Context: I don't know much about Mirzakhani's work, but I'm a PhD student in algebraic geometry and I studied Hodge theory and complex manifolds.

> These smooth surfaces, not obviously connected to algebraic geometry

According to Wikipedia:

> Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".

But Riemann surfaces are the simplest complex manifolds, and the compact ones are algebraic curves. I don't see how the connection to algebraic geometry is "not obvious". Is there something deeper going on? I would like to learn a bit more :)