Maximal Multiplicity of Laplacian Eigenvalues in Negatively Curved Surfaces

Explore a seminar on spectral geometry focusing on the maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces. Delve into the historical context of this mathematical problem, tracing its roots to the 1970s and examining key contributions from researchers like Colin de Verdière, Cheng, and Besson. Learn about recent advancements in the field, including a collaborative study that established the first sublinear upper bound on multiplicity for negatively curved surfaces. Discover how this research combines heat kernel trace arguments with r-net surface control techniques, drawing inspiration from methods used in bounded degree graph analysis. Gain insights into the concept of "approximate multiplicity" and its implications for eigenvalue distribution. Examine how this work sheds new light on Colin de Verdière's conjecture and offers a novel approach to transferring spectral results from graphs to surfaces.