Deformations of functions and F-manifolds

In this thesis we study deformations of functions on singular varieties with a view toward Frobenius manifolds. Chapter 2 is mainly introductory. We prove standard results in deformation theory for which we do not know a suitable reference. We also give a construction of the miniversal deformation of a function on a singular space that to the best of our knowledge does not appear in this form in literature. In Chapter 3 we find a sufficient condition for the dimension of the base space of the miniversal deformation to be equal to the number of critical points into which the original singularity splits. We show that it holds for functions on smoothable and unobstructed curves and for function on isolated complete intersections singularities, unifying under the same argument previously known results. In Chapter 4 we use the previous results to construct a multiplicative structure known as F -manifold on the base space of the miniversal deformation. We relate our construction to the theory of Frobenius manifolds by means of an example: mirrors of weighted projective lines.The appendix is joint work with D. Mond. We study unfolding of composed functions under a suitable deformation category. It also yields an F-manifold structure on the base space, which we use to answer some questions raised by V. Goryunov and V. Zakalyukin on the discriminant on matrix deformations