Real-normalized differentials and geometry of the moduli space of Riemann surfaces -- Igor Krichever, February 10, 2012

The widely accepted by experts, but still conjectural, "geometric explanation" of curious vanishing properties of the moduli space Mg,k of smooth genus g algebraic curves with punctures is the existence of its stratification by a certain number of affine strata or the existence of a cover of Mg,k by a certain number of open affine sets.

Recently, the author jointly with S. Grushevsky proposed an alternative approach for geometrical explanation of the vanishing properties of Mg,k motivated by certain constructions of the Whitham perturbation theory of integrable systems. These constructions has already found their applications in topological quantum field theories (WDVV equations) and N=2 supersymmetric gauge theories. In the talk I'll present the key ideas of this approach and its latest applications: the proof of Arbarello's conjecture and a new upper bound for the dimension of complete complex subvarieties in the moduli space of stable curve of compact type.