Embedded Files
If a picture is worth 1000 words...
5tiles (wider).avi... then how much is an animation worth?
Ideas vs Proofs: While formal proofs are necessary for a complete and precise argument, they are incredibly inefficient for spreading ideas. A well chosen example presented in a visually pleasing way can spread the beauty of one's research to a wider audience, and make the overall behavior of an object much easier to understand. Furthermore, a picture or animation doesn't have to be professional quality to be vastly superior to a wordy description.
Toggle Groups.aviFor the 2022 FPSAC, each group presenting a poster was required to submit a 1 minute preview video with no other guidelines. I challenged myself to show the idea for our topic without using words. This restriction helped me discover a "growing and shrinking disk" perspective to demonstrate the rules for sequentially toggling independent sets on a cycle graph. Audio cues also help draw a link between the toggling operation and the orbit table which keeps track of when nodes are live. The repetition in the audio also suggests the importance of the period of the table. Finally, the appearance of snakes at the end of the video hint at the structure of the table. Even though this video does not state any results, it is meant to spread ideas and inspire curiosity.
Making Talks Easier: Designing effective visuals is certainly challenging and time-consuming. However, a good visual aid makes the task of effectively presenting one's work significantly easier. Before I started working with animations, I found it almost impossible to effectively communicate my ideas, especially when many talks last 30 minutes or less. This became much easier after I adopted a new talk philosophy: I take what I consider the most beautiful aspect of the research and then find a way to show it to people. After this change in perspective, even a one minute presentation becomes possible to do effectively, and a 30 or 50 minute talk is able to go much deeper without losing the audience.
Animation.movHere is a periodic tiling of n-space that I constructed. This tiling is used to associate elements of the sandpile group of a particular matrix to bases of the associated arithmetic matroid. I made the animation using Blender. I also have a 3D print of this tile thanks to Mariel and Carlo Supina! Here's a blog post on how it was made.
Defeating the AI Overlords: As automated proof systems get more powerful, there is speculation that aspects of the job of mathematicians will become obsolete. Computers are already excellent for reducing the burden of tedious calculations, but over time they have become more and more effective at determining which calculations can be used to prove or disprove results. However, while computers are well-suited for precise calculations, I suspect that they will lag behind humans for the foreseeable future in terms of sharing ideas. One way to prepare for a future paradigm shift is to become familiar with an automated theorem prover such as Lean. Otherwise, our best hope in staying relevant is to focus on what humans do best by sharing beautiful ideas in the most elegant way possible.
Contraction(clip).aviThis is a clip from a longer video made up of visual aids for my talk on sandpile torsors. Ever since I first learned about rotor-routing, I wished that there was an animation I could watch to help clarify the action. I hope that these visuals will inspire many more discoveries on the topic!
Building a Deeper Understanding: Making a good visual aid can take a lot of time, but I find that it inevitably gives me a deeper understanding of a topic and many new ideas to think about. After making the previous animation above, I thought of alternative ways to frame the main ideas in order to reach a more general group of mathematicians. The way sandpile torsors are typically introduced requires understanding of something called the sandpile group as a prerequisite, something that takes quite some time to cover. With the animation below, I put spanning trees on center stage, and phrase the main operation in terms of mathematicians looking for coffee. This not only gives a shortcut to understand sandpile torsors, but also an alternate motivation for studying the sandpile group.
Software Recommendations:
SageMath is a free open-source Python-based programming language which is popular among mathematicians. I use it to create most of the objects in my animations.
VSDC is a free video editing software (with a $20 a year pro version). It is incredibly versatile but has a bit of a learning curve. It is excellent for combining objects, text, and/or video, and is what I use to turn objects into a full video.
Blender is a free open-source computer animation software that is popular among professional animators. Its potential to make beautiful 3D animations is unmatched, but it takes some time to get used to.
Manim is a community maintained python library specifically for mathematical animations that was created by Grant Sanderson of 3Blue1Brown. I never worked with it since I want to develop my own style, but videos using it tend to look incredible.
Notability is a free iPad note app which also allows for an appealing alternative to a traditional slideshow.
Not all useful visual aids need to be animations! Here is a slide from a talk I gave on a multijections of cokernels. I completely remade this talk after deciding to focus on tilings I constructed instead of my initial chip-firing inspiration. This decision was partly motivated by the fact that I was able to generalize these tilings to contexts unrelated to chip-firing, and I think it lead to a more captivating talk. Here are the original slides for comparison.
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