Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V . The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V...

We introduce the notion of Zwiebach invariants that generalize Gromov-Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach on the subbicomplex, that gives the structure of Gromov-Witten invariants on subbicomplex with zero diffferentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach inva...

A. In this note we give new characterizations of metric trees and Gromov hyperbolic spaces and generalize recent results of Chatterji and Niblo. Our results have a purely metric character, however, their proofs involve two classical tools from analysis: Stokes’ formula in R2 and a Rademacher type differentiation theorem for Lipschitz maps. This analytic approach can be used to give chara...

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of r-th root constructions on M0,n. We prove a closed formula for the total Chern class of μreigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C/μr ]. We deduce linear recursions for all genus-zero ...

In this paper we define the notion of (p, δ)–Gromov hyperbolic space where we relax Gromov slimness condition to allow that not all but a positive fraction of all the triangles are δ–slim. Furthermore, we study their traffic congestion under geodesic routing. We also construct a constant degree family of expanders with congestion Θ(n) in contrast to random regular graphs that have congestion O(...

We introduce a new quantum Gromov-Hausdorff distance between C∗-algebraic compact quantum metric spaces. Because it is able to distinguish algebraic structures, this new distance fixes a weakness of Rieffel’s quantum distance. We show that this new quantum distance has properties analogous to the basic properties of the classical Gromov-Hausdorff distance, and we give criteria for when a parame...

In this talk, we will introduce a real analogue of Noether-Lefschetz number which captures the existence of holomorphic discs with boundary on special Lagrangians. We will study the holomorphic discs counting on Calabi-Yau 3-folds with K3 fibration. The open Gromov-Witten invariant on such Calabi-Yau 3-folds (when it is well-defined) can be expressed in terms of reduced open Gromov-Witten invar...

In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the for...