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.

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