Explore the intricacies of ellipses with small eccentricity in this analysis seminar presented by Steven Morris Zelditch from Northwestern University. Delve into the fascinating world of Dirichlet and Neumann spectra as they relate to these elliptical shapes. Begin with an introduction to the well-known problem, then progress through a comprehensive proof involving key definitions. Examine caustic billiard tables, periodic orbits, and nearly circular domains. Investigate rational integrals, dynamical results, and spectral outcomes. Analyze the length spectrum, band gap structure, and wave trace. Review the loop length function and Q bounce before concluding with a theorem addressing the perennial problem in this field. Gain valuable insights into mathematical analysis and geometry during this 1-hour and 15-minute lecture hosted by the Institute for Advanced Study.