In this post we will see how to extend reverse mode automatic differentiation to a language with first class function types, function application and lambda-abstraction. This method is not new, but we will give a new derivation of it by showing how it arises universally from noticing that the category of “additive lenses” is cartesian closed. In the end we will see that this idea sounds like it should revolutionise machine learning, but then doesn’t.

Some interesting ideas, although I won’t claim that I understand them all.

Circles and hyperbolas are, in a sense, surprisingly similar shapes. In this video, we explore the mystery behind their strange connection and step into a world of “rotation” and “angle” with meanings quite different from the usual ones. The key to this story lies in hyperbolic functions, which share many properties with trigonometric functions.

This video was really helpful in understanding the hyperbolic functions! Was happy to see that a connection I saw to spacetime is actually an application of them.

Ed Copeland goes deep as we explore how we cracked an infinite sum relating to black holes.

Lots of great techniques.

Ed Copeland continues from the main video

Lots of great techniques.

This summer at the Topos Institute, under the supervision of Dr. Sophie Libkind, I studied the composition of attractors. The project itself started earlier with my advisor, Dr. William Kalies, who asked me the following question: how do attractor lattices behave when we combine dynamical systems? In this post, I explain how attractor lattices in decoupled product systems can be characterized algebraically in terms of the lattices of their component systems.

Great way of explaining this classis paradox in analysis using uniform convergence.

In this video, I present the story of phase space and one of the most fundamental theorems of classical physics — Liouville’s theorem. This is a walk through the birth of phase space and how the discovery of Liouville’s theorem involves not only Liouville but also Jacobi and Boltzmann.

This is a playlist covering various topics in Measure Theory

Intro to Measure Theory covering Sigma Algebras, Measures, Measurable Spaces, and Measure Spaces.

Very slick technique.

Such an aesthetic problem.

Zvezdelina Stankova discusses the raffle function - and her epic proof ends with an interesting connection.

Beautiful, beautiful problem. Abstract algebra, calculus, number theory, and combinatorics all wrapped up into a bow. :)

This is how we do math in the 21st century.

The images on this page are created using the standard iterative series of the Mandelbrot, that is, iterate the function zn+1 = zn2 + z0 where z0 is each point in the image plane (complex plane). However, instead of recording the behavior of the series at each point z0 we now consider only those points that escape to infinity and we create a density plot of the terms in the series. The result then is a 2D density plot of the trajectories that escape to infinity. The following shows the buddhabrot for that part of the complex plane that is interesting.

Beautiful illustrations.

This video singlehandedly helped me understand coverings and Lie algebras way better than any Wikipedia article I’ve ever read. :P

Notes on Benn Stancil’s post in 2021 Tilt and tilted, about the objectivity of data-driven decisions.