I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem. In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms.

The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

Spectro-microscopy is an experimental technique with great potential to science challenges such as the observation of changes over time in energy materials or environmental samples and investigations of the chemical state in biological samples. However, its application is often limited by factors like long acquisition times and radiation damage. We present two measurement strategies that significantly reduce experiment times and applied radiation doses. These strategies involve acquiring only a small subset of all possible measurements and then completing the full data matrix from the sampled measurements. The methods are data-driven, utilizing spectral and spatial importance subsampling distributions to select the most informative measurements. Specifically, we use data-driven leverage scores and adaptive randomized pivoting techniques. We explore raster importance sampling combined with the LoopASD completion algorithm, as well as CUR-based sampling where the CUR approximation also serves as the completion method. Additionally, we propose ideas to make the CUR-based approach adaptive. As a result, capturing as little as 4–6% of the measurements is sufficient to recover the same information as a conventional full scan.

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 $h$.
 

Low-rank approximation of parameter-dependent matrices A(t) is an important task in the computational sciences, with applications in areas such as dynamical systems and the compression of series of images. In this talk, we introduce AdaCUR, an efficient randomised algorithm for computing low-rank approximations of parameter-dependent matrices using the CUR decomposition. The key idea of our approach is the ability to reuse column and row indices for nearby parameter values, improving efficiency. The resulting algorithm is rank-adaptive, provides error control, and has complexity that compares favourably with existing methods. This is joint work with Yuji Nakatsukasa.