Daniel Williams will present his General Exam "Daniel Williams will present his General Exam "An Efficient and Scalable Language Design for High-Dimensional Clifford Algebras on GPU" on Wednesday, April 22, 2026 at 3:00 PM in E-Quad J223 and via zoom. Zoom link: https://princeton.zoom.us/j/6762826001 Committee Members: Ryan Adams (advisor), David Walker, Christine Allen-Blanchette Abstract: Many of the most challenging problems in computational physics and engineering, ranging from electromagnetism to inverse design of complex mechanical systems, are fundamentally geometric. However, conventional linear algebra based numerical methods do not natively encode geometric structure, often leading to ill-conditioned optimization, constraint drift, and poor scaling in high-dimensional settings. While Clifford Algebras provide a principled and coordinate-free mathematical framework for representing symmetries, high dimensional spaces, and differential geometry on curved manifolds, its practical adoption in large-scale scientific computing has been limited by exponential memory requirements and lack of GPU-accelerated infrastructure. This work introduces a new framework for accelerated computational geometry in high dimensions using Clifford Algebras. We develop a static type system that automatically infers and reduces memory constraints, allowing numerical execution to operate on dramatically reduced representations without approximation. Building on this foundation, we implement a Python-based Just-in-Time (JIT) compiler integrated with JAX, the accelerated array computation library, that traces user-defined geometry programs of various clifford algebra signatures and lowers them into optimized GPU kernels, while preserving precision and full compatibility with automatic differentiation. Preliminary results demonstrate substantial improvements in tractable dimensionality and GPU throughput under batching, extending feasible computation from n < 6 to up to n = 20 dimensions on a single GPU by exploiting known sparsity before execution. Reading List: https://docs.google.com/document/d/1eSBhk0Irmfqf26QOlY5Jrbb_kUmhOQgN4Jbay6_Z... Everyone is invited to attend the talk, and those faculty wishing to remain for the oral exam following are welcome to do so.