Here are some suggestions and references. For the second and the third topics, I am looking for students who have some experience in ML and who can use developer tools like GitHub and VS Code.
Logics Introduction to logics and Gödel's incompleteness theorem
We start from learning basic notion of set theory and move on to introduction to logics/model theory. Our goal is to understand Gödel's incompleteness theorem. If we have time, we will go through the formalizaiton of mathematical concepts using Lean.
References: Set Theory, Logics, Lean Tutorial
Applied Math Grokking
Grokking in machine interpretability is a phenomenon where a machine learning model, after a long period of simply memorizing the training data, suddenly and rapidly generalizes to understand the underlying patterns and perform well on new, unseen data.
References: Grokking
Applied Math Graph Learning
This topic is inspired by a talk of Professor Krishnagopal(UC Santa Barbara) in Applied Math Seminar at UC Berkeley. Graph learning models train data with their relationships as an interconnected graph. We will learn some concepts in graph learning such as graph signal processing using graphon and simplicial complex framework.
Keywords: Graphon, Neural Tangent Kernel, Spectral Convergence, Simplicial Complexes, Spectral Hodge Theory, Laplacians, Simplical Complex Framework, Controlling Signals, GNN, WNN
References: 2301.10808 1806.07572 2108.06547 2308.14189
We read M. Nakahara’s Geometry, Topology and Physics to learn the mathematical language commonly used in mathematical physics, and A. Yu. Kitaev’s paper Fault-tolerant quantum computation by anyons (Annals of Physics, 303(1):2–30, January 2003) to study the toric code.