Geometry of oscillating integrals and Dubrovin conjecture

We consider the oscillating integral defined by the polynomial phase function $f(x)$ with non-degenerate singular points. So called "Lefschetz thimble" can be constructed for each singular point of $f(x)$. This integral can be regarded as the Laplace transform of the fibre integral associated to the non-singular variety $f^{-1}(c)$. It turns out (F. Pham) that the intersection indices of vanishing cycles of the variety $f^{-1}(c)$ coincide with those of Lefschetz thimbles (regarded as one dimension higher cycles in a relative homology). We show that the elements of the Stokes matrix defined for the oscillating integral calculate exactly the intersection indices mentioned above.
As an application we shall discuss the question on the Stokes matrix $S$ for the quantum cohomology of a weighted projective space $P$. Namely we shall present a positive answer to the hypothesis proposed by Boris Dubrovin who predicted that the Stokes matrix $S$ coincides with the Gram matrix of the exceptional collection of coherent sheaves on $P$. This is a collaboration with Kazushi Ueda.