Non-commutative Hodge structures

The talk will give a survey of recent results on a generalization of
the notion of a Hodge structure. The main example is related to the
Fourier-Laplace transform of a variation of polarizable Hodge
structure on the punctured affine line, like the Gauss-Manin systems
of a proper or tame algebraic function on a smooth quasi-projective
variety. Variations of non-commutative Hodge structures often occur
on the tangent bundle of Frobenius manifolds, giving rise to a tt*
geometry.