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Mathematics Colloquium


Date:  Tuesday, February 14, 2012

Title:  Real-normalized differentials and geometry of the moduli space of Riemann surfaces

Abstract:  Widely accepted by experts, but still conjectural, "geometric explanation" of curious vanishing properties of the moduli space ${\cal M}_{g,k}$ of smooth genus $g$ algebraic curves with punctures is the existence of its stratification by certain number of affine strata or the existence of a cover of ${\cal M}_{g,k}$ by 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 $M_{g,k}$ 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 the new upper bound for the dimension of complete complex subvarieties in the moduli space of stable curve of compact type.

Speaker:  Igor Krichever
Institution:  Columbia University


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