Georgia Institute of Technology College of Sciences

Can you hear the shape of a drum? A classical inverse problem in mathematical physics is to determine the shape of a membrane from the resonant frequencies at which it vibrates. This problem is very much still open for smooth, strictly convex planar domains and one tool in that is often used in this context is the wave trace, which contains information on the asymptotic distribution of eigenvalues of the Laplacian on a Riemannian manifold. It is well known that the singular support of the wave trace is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. In a recent work together with Vadim Kaloshin and Illya Koval, we construct large families of domains for which there are multiple geodesics of a given length, having different Maslov indices, which interfere destructively and cancel arbitrarily many orders in the wave trace. This shows that there are potential obstacles in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned.

Recent advancements in operator-type neural networks, such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet), have shown promising results in approximating the solutions of spatial-temporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new operator learning framework to address these issues. The proposed paradigm leverages the traditional wisdom from numerical PDE theory and techniques to refine the pipeline of existing operator neural networks. Specifically, the proposed architecture initiates the training for a single or a few epochs for the operator-type neural networks in consideration, concluding with the freezing of the model parameters. The latter are then fed into an error correction scheme: a single parametrized linear spectral layer trained with a convex loss function defined through a reliable functional-type a posteriori error estimator.This design allows the operator neural networks to effectively tackle low-frequency errors, while the added linear layer addresses high-frequency errors. Numerical experiments on a commonly used benchmark of 2D Navier-Stokes equations demonstrate improvements in both computational time and accuracy, compared to existing FNO variants and traditional numerical approaches.

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.