From the work of Lian, Liu, and Yau on" Mirror Principle", in the explicit computation of the Euler data Q=\{Q_0, Q_1,...\} for an equivariant concavex bundle Q=\{Q_0, Q_1,...\} over a toric manifold, there are two places the structure of the bundle comes into play:(1) the multiplicative characteric class Q=\{Q_0, Q_1,...\} of Q=\{Q_0, Q_1,...\} one starts with, and (2) the splitting type of Q=\{Q_0, Q_1,...\} . Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of Q=\{Q_0, Q_1,...\} to the classifying data of Kaneyama. From these relations, we compute the splitting type of a couple of nonsplittable equivariant vector bundles over toric manifolds that may be of interest to string theory and mirror symmetry. A code in Mathematica that carries out the computation of some of these examples is attached.

This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.

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Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given positive integer n? We give an affirmative answer to the question when n=9 and 18 (the smallest open cases), and a negative answer for infinitely many n's.

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.