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Alessandro Chiodo - Towards global mirror symmetry (Part 3)


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Alessandro Chiodo - Towards global mirror symmetry (Part 3)

Mirror symmetry is a phenomenon which inspired fundamental progress in a wide range of disciplines in mathe-

matics and physics in the last twenty years; we will review here a number of results going from the enumerative
geometry of curves to homological algebra. These advances justify the introduction of new techniques, which
are interesting in their own right. Among them, Gromov{Witten theory and its variants allow us to provide a
re ned statement of mirror symmetry. Of course this leads to further open questions (despite much e ort and
progress, Gromov{Witten theory remains unknown in high genus for the quintic threefold). In this course, we
will illustrate the natural problem of moving beyond the local mirror symmetry statement and completing a
framework of global mirror symmetry which is gradually taking shape. We will show how the missing piece in
this picture comes unexpectedly from a classical subject in algebraic geometry: the theory of curves with level
structures.
 

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