Seminar

Untangling unknots has been a fun pastime since at least the Victorian era. (And there are some great phone games as well, now!) But now AI can play, and it turns out to be surprisingly bad at the game. In this talk, we talk about some ways to look at knot and unknot puzzles mathematically and computationally and talk about why you might seriously want to solve a lot of them in a short period of time. The key turns out to be new and interesting ways to find and visualize moves that you can make on the diagrams. 

For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics.  In this talk I'd like to show you some of those images and explain some of the mathematics they illustrate.

We develop the idea that we can learn about special relativity, not through physics, but through a relativistic dot product. It's the ordinary dot product, but with a minus sign stuck in front of the final term. What difference could that make? You will see nonzero vectors of length zero, events that are both before and after other events, depending on who is asking, and even the twin paradox, where two travelers start and end at the same coordinates in spacetime, but one is actually older. We illustrate all this with mathematical art, as in the attached image. The pearls at the ends of curves are places where various travelers could arrive in spacetime, having traveled paths that are indistinguishable in any physical way!

I'll share some personal experience using mathematical illustration as an educational tool with art & design students in a course about math and papercraft at The Fashion Institute of Technology. Some topics included are flat foldable origami, pop up cards, dissections, and flexagons.

We survey what we know about the dynamics of billiards in rational prisms, that is, right prisms over rational polygons. We will discuss how to use beautiful ideas of Furstenberg and Veech to make connections between mixing properties of billiards in polygons to ergodic properties of billiards in prisms. There will be lots of examples, and lots of pictures, and no prior knowledge of ergodic theory is needed. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

The exponential map is one of the most common functions in many areas of mathematics. However, when it comes to illustration, it is often a source of struggle. Even when we draw its graph, we have only a limited window, which makes it difficult to grasp all the features of this function. When we represent fractals, we very quickly create a mass of details that we are unable to render properly. In computations, it can be a source of numerical errors due to the limited precision of floating-point numbers, etc. In this talk, I will present some attempts (failures?) of mathematical illustration where, to my own misfortune, I came across the exponential function. 

How do you visualize something which doesn't fit on a page, like a high dimensional space or an entire field of math? You lie and cheat, use shortcuts and shorthand.  In this talk, I'll share my favorite tips and tricks for bringing math to life in digital painting.

The Illustrating Mathematics Seminar Online celebrates the life and work of Roger Antonsen.

The theoretical discovery of hyperbolic geometry first got its actual tactile example in 1868 when Eugenio Beltrami created a negatively curved surface from paper annuli and named it a pseudosphere. Later the name pseudosphere got attached to a surface created by a tractrix rotating around its axis. However, mathematicians found more useful for theoretical purposes using different, non-tactile models such as Klein or Poincare disc models or half-plane model. Those are traditionally used in college textbooks. However, to experience deeper understanding of hyperbolic geometry, these models were not enough for Bill Thurston when he was a college student. Since in 1901 Hilbert proved that hyperbolic plane cannot be described analytically in 3-space, Thurston together with his peers at informal seminar decided to make a tactile model of hyperbolic plane and created it by gluing together paper annuli without knowing about Beltrami’s paper model created hundred years earlier. I learned about Thurston’s model in 1997 and decided to make it more durable by crocheting it. Crocheted hyperbolic planes have turned out to be a useful tool in tactile explorations of hyperbolic geometry giving to theoretical knowledge a different perspective. 
 

I'll tell a few mathematical stories from my personal experience with mathematical illustration as a research tool in number theory, sharing some of my experience with the process, not just results.  Topics include Apollonian circle packings, Mobius transformations, continued fractions, and algebraic integers.