It is a very good time to be a mathematician. This millennium, while only a teenager, has already seen spectacular breakthroughs on problems like the Poincar´e Conjecture (solved by Grisha Perelman, who declined both a Fields Medal and a million dollar Clay Prize) and the near-resolution of the Twin Primes Conjecture (by Yitang Zhang). Much of the time, a breakthrough occurs as the result of an unexpected synthesis of two (or more) fi for example, Perelman’s solution connects Topology to Geometry, and proceeds through Dynamical Systems and Partial Differential Equations by understanding certain “flows,” analogous to the heat equation.

This session will feature a number of such unexpected connections, most notably the interaction between the most rigid and discrete of objects – the whole numbers – and the most wild and chaotic: fractals (think the Mandelbrot set, which still graces many a screensaver). One of many connections between these two fields comes from studying so-called “Apollonian circle packings.” We will fi describe what these geometric fractal-like objects are and how they’re constructed, before discussing what they have to do with number theory, geometry, dynamics, differential equations, and even piles of sand.


https://vimeo.com/204833877