Geometric Laplacians on Self-Conformal Fractal Curves in the Plane

Explore geometric Laplacians on self-conformal fractal curves in the plane through this 51-minute talk by Naotaka Kajino from Kyoto University, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the speaker's ongoing research on constructing a family of Laplacians whose heat kernels and eigenvalue asymptotics respect the fractal nature of the curve's Euclidean geometry. Discover how this work extends from previous studies on circle packing fractals, where a Dirichlet form was explicitly defined using a weighted sum of standard one-dimensional Dirichlet forms on constituent circles. Learn about the uniqueness of this form for classical Apollonian gaskets and its satisfaction of Weyl's eigenvalue asymptotics. Understand the key aspects of constructing Laplacians for self-conformal fractal curves, including the use of harmonic measure in defining the Dirichlet form and fractional-order Besov seminorms for the L^2-inner product. Gain insights into this extension of geometric analysis to non-circle packing self-conformal fractals, advancing the field of fractal geometry and analysis.