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Explore the geometric behavior of multisummable series in this mathematical lecture from Harvard CMSA's "Geometry Meets Physics: Finiteness, Tameness, and Complexity" series. Delve into sophisticated summability theories that transform divergent series solutions of differential or functional equations into holomorphic solutions within sector-like domains. Learn about the groundbreaking work of Van den Dries and Speissegger, who demonstrated that functions derived from real multisummable power series exhibit tame geometric behavior when restricted to real numbers. Discover recent findings by the speaker and Speissegger revealing that these functions are generally tame only on portions of their domains, contrary to the desired property of complete domain tameness. Examine specific examples including the Stirling series from the asymptotic expansion of the Gamma function, and understand how tameness is defined through o-minimal structures. Gain insights into the intersection of mathematical analysis, differential equations, and geometric complexity theory through this advanced mathematical discourse.
