In this companion piece to [9], some variations on the main results there are sketched. In particular, • the recursions in [9], which we interpreted as the quantum Lefschetz, is re- formulated in terms of Givental’s quantization formalism, or equivalently, a summation of finitely many graphs; • varietiesofmodificationoftheauxilliaryspaces(masterspaces)forthefixed point localization are given, leading to different (looking) recursions; • applications of this circle of ideas to derive (apparently) new relations of Gromov–Witten invariants.

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs , the Gosset graph J(2n, n) and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness and classify all distance-regular Lichnerowicz sharp graphs under the additional condition θ_1=b_1-1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature.

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar—possibly infinite—graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group.

We prove that all maximal-tb positive Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m), (3,3), (3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2,Z) and the mapping class group M0,4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms λ_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise numerical bounds on λ_1^* for several hyperbolic rational homology spheres.