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Explore the intersection of polynomial dynamics and Fuchsian groups in this mathematics department colloquium lecture delivered by Sabyasachi Mukherjee from the Tata Institute of Fundamental Research. Discover the connections and philosophical analogies between rational dynamics on the Riemann sphere and actions of Kleinian groups, two fundamental branches of conformal dynamics. Learn how to construct combinations and matings of complex polynomials and Fuchsian groups through iterated algebraic correspondences on compact Riemann surfaces, creating a unified framework for studying these mathematical structures. Examine how these constructions lead to products of Teichmuller spaces of genus zero orbifolds and parameter spaces of polynomials within Hurwitz spaces, which are moduli spaces of ramified covers of the Riemann sphere. Delve into the analytic and algebraic aspects of these mathematical frameworks while considering numerous open questions that emerge from this research area, providing insight into current frontiers in conformal dynamics and algebraic geometry.
