I am a Research Assistant for the Optimization and Algorithmic Theory (OAT) Lab, a grant-funded research endeavor in theory and application of optimization algorithms. Aside from the specific technical aspects previously listed, my work revolves around projection-free optimization methods to find numerically stable and fast algorithms for finding solutions to constrained optimization problems.

The information bottleneck characterizes the tradeoff between information compression and learning. It has notable uses in neural network training and, more recently, a potential ability to explain the generalization abilities of deep neural networks (DNNs). This is a proprietary project which explores a new view of a time-dependent tradeoff parameter for time-series learning, neural network training, and generalization guarantees through PAC-Bayes bounds. This project is under my work as Lead Quantitative Researcher At the Madison Institute for Mathematical Finance.

Perlin noise is a gradient noise algorithm used to generate smooth, seamless transitions across a plane. While typically used for virtual terrain generation, layering Perlin noise generations on top of one another (called octaves) approximates fractional Brownian motion (fBm). This project focuses on the computational methods for calculating the dimension of such processes, with potential implications to come. In the GitHub Repository linked below, you can find code to generate perlin noise with any fractal dimension between 2 and 3.

This project is slightly different from the rest; it is centered around the idea that visualizing advanced mathematics can assist students in learning most topics. This project was programmed in Manim, a Python library developed by Grant Sanderson (3Blue1Brown), and includes 30+ minutes of tailored animations for students learning linear algebra, real analysis, probability, stochastic processes, and multivariate calculus.

I've always been a player of chess at some level. Exploring move sequences and tactical positions with friends was a rewarding past-time. However, the purpose of this project was to challenge myself and push my ability to write highly optimized code. This project uses no libraries and no languages outside of Python; the game engine, neural network architecture, training process, pruning logic, and game algorithms are all hand-crafted.

In particular, Python integers are arbitrary-precision (they never overflow). Standard chess engine magic numbers, including those attributed to Pradyumna Kannan, are discovered by random search under C's implicit 64-bit multiplication overflow, and silently fail for certain squares when ported to Python. This project contains what appears to be the first published set of magic numbers found and validated under non-overflowing arithmetic, verified by perft testing against known node counts.

Combining my experience with linear algebra and my desire to master C++, I created a linear algebra library in C++. This includes simple computations such as inner products, L2 norms, and matrix products; it also includes efficient methods to compute eigenvectors/eigenvalues, decompositions (spectal, SVD, LU), and more. Most importantly, this project focuses on efficient storage of data.

Topological Data Analysis (TDA) is an explored area, but it still has plenty of benefits in nearly every industry. This project is an exploration for my own knowledge of the inner workings of persistent homology, a tool ubiquitous with TDA.