相关论文: Geometry of Carnot--Carath\'{e}odory Spaces, Diffe…

Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The…

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carath\'{e}odory distance. We show that the uniform…

In $\mathbb{R}^3$ we consider the vector fields \[ X_1 =\frac{ \partial }{\partial x},\qquad X_2 =\frac{ \partial }{\partial y}+ |x|^\alpha \frac{ \partial }{\partial z}, \] where $\alpha\in\left[1,+\infty\right[$. Let $\mathbb{R}^3_+…

In this paper, we study the differential geometry of null Cartan curves under the similarity transformations in the Minkowski space-time. Besides, we extend the fundamental theorem for a null Cartan curve according to a similarity motion.…

These notes grew out of an expose on M. Gromov's paper "Convex sets and K\"ahler manifolds'' ("Advances in Differential Geometry and Topology,'' World Scientific, 1990) at the DMV-Seminar on "Combinatorical Convex Geometry and Toric…

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath\'eodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a…

The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to an isometry and endowed with the Gromov-Hausdorff metric. The main goal is…

The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where…

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete…

This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of…

In the setting of Carnot groups, we prove the $q-$Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carath\'eodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot…

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale…

We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an…

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent…

The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient…

The purpose of this paper is twofold: 1. we prove the triangulability of smooth orbifolds with corners, generalizing the same statement for orbifolds. 2. based on 1, we propose a new homology theory. We call it geometric homology theory…

Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this…

We formulate and prove a generalized Albanese property for families of maps from a smooth curve over an arbitrary field into a commutative group stack. Our proof, which is mostly self-contained, employs local-to-global techniques and some…

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this…

We prove a Carath\'eodory-type extension of BQS homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carath\'eodory-type extension of geometric…