Determining a Collection of Boundary Points on the p-adic Mandelbrot Set of degree d Polynomials (24rit028)

The Banff International Research Station will host the "Determining a Collection of Boundary Points on the p-adic Mandelbrot Set of degree d Polynomials" workshop in Banff from October 13 - 20, 2024.

The Mandelbrot set is one of the most famous and inspiring images in mathematics. It has an intricate, fractal-like boundary which encodes key information about the dynamics of quadratic polynomials defined over the complex numbers. First defined in 1978, this set has been studied extensively over the past half-century, and much is known about its structure and what it can tell us about the dynamical behavior of quadratic polynomials.

We aim to explore generalizations of the Mandelbrot set for polynomials defined in a different setting, specifically over a $p$-adic field. While $p$-adic numbers are structured very differently from complex numbers, we have evidence of some similarities between the classical complex Mandelbrot set and our newly-defined $p$-adic counterparts. In this project, we will investigate these structures further and build a better understanding of what $p$-adic Mandelbrot sets look like and what they can tell us about dynamical systems.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.