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In recent joint work with J. Merikoski, we developed a new way to employ $\mathrm{SL}_2(\mathbb{R})$  spectral methods to number-theoretical counting problems, entirely avoiding Kloosterman sums and the Kuznetsov formula. The main result is an asymptotic formula for an automorphic kernel, with error terms controlled by two new kernels. This framework proves particularly effective when averaging over the level and leads to improvements in equidistribution results involving quadratic polynomials. In particular, we show that the largest prime divisor of $n^2 + h$ is infinitely often larger than $n^{1.312}$, recovering earlier results that had relied on the Selberg eigenvalue conjecture. Furthermore, we obtain, for the first time in this setting, strong uniformity in the parameter $h$.
 

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Many applications in electromagnetism, magnetohydrodynamics, and pour media flow are well-posed in spaces from the 3D de Rham complex involving $H^1$, $H(curl)$, $H(div)$, and $L^2$. Discretizing these spaces with the usual conforming finite element spaces typically leads to discrete problems that are both structure-preserving and uniformly stable with respect to the mesh size and polynomial degree. Robust preconditioners/solvers usually require the inversion of subproblems or auxiliary problems on vertex, edge, or face patches of elements. For high-order discretizations, the cost of inverting these patch problems scales like $\mathcal{O}(p^9)$ and is thus prohibitively expensive. We propose a new set of basis functions for each of the spaces in the discrete de Rham complex that reduce the cost of the patch problems to $\mathcal{O}(p^6)$ complexity. By taking advantage of additional properties of the new basis, we propose further computationally cheaper variants of existing preconditioners. Various numerical examples demonstrate the performance of the solvers.

A beautiful example of a nonlinear eigenvalue problem is the determination of complex eigenvalues for wave scattering. This talk will show how nicely this can be done by applying AAA rational approximation to a scalarized resolvent sampled at a few real frequencies.  Even for a domain as elementary as a circle with a gap in it, such computations do not seem to have been done before. This is joint work with Oscar Bruno and Manuel Santana at Caltech.