Renormalization Group Flows, Embedding Theorems, and Applications

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A central theme in mathematics is to understand the nature of space. Mathematicians use the term manifold to formalize the notion of a curved space built by piecing together patches of more familiar Euclidean space. This idea dates to the work of Riemann in the mid-1800s, yet several questions about manifolds remain poorly understood. In the 1950s, Nash proved striking embedding theorems identifying two distinct ways of thinking about manifolds, intrinsic and extrinsic. One can imagine for example, the measurement of lengths on earth by an ant crawling around (intrinsic) and the measurement of lengths on earth by an observer on the moon (extrinsic). The main purpose of this project is to revisit Nash's embedding theorems from a modern perspective, expanding their range of applications to problems in science and technology. Approaches of this project have applications to the description of disordered systems in physics (such as turbulent fluid flows) and the design of algorithms for artificial intelligence (such as manifold learning). The project contributes to the development of the scientific workforce through the mentoring of graduate students in PhD projects that include both rigorous mathematics and numerical computations. A particular goal of the project is to visualize the embedded manifolds in order to communicate Nash’s vision to a broad scientific audience and the public at large.The main technical goal of this project is to develop a new method to construct Gibbs measures for nonlinear partial differential equations. The method relies on lifting the PDE into a space of Gaussian measures and designing new heat flows, termed renormalization group flows, that improve subsolutions of the PDE into solutions through band-limited approximations at increasingly fine scales. Both numerical and analytical approaches will be pursued. The analytical method has its roots in Nash’s techniques, but its conceptual foundation is a Bayesian description of the measurement of length, not the nature of the underlying space. This shift in emphasis provides a framework that unifies the embedding problem for Riemannian manifolds with that of metric spaces and graphs. It identifies the problem of critical exponents for embeddings with the study of critical exponents in statistical physics. The numerical counterpart to these analytical methods is a new class of stochastic interior point methods for the solution of nonlinear PDE. The primary goal here is to explore the idea that PDE may be solved by optimization techniques, such as relaxation, semidefinite programming, and Markov chain Monte Carlo methods, rather than the traditional methods based on spatial discretizations.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.