Summer of Math Exposition

This video is about how Turing patterns work: how repeated blurring and sharpening produce a Turing Pattern by making a band-pass. I show an alternative method of making a band-pass using two blurs. I look at the reaction-diffusion / Gray-Scott model and where its hidden band-pass is. I cover multi-scale patterns, where several band-passes are made using a cascade of bigger and bigger blurs, and how to efficiently run a large blur using FFT. Bonus: a Turing pattern wavetable, and Turing patterns in 3D.

This video describes a way to hear light, as in, shine a light into your ear and hear music. When I first started this video I actually didn’t realize that this was possible, but as I experimented with various items I discovered it was. There are two key ideas I try to explain in this video: radio transmission and photoacoustic transduction. These two ideas are what lead to the phenomenon I discuss in the video.

Although the video is geared towards a general audience, and there’s little math in terms of equations and such, it is still somewhat technical, so I would say the target audience is undergraduate/graduate students, along with high-school students and teachers/professors. But my hope, of course, is that anyone with a genuine curiosity about science will appreciate the video and understand most of what’s discussed.

How can you randomly place objects in a space and ensure that no two overlap? This blog post is an introduction to Poisson disk sampling, an important idea in computer graphics, and describes some lesser known improvements and extensions of it.

One of the most important theorems in my area of research is the Riemann—Hilbert correspondence. Roughly, it tells you that you can convert differential equations into certain geometric objects, and that this conversion process loses no information. In particular, one can convert questions about differential equations into geometric questions, and conversely one can convert certain geometric questions into problems about differential equations.

We wrote a blog post showing an example of this phenomena in a very simple case. The differential equation in question is very simple: f’(x) = f(x)/2x, and the resulting geometric object is related to the Möbius strip!

We assume the reader knows very basic calculus (what a derivative is and how to differentiate polynomials), but nothing else.

In this video, we investigate mathematical knots, and our guiding question is: How can you tell when two knots are the same or different? In pursuit of an answer, we argue that two knots are the same only when one can be transformed to look exactly like the other using a sequence of three basic moves, called Reidemeister moves. Then, we prove that a quantity called the linking number is an oriented link invariant using the fact that it doesn’t change under any of the Reidemeister moves. Finally, we stumble upon the most famous knot invariant – the Jones polynomial – and show how one could’ve devised it from scratch, along with several applications.

Most of the exposition and practice problems come from The Knot Book by Colin C. Adams. However, we have a more elaborate discussion about why the three Reidemeister moves are all you need, and we made modifications to the definition of the X polynomial where we felt was appropriate.

References: The Knot Book by Colin C. Adams https://katlas.org/wiki/K11n34 https://katlas.org/wiki/K11n42