Playing Dice with the Universe

About two years ago, around this time when the weather started getting nice, I was hospitalized at Princeton Medical Center—the same place where Albert Einstein died, exactly 70 years ago today.

Einstein had an enormous legacy. To commemorate, I wanted to reflect on how that legacy shaped my own academic interests.

Growing up, I always wanted to be a physicist. I studied special relativity at the age of 12 and was comfortable with calculus and knew the Lorentz transformation. In 2014, when Interstellar was a major hit. I was fascinated by the nature of spacetime. I would copy down the Einstein equation on my science notebook back then without even knowing what a tensor was.

At age 19, I entered Carnegie Mellon. In my first year I was fascinated by computer graphics and geometry. That beautiful language of geomery propelled me to take a course in general relativity from the physics department in my second year. I had an influence from my friends in college who were absolute geometry fanatics—whether it was algebraic geometry, differintal geometry, or discrete geometry.

Further pursuing computer science I encountered the name Einstein in machine learning. It was from the einsum operation, which is the Einstein summation notaiton I learned and used extensively throughout GR. ML also extensively manipulates multidimensional arrays which called tensors: A famous multi-linear map central in geometry.

Last summer, I studied deeper into differential geometry by watching the entire series of Frederic Schuller’s lectures on The Geometric Anatomy of Theoretical Physics. Where at the end it touched on some guage theory through principal bundles and associated bundles. That love for geometry put me on a trajectory towards stochastic analysis. Many variational methods could be found in GR, examples like the Einstein-Hilbert Action. Light follows a geodesic in spacetime. It was fascinating to learn that geodesics minimizes the dirichlet energy and therefore harmonic. This led me to study deepenr into the Laplace-Beltrami operator. Studying this second order operator it had a natural connection with curvature in geometry and led me to study the heat equation and their probabilistic formulations, getting the taste of PDEs and SDEs.

I’ll write more about stochastic analysis later on but this is what led me to my study in diffusion models where Einstein’s legacy lives on in de novo drug design. Incorporating geometry as an inductive bias in models were effecitve from both data and compute perspective.

The natural connection of probability theory and geometry fascinates me. It feels like this elegant bridge that connects the microscopic with the macroscopic, and everything in between. As Einstein also studied the Brownian motion (independently of Bachelier, later formalized by Wiener).

Einstein’s legacy will live on as ML progresses in biopolymer modelling, geoscience, computer graphics and more. Standing long with other notable physicsits like Schrödinger, Feynman, Planck.

** Written on a Friday night at a hotel lobby in Tokyo by Shawn Park. **