Generating and approximating special geometries with machine learning

• Grant number: 24/18994-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: November 01, 2025
• End date: October 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Henrique Nogueira de Sá Earp
• Grantee: Edward George Hirst
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Company: Universidade Estadual de Campinas (UNICAMP). Faculdade de Engenharia Elétrica e de Computação (FEEC)
• Associated research grant: 20/09838-0 - BI0S - Brazilian Institute of Data Science, AP.PCPE

Abstract

Riemannian geometry is one of the most impactful domains in mathematics, being essential for the description of spacetime in General Relativity and widely used across fields such as economics, biomedicine, data science, and computer graphics. The classification of Riemannian manifolds, completed by Berger, highlights seven classes of holonomy groups, including Calabi-Yau and $G_2$ varieties, which are particularly important in String Theory as they describe extra-dimensional spaces that affect the vibrational patterns of strings, where the shape of these geometries sets limits on the types of universes that these theories can model.The varieties with each type of holonomy can be constructed in many ways, but there is an enormous number of such geometries. For example, with just one construction method, there are about 0.5 billion starting points for constructing even more Calabi-Yau varieties. Thus, to study these geometries on a large scale, statistical methods are essential, and this is where Machine Learning techniques play a crucial role, providing new ways to generate, search, and intelligently approximate these structures.In addition to topological properties, where Machine Learning has already shown significant success, the metrics that define the shape of these geometries are also important, though they are analytically unknown in most non-trivial cases. Recently, semi-supervised Machine Learning methods have been developed to approximate these metrics using losses derived from the highly non-linear geometric differential equations that define them.These new methods have the potential to model these geometries with greater precision than was previously possible, using pre-existing databases of Calabi-Yau and $G_2$ varieties, whose metrics can now be approximated for the first time. Furthermore, the implementation of geometric flows using Machine Learning, guided by the experience of the mentor of this proposal, will allow for the construction of new geometries with desired properties and open new avenues for studying geometric flows.These Machine Learning-approximated metrics may provide mathematical insights into their true forms, which are still analytically unknown. The existence of Calabi-Yau metrics was proved by Yau's work, which was awarded the Fields Medal, but their actual structure remains undiscovered. Through these methods, Machine Learning holds the potential to uncover this structure.The recent growth of AI and data science is facilitating their emerging use in pure mathematics. This proposal also includes the design and organisation of a graduate course in AI applied to Mathematics, training the local academic community in these methods. The results generated will include the first databases of approximated metrics for Calabi-Yau and $G_2$ varieties using neural networks, as well as open-source code packages for analysis. These differential flows will be numerically modelled for the first time using Machine Learning, offering new perspectives on previously inaccessible mathematical problems. (AU)