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Lectures on Numerical and Machine Learning Methods for Approximating Ricci-flat Calabi-Yau Metrics

Calabi-Yau (CY) manifolds play a ubiquitous role in string theory. As a supersymmetry-preserving choice for the 6 extra compact dimensions of superstring compactifications, these spaces provide an arena in which to explore the rich interplay between physics and geometry. These lectures will focus on compact CY manifolds and the long standing problem of determining their Ricci flat metrics. Despite powerful existence theorems, no analytic expressions for these metrics are known. In this lecture series we review numerical approximation methods for Ricci flat CY metrics. Our first aim is to give a brief overview of the mathematical framework underlying CY geometry, and the various metrics that CY manifolds admit. We will then discuss the three types of numerical methods that have been developed to compute Ricci-flat CY metrics: Donaldson's algorithm, functional minimization methods, and machine learning methods. Due to the limited time/space we have, this will not be a comprehensive review, but instead we hope to give a brief survey and illustrate the essential tools, key ideas, and implementations of this rapidly advancing field.