Number Theory Through Geometry, Dynamics and Illustration - Lecture 3

Explore number theory through a dynamic lens in this third lecture of a series on geometry, dynamics, and illustration. Delve into continued fractions, Diophantine approximation, and Apollonian circle packings, emphasizing both theoretical concepts and experimental tools. Experience the crucial role of visualization in mathematical research as the lecture presents background material with a focus on visual and dynamical aspects. Examine continued fractions, quadratic forms, and Diophantine approximation, followed by an exploration of hyperbolic geometry, Minkowski space, and Kleinian groups. Investigate current research areas, including the geometry of Diophantine approximation and continued fractions in the complex plane, with a focus on algebraic starscapes and Schmidt arrangements. Discover the surprising relationships between Apollonian circle packings and the preceding topics. Recorded during the thematic meeting "Renormalization and Visualization for packing, Billiard and Surfaces" at the Centre International de Rencontres Mathématiques in Marseille, France, this lecture offers a rich, visual approach to advanced mathematical concepts.