4 ms·

Hodge theory is both beautiful and extremely powerful; it was used in ingenious ways by another prominent mathematician, the Fields medalist Maryam Mirzakhani,

Hodge theory is both beautiful and extremely powerful; it was used in ingenious ways by another prominent mathematician, the Fields medalist Maryam Mirzakhani, to study the deformations of flat surfaces. These smooth surfaces, not obviously connected to algebraic geometry, were found woven into a tapestry of rich algebraic structures by work of Eskin, Mirzakhani and Wright. The Hodge conjecture has a similar philosophy: you define certain data in terms of "harmonic analysis" on the geometric object, and the conjecture says that they all correspond to the algebraic data out of which the object can be rebuilt.

red_trumpet 3y ago
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 :)
- tacomonstrous 3y ago
  Riemann surfaces are algebraic, but their moduli can mean two different things: a) the (algebraic) moduli space of complex Riemann surfaces of a fixed genus (number of donut holes). This is an object in the realm of algebraic geometry in that it is possible to embed it into complex projective space, using for instance theta functions. b) the moduli of complex structures on a fixed topological surface of genus g: this is Teichmuller space, which as far as we can tell only an object in differential geometry. However, Eskin, Mirzakhani, Filip and others have discovered that various subspaces of this non-algebraic space are 'naturally' algebraic (or more precisely quasi- projective). This is the surprising part.

red_trumpet 3y ago

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 :)

tacomonstrous 3y ago
Riemann surfaces are algebraic, but their moduli can mean two different things: a) the (algebraic) moduli space of complex Riemann surfaces of a fixed genus (number of donut holes). This is an object in the realm of algebraic geometry in that it is possible to embed it into complex projective space, using for instance theta functions. b) the moduli of complex structures on a fixed topological surface of genus g: this is Teichmuller space, which as far as we can tell only an object in differential geometry. However, Eskin, Mirzakhani, Filip and others have discovered that various subspaces of this non-algebraic space are 'naturally' algebraic (or more precisely quasi- projective). This is the surprising part.