchin and Singer. We will deduce new correspondences betwenn differential geometry and (generalized) complex geometry. In the last se…

Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.

We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function algebras in several variables.

On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its …

New geometries explain solutions to differential equations.

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we o…

We introduce a program aimed to studying problems arising from the theory of complex networks with differential geometric means. We study the propagation of influences on manifolds assuming that at each point only a finite number of propagation velocities are allowed. This leads to the computation of the volume of the …

Defines discrete differential geometry concepts in homotopy type theory.

This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

We show how the machine of PROP profiles invented by S. Merkulov can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natura…