Conformal Field Theory, Cryo-Electron Microscopy, and Neural Networks

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This project applies probability to three separate areas of mathematical physics, statistics, and machine learning. The first two areas are conformal field theory (CFT) and cryo-electron microscopy. The former is instrumental for the theoretical high-energy physics, while for the latter recent advances in detector technology and software algorithms have allowed to achieve the determination of biomolecular structures at near-atomic resolution. The project combines techniques from probability and representation theory to study them. The third area, neural networks, uses methods which are extremely effective in practice of Big Data, yet theoretically are mysterious. The project applies probabilistic tools to theoretically ground and improve these methods. The award will support the educational, mentoring, and outreach activities including undergraduate and masters research projects in empirical machine learning and training of the US national math olympiad team. In more detail, the project focuses on the conformal bootstrap in CFT, non-convex optimization for cryo-electron microscopy, and data augmentation for neural networks. The first part of the project applies prior work of the PI giving a probabilistic construction of Liouville conformal blocks to characterize their properties and complete the conformal bootstrap program for Liouville theory. The second part of the project applies prior work of the PI to characterize and improve first-order optimization algorithms for maximum-likelihood estimation in cryo-EM in practice. The third part of the project applies prior work of the PI connecting data augmentation and stochastic optimization to create theoretically principled data augmentation methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.