1. Math and Art in Prison, a Collaborative Effort Across the Ocean - 2. A smooth 1-manifold with many, many, many, many folds

1. A story of inclusiveness and collaboration through mathematics and art.
A inmate of a North American Jail taught himself mathematics, then went on to study more and more and contacted the second author about one of her papers. The contact was made possible by the papers that Aplimat allows to store online. He then planned and realized together with other inmates, a piece of art reproducing a medieval pattern as studied mathematically. As the art is modular, several other inmates could collaborate. An interesting issue arose about the expansion of errors.

2. (in collaboration with C. Falcolini) In 1972 Sheldon Newhouse constructed a dynamical system in the real plane, showing that infinitely many attracting periodic orbits can be contained in a compact set. It is a quite counterintuitive situation and it contradicted widely held conjectures. Moreover in the following years he also proved this property is ubiquitous in a space of maps, thus prompting the unpostponable need to update personal and collective intuition... Each of these periodic points attracts an open neighborhood, and there are infinitely many in a compact region. Mathematicians in dynamical systems do rely on visual representation and intuition to proceed, but they do not seem to share a repertoire of images to pilot and modulate their intuition. Certainly, not in this case.
One wonders how this horde of "sinks" manage to coexist and interact with each other in a limited space.
On our road to understanding some of the features of such systems, we relied on graphic visualization, obtaining pictures that people tell us are beautiful to look at. We lived with them for years.
We will show images picturing some actual coexistence and the corresponding organization of basins of attraction.
The most beautiful image, we believe, is the graphic animation of a smooth curve that we have been studying for years. As a parameter is varied, this curve creates successive weaves and weaves of weaves, to accomodate and harbor the sinks, building a co-housing and welcoming them home.
...welcome home, as many as you deem to be.