We introduce the notion of the Gromov-Hausdorff fuzzy distance between two non-Archimedean fuzzy metric spaces (in the sense of Kramosil and Michalek). Basic properties involving convergence and the fuzzy version of the completeness theorem are presented. We show that the topological properties induced by the classic Gromov-Hausdorff distance on metric spaces can be deduced from our approach.

The J-function in Gromov-Witten theory is a generating function for one-point genus zero Gromov-Witten invariants with descendants. Here we give formulas for the quantum K-theoretic J-functions of type A flag manifolds. As an application, we prove the quantum K-theoretic J-function version of the abelian-nonabelian correspondence for Grassmannians and products of projective space.

We give an introduction to moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet [55] and Schmitt [61], and the associated integrals giving rise to gauged Gromov-Witten invariants. We survey various applications to cohomological and Ktheoretic Gromov-Witten invariants.

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on simplicial trees. This implies that their Gromov boundaries, defined at density less than 1 2 , are Menger curves. Mathematics Subject Classification (2000) 20F65 · 20F67 · 20E08

It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(−1) space can fail to be either injective or surjective.

This paper wishes to foster communication between mathematicians and physicists working in mirror symmetry and orbifold Gromov-Witten theory. We provide a reader friendly review of the physics computation in [ABK06] that predicts Gromov-Witten invariants of [C/Z3] in arbitrary genus, and of the mathematical framework for expressing these invariants as Hodge integrals. Using geometric properties...

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.

Quantum Lefschetz theorem by Coates and Givental [4] gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semisimplicity and the Virasoro conjecture for the Gromov-Witten theories of (not ...

We study sequences of conformal deformations a smooth closed Riemannian manifold dimension $n$, assuming uniform volume bounds and $L^{n/2}$ on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such underlying metric spaces are pre-compact Gromov-Hausdorff topology. Our is based use $A_\infty$-weights from harmonic analysis, provides geometric cont...

For each compact almost Kahler manifold (X,ω, J) and an element A of H2(X ;Z), we describe a natural closed subspace M 0 1,k(X,A; J) of the moduli space M1,k(X,A; J) of stable J-holomorphic genus-one maps such that M 0 1,k(X,A; J) contains all stable maps with smooth domains. If (P, ω, J0) is the standard complex projective space, M 0 1,k(P , A; J0) is an irreducible component of M1,k(P, A; J0)...