In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural network trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional Hölder class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the partition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural networks without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.

People have used tiling patterns in the decorative arts for thousands of years. In the 19th century chemists classified the crystallographic groups, which describe all ways of tesselating 3-dimensional space by repeating shapes. Subsequently, mathematicians realised that the same arguments could be used to classify the 2-dimensional crystallographic groups, that is, the wallpaper groups. There is a close connection between crystallographic groups and affine Coxeter groups, which are collections of Euclidean isometries generated by reflections. In this talk, I will describe the geometry of conjugation in affine Coxeter groups and many other Euclidean isometry groups. This is joint work with Elizabeth Mili?evi? and Petra Schwer, and I will also describe an app for visualisation developed together with Amy Herron.

The critical Abelian Higgs model (AHM) is a system of nonlinear wave equations arising in particle physics. We construct solutions of this system in 3+1 dimensions that exhibit a number of slowly-moving nearly parallel vortex filaments. The leading-order dynamics of this ensemble of filaments are described by a wave map into the moduli space, a manifold carrying a natural Riemannian structure that parametrizes stationary 2D solutions of the AHM. These results allow for the study of the poorly-understood phenomenon of vortex reconnection in this setting. In particular, it is shown that in the regime studied, reconnection is the generic outcome of collisions of pairs of vortex filaments. Extremely similar results are also proved for the critical Abelian Higgs heat flow, modeling certain superconductors, near-critical versions of these equations, and in higher dimensions. This work is joint with Masoud Geevechi.