Mathematics

Calabi-Yau manifolds play a central role in algebraic/differential
geometry and mathematical physics.
Log Calabi-Yau manifolds with maximal boundary form a class of
noncompact examples with especially nice properties. The course aims
at presenting an introduction to the theory of log Calabi-Yau
surfaces, their moduli, and their mirrors following Looijenga,
Friedman and Gross-Hacking-Keel, with some applications to
mathematical physics. It could serve as a second graduate course in
algebraic geometry.

Sketch of the course:
1) Some preliminary results from the theory of algebraic surfaces.
2) Affine log Calabi-Yau surfaces and their moduli.
3) Construction of mirrors in the affine case in the sense of
Gross-Hacking-Keel.
4) Some connections to cluster varieties.
5) Some applications to mathematical physics (e.g. the role of
Jeffrey-Kirwan residues in mirror symmetry).

References:
[1] Beauville, Arnaud: Complex algebraic surfaces. Cambridge University Press, Cambridge, 1996.
[2] Gross, Mark; Hacking, Paul; Keel, Sean: Moduli of surfaces with an anti-canonical cycle. Compos. Math. 151 (2015), no. 2, 265–291.
[3] Gross, Mark; Hacking, Paul; Keel, Sean: Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168.
[4] Gross, Mark; Hacking, Paul; Keel, Sean: Birational geometry of clusteralgebras. Algebr. Geom. 2 (2015), no. 2, 137–175.