For each compact almost Kahler manifold (X,ω, J) and an element A of H2(X ;Z), we describe a 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). We als...

It is well known that relations in the tautological ring of moduli spaces of pointed stable curves give partial differential equations for Gromov-Witten invariants of compact symplectic manifolds. These equations do not depend on the target symplectic manifolds and therefore are called universal equations for Gromov-Witten invariants. In the case that the quantum cohomology of the symplectic ma...

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 trees. This implies that their Gromov boundaries, defined at density less than 1 2 , are Menger curves. MSC: 20F65; 20F67; 20E08

Given two pointed Gromov hyperbolic metric spaces (Xi, di, zi), i = 1, 2, and ∆ ∈ R+0 , we present a construction method, which yields another Gromov hyperbolic metric space Y∆ = Y∆((X1, d1, z1), (X2, d2, z2)). Moreover, it is shown that once (Xi, di) is roughly geodesic, i = 1, 2, then there exists a ∆′ ≥ 0 such that Y∆ also is roughly geodesic for all ∆ ≥ ∆ ′.

We prove that the Gromov-Witten moduli space of a compact symplectic manifold carries a unique virtual fundamental class that satisfies certain naturality conditions. The theorem also applies to moduli spaces of relative J-holomorphic maps. The virtual fundamental class is constructed using only Gromov-type perturbations; it is based on introducing stabilizing divisors and systematically applyi...

We prove that the Gromov-Witten moduli space of a compact symplectic manifold carries a unique virtual fundamental class that satisfies certain naturality conditions. The theorem also applies to moduli spaces of relative J-holomorphic maps. The virtual fundamental class is constructed using only Gromov-type perturbations by introducing stabilizing divisors and systematically applying the natura...

We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2] for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is E × S where E is a smooth elliptic curve and S is a smooth surface with numerically trivial canonical class.

The concept of intuitionistic fuzzy β-almost compactness and intuitionistic fuzzy β-nearly compactness in intuitionistic fuzzy topological spaces is introduced and studied. Besides giving characterizations of these spaces, we study some of their properties. Also, we investigate the behavior of intuitionistic fuzzy β-compactness, intuitionistic fuzzy β-almost compactness, and intuitionistic fuzz...