Spring 2026

Solving partial differential equations (PDEs) often involves solving a linear system \(Ax = b\) where \(A\) is a highly structured stiffness matrix. We seek to precondition PDEs effectively and cheaply by taking advantage of this structure. A displacement operator, \(\nabla (A) = A – Z R Z^{\top}\), for certain unitary matrices \(Z\), can detect a matrix’s structure by examining the rank of \(\nabla (A)\). It turns out that for a specified \(Z\), the kernel of these operators is exactly the space of matrices diagonalized by one of the discrete sine and cosine transforms. In this talk I will overview these results and discuss how are they are applied to precondition stiffness matrices with increasingly complex structures.

I will outline a series of results about finite time explosion for SPDEs. In some settings, the addition of strong stochastic forces can cause solutions to explode in finite time. In other settings, strong stochastic perturbations have a regularizing effect and can prevent solutions from exploding.

Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to  static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product of Euclidean space and graph \(G\), where \(G\) is a weighted graph over a finite set of nodes, and the graph geometry strongly influences the associated dynamic and static distances. In this talk, we will present sharp inequalities relating four notions of vector valued optimal transport metrics, which show that the metrics are mutually bi-Hölder equivalent. We will also discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis.

The asymptotics behavior of the eigenvalues of Toeplitz matrices is a well-developed field, dating to the early 20th century. In this talk, we will discuss some recent results and conjectures related to the eigenvalue behavior of a large class of Berezin-Toeplitz operators, which include Gabor-Toeplitz and Calderon-Toeplitz operators as special cases. This is partly joint work with Mishko Mitkovski.

We look at a finite collection of time–frequency shifts built from two parts:

Lattice part – all but one shift lie on the usual integer grid;
Rogue shift – one extra shift sits off that grid.

For any non-zero Schwartz function \(f\), the vectors obtained by applying those shifts to \(f\) are linearly independent. Assuming a dependence leads, via the Zak transform, to a torus equation tying size and phase. An orbit trichotomy for the induced translation then shows: dense orbits force \(f=0\); finite orbits invoke Linnell’s lattice theorem; infinite but non-dense orbits are excluded by a new rigidity argument blending ergodic averages with phase analysis. These three mutually exclusive cases cover all possibilities, settling the Heil–Ramanathan–Topiwala conjecture for mixed-integer configurations with Schwartz windows.

We will analyze a spherical Radon transform, \(\mathcal{R}\), which integrates a function over spheres in Euclidean space with arbitrary centers \(\mathbf{y}\) and radii \(r(\mathbf{y})\) that vary smoothly with \(\mathbf{y}\). We prove conditions under which the normal operator, \(R^*R\), (or a localized version) is an elliptic pseudodifferential operator. This gives us Sobolev regularity and stability for inversion, and we explain in concrete terms why these are important properties. We apply our results to problems including Compton Scatter Tomography and Ultrasound Reflection Tomography. We show reconstructions from my REU student Kaden Rajaniemi illustrating the theoretical results.

This is joint work with James W. Webber (Cleveland Clinic).