# Solitons, Contours, and Thinking Sdrawkcab

Welcome to this week’s Math Munch!

Meet Nalini Joshi, a mathematician at the University of Sydney in Australia. I’ll let her introduce herself to you.

Nalini has an amazing story and amazing passion. What does her video make you think? To hear more from Nalini, you can watch this talk she gave last month at the Women in Mathematics conference at the Isaac Newton Institute in Cambridge, England. Her talk is called “Mathematics and life: a personal journey.” You might also enjoy reading this interview or others on her media page.

I’d like to share three clumps of ideas that might give you a flavor for the math that Nalini enjoys doing. Most of it is way over my head, but I’m reaching for it! You can, too, if you try.

Here’s clump number one. Two of the main objects that Nalini studies are dynamical systems and differential equations. You can think of a dynamical system as some objects that interact with each other and evolve over time. Think of the stars that Nalini described in the video, heading toward each other and tugging on each other. Differential equations are one way of describing these interactions in a mathematically precise way. They capture how tiny changes in one amount affect tiny changes in another amount.

Vlasov billiards.

To play around with some simple dynamical systems that can still produce some complex behaviors, check out dynamical-systems.org. Vlasov billiards was new to me. I think it’s really cool. The three-body problem is one of the oldest and most famous dynamical systems, and you can tinker around with examples of it here and here. There’s even a three-body problem game you can try playing. I’m not too crazy about it, but maybe you’ll enjoy it. It certainly gives you a sense for how chaotic the a three-body system can be!

Nalini doesn’t study just any old dynamical systems. She’s particularly interested in ones where the chaotic parts of the system cancel each other out. Remember in the video how she described the stars that go past each other and don’t destroy each other, that are “transparent to each other”? Places where this happens in dynamical systems are called soliton solutions. They’re like steady waves that can pass through each other. Check out these four videos on solitons, each of which gives a different perspective on them. If you’re feeling adventurous, you could try reading this article called What is a Soliton?

 Making a water wave soliton in the Netherlands. A computer animation of interacting solitons. Japanese artist Takashi Suzuki tests a soliton to be used in a piece of performance art. Students studying and building solitons in South Africa.

Level curves that are generalized Cassini curves.
Also, it kind of looks like a four-body problem.
(click for video)

The second idea that Nalini uses that I’d like to share is level curves, or contours. Instead of studying complicated differential equations directly, it’s possible to get at them geometrically by studying families of curves—contours—that are produced by related algebraic equations. They’re just like the lines on a topographic map that mark off areas of equal elevation.

Here’s a blog post by our friend Tim Chartier about colorful contour lines that arise from the differential equation governing heat flow. The temperature maps by Zachary Forest Johnson from a few weeks ago also used contour lines. And I found some great pieces of art that take contours as their inspiration. Click to check these out!

The last idea clump I’ll share involves integrable systems. In an integrable system, it’s possible to uniquely “undo” what has happened—the rules are such that there’s only one possible past that could lead to the present. Most systems don’t work this way—you can’t tell what was in your refrigerator a week ago by looking at it now! Nalini mentions on her research page that “ideas on integrable differential equations also extend to difference equations, and even to extended versions of cellular automata.” I enjoyed reading this article about reversible cellular automata, especially the section about Critters.

What move did Black just play?
A puzzle by Raymond Smullyan.

And this made me think of a really nifty kind of chess puzzle called retrograde analysis—a fancy way of saying “thinking backwards”. Instead of trying to find the best chess move to play next, you instead have to figure out what move was made to get to the position in the puzzle. Most chess positions could be arrived at through multiple moves, but the positions in these puzzles are specially designed so that only one move will work. There’s a huge index of this kind of problem at The Retrograde Analysis Corner, and there are some great starter problems on this page.

Maurice Ashley

And perhaps you’d like to hear a little bit about thinking backwards from one of the greatest teachers of chess, Grandmaster Maurice Ashley. Check out his TED video here.

I hope you’ve enjoyed finding out about Nalini Joshi and the mathematics that she loves. I asked Nalini if she would do a Q&A with us, and she said yes! Do you have a question you’d like to ask her? Send it to us below and we’ll include it in the interview, which I send to Nalini in about a week.

UPDATE: We’re no longer accepting questions for Nalini, because the interview has happened! Check it out!

Bon appetit!