In 1966, Mark Kac famously posed the question "Can one hear the shape of a drum?" In mathematical terms, this is an inverse eigenvalue problem: do the vibrational frequencies of a planar domain (its "Dirichlet eigenvalues") determine the shape of the domain? In 1994, Gordon, Wolpert, and Webb showed that one cannot hear the shape of a drum. The constructed pairs of "isospectral" domains with different shapes.

There is still much that we don't know about this inverse problem. While more examples of isospectral domains have been found, and some domains have been shown to be "spectrally unique", there is not a general understanding of precisely which aspects of shape are determined by the Dirichlet eigenvalues.

The goal of this project is to develop a Bayesian computational approach for solving a version of this inverse problem: given finitely many "measured" eigenvalues, what is the shape (or family of shapes) that best match the data?

In the dairy industry, feed costs make up roughly 75% of the total production costs for milk. The advent of robotic milking and automated feeding systems gives dairy farms the ability to provide individualized diets to each cow. We are developing a unified mathematical framework for the simultaneous optimization of dairy cattle diets and feed mixtures. This mathematical framework is flexible, and can be adapted to utilize any predictive milk production model. The resulting optimization problem is nonlinear, constrained, high-dimensional, and generally ill-posed (due to simultaneous consideration of diets and mixtures). This work is a product of SL-Dairy, which is a self-learning, precision feeding and performance diagnostic system for dairy cattle.

Inference through data and mathematical modeling is particularly challenging for dynamical systems with noisy data, model uncertainties, and unknown mechanisms. Here, parameter and uncertainty estimation problems are typically ill-posed, meaning solutions do not exist, are not unique, or do not depend continuously on the data. Inference depends on the proper inclusion of prior knowledge. We are developing surrogate data techniques to regularize ill-posed problems.

How can the size, location, and the geometry of a megathrust earthquake be determined without the use of modern instruments? This is an important question for fault zones that can support these dangerous earthquakes, but haven't done so for hundreds of years. Using Markov Chain Monte Carlo methods, sparse anecdotal observations of the ground motion and possible tsunami can be used to infer the source earthquake parameters. This will allow for a more global survey of fault zones, aiding disaster mitigation efforts. See our GitHub page.

There are many methods for finding multiple roots of univariate functions, and there are also many methods for finding single roots of multivariate functions. However, there are relatively few methods for finding multiple roots of multivariate functions. Similar to the popular Matlab software Chebfun, our method uses Chebyshev polynomial interpolation to approximate functions to a high degree of accuracy. Algebraic-geometric methods (Macaulay matrices) can convert these polynomial root-finding problems into eigenvalue problems (Möller-Stetter matrices). See our GitHub page.