Constrained Changes in Approximate Symmetries of a Turing Pattern
Symmetry is ubiquitous in biology and is an important predictor of survivability, fecundity, and evolvability. Although biological symmetry is often intuitive, it doesn’t fit idealized definitions of symmetry that are used in mathematics. Transformation Information introduced by Gandhi et al. (2021) is a useful tool to measure approximate symmetries in biology. I tracked approximate symmetries of a Turing PDE over time, noticing that the way that symmetries can change is constrained. More details can be found in this infographic: Changes in Approximate Symmetries of a Turing Pattern.
The image on the left is a depiction of the steady state of the Turing PDE. The image on the right is the transformation information of the image, which can tell us the approximate symmetries.
Modeling Hematopoiesis with Machine Learning
My research at UCI in Dr. Qing Nie’s Lab group worked towards using machine learning to model cell-fate. In particular, I wanted to discover if a graph neural ODE could mimic a stochastic model of hematopoiesis. My github page contains the code for both (1) creating toy datasets based on a model of hematopoiesis and (2) modeling the toy datasets with a neural ODE and graph neural ODE.
A neural ODE’s fitting to a stochastic model of hematopoiesis
Intro to Fractals, Barnsley’s Collage Theorem, and Image Compression
A fractal leaf defined by just three linear functions
In the linked paper, I introduce fractals. Assuming only knowledge from undergraduate real analysis, we build the theory required to understand Barnsely’s Collage Theorem. The theorem allows us to approximate extremely complex shapes and patterns with only a few functions. In the end of the paper, I briefly discuss how the Collage Theorem can be used for image compression. (Disclaimer: I wrote this as an undergraduate and am by no means an expert on fractals.)
A History of Sphere Packing
The optimal sphere packing in 3 dimensions
In the linked paper, I explain the history of finding an optimal sphere packing in n-dimensions — accessible to undergraduates in mathematics. I provide the proof of the optimal packing in two dimensions. Then I discuss the intuition behind the proof in 3 dimensions (from Thomas Hales); and in 8 and 24 dimensions (from Maryna Viazovska). Again, I am certainly no expert in this topic, but I find it fascinating.