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Explore the mathematical foundations of population genetics through this 45-minute conference talk examining the Kimura diffusion equation and its applications in modeling genetic drift, selection, and mutation. Delve into the Wright-Fisher model's infinite population limit and understand how the Kimura diffusion operates on unit intervals for single genes and N-1 simplexes for multiple alleles. Learn about the unique analytical challenges posed by the degenerate nature of the Kimura operator at simplex boundaries, which places it outside standard elliptic and parabolic theory frameworks. Discover recent collaborative research on the analytic properties of this diffusion operator and examine cutting-edge numerical methods that achieve highly accurate solutions even in computationally challenging scenarios. Gain insights into the mathematical complexities of allele frequency spectrum analysis and the theoretical underpinnings that connect pure mathematics to biological population dynamics.
