Abstract. We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a rational projective surface. This characterization is based on the singularity analysis of the product of evaluation maps and deformation properties of the pointed stable maps. If a target space is equipped with a real structure, i.e, anti-holomorphic involution, then this transversality result has a real enumerative implication. One of the natural consequence of this result is Welschinger’s invariant in an algebraic geometry category, which is about the global minimum bound of the number of real stable maps passing through certain real points in a general position on the target space. 1.

Abstract. We show that for each n ≥ 2 there are 1 n22n−2 ()

We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil’s geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.

We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ ◦ g is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics. One of the many equivalent formulations of the Shapiro conjecture is the following. Let f =(f1,...,fp) be a vector of polynomials in one complex variable, and assume that the Wronski determinant W(f) =W(f1,...,fp) has only real roots. Then there exists a matrix A ∈ GL(p,C) such that fA is a vector of real polynomials. This conjecture plays an important role in real enumerative geometry [15, 16], theory of real algebraic curves [8] and has applications to control theory [10, 4]. There is a substantial numerical evidence [16] in favor of the

Real Gromov-Witten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space

Abstract. We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space. 1.

Abstract. We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We introduce the real version of the Gromov-Witten theory proposed by Gang Tian. 1.