** Title: **
Mirror Symmetry partners via vanishing cycles

**Abstract:**
In a joint work with Mark Gross and Ludmil Katzarkov, we propose a
construction for mirror symmetry partners which works for any Kodaira
dimension, e.g. this allows us to produce the mirror of a complex
curve of any genus. In general, the mirror will be very singular and
equipped with a sheaf of vanishing cycles. We associate Hodge numbers
to the singular space with this sheaf and show that these fulfil the
mirror duality h^{p,q}(X) = h^{d-p,q}(Y) for X,Y a mirror pair of our
construction. In this talk, I intend to introduce the construction
and demonstrate a few examples.