 # Projects

For the most part, math papers are dry and technical documents, written to convey theorems and proofs in an efficient way. Oftentimes, they don't give any indication how results were discovered or how much editing was necessary to get the paper into its final format. One of my goals in math is to break the illusion that research is a polished and streamlined process.

In order to do that, I try to give some insight into the process behind the scenes. For me, doing math is a very human process, filled with messy sketches, rough heuristics. While working on a problem, my desk is filled with doodles and calculations, many of which end up being dead ends.

On this page, I've tried to provide some "semi-polished doodles" in Geogebra, which can be used to reproduce some of my sketches. I've also provided some Mathematica notebooks which can be used to reproduce some of the more involved computations.

• ## Hall's Conjecture

Hall's conjecture is a question put forth by Glen Hall, who was my undergraduate advisor. While working on this conjecture, I made some notebooks in Geogebra and Mathematica, which might be useful to others.

• ## Lecture Notes

Here are some lecture notes for some of my courses and lectures.

• ## Optimal Transport

In optimal transport, a quantity known as the MTW tensor plays an important role in the regularity theory. However, calculating the MTW tensor is very tedious. These are some Mathematica notebooks that can be used to compute the MTW tensor for various different cost functions.

• ## What does the Sasaki metric look like?

In 1958, Sasaki found a way to make the tangent bundle of a Riemannian manifold (with an affine connection) into an almost Hermitian manifold. In order to better understand the geometry of these spaces, I wrote some Geogebra notebooks which helped me to visualize what was going on.