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Explore wild Hurwitz moduli spaces and level structures in this advanced mathematics seminar lecture from the Institute for Advanced Study. Delve into the classical theory of Hurwitz moduli spaces of covers of curves of degree d when d! is invertible, then examine the challenging wild case where wild ramification phenomena occur. Begin with the foundational work of Abramovich and Oort on the classical space H_{2,1,0,4} of double covers of P^1 ramified at four points, following Kontsevich and Pandariphande's approach to describe its schematic closure H in the space of stable maps over Z. Analyze the strange yet informative results over F_2 and understand why they lacked a modular interpretation. Learn about Hippold's work in progress constructing a logarithmic modular version of Hurwitz space of degree p when only (p-1)! is invertible, which provides conceptual explanations for the phenomena observed by Abramovich-Oort. Discover how these same ideas lead to refinements of wild level structures of Drinfeld and enable construction of modular interpretations of minimal modifications of curves X(p^n) that separate ordinary branches at supersingular points. Examine the connection between H as the blowing up of the modular curve X(2) and its broader implications for understanding wild ramification in algebraic geometry.
