# Abstracts

Lucia de Luca (TUM)

Title: *Variational analysis for dipoles of topological singularities in two dimensions*.

Abstract: We present two continuous models for the study of topological singularities in 2D: the core-radius approach and the Ginzburg-Landau theory.

It is well known that - at zero temperature and under suitable regimes - the energies associated to these models tend to concentrate, as the length scale parameter epsilon goes to zero, around a finite number of points, the so-called vortices.

We focus on low energy regimes that prevent the formation of vortices in the limit as epsilon tends to zero, but that are compatible (for positive epsilon) with configurations of short (in terms of epsilon) dipoles, and more in general with short clusters of vortices having zero average.

By using a Gamma-convergence approach, we provide a quantitative analysis of the energy induced by such configurations on a continuous range of length scales.

Oliver Sheehan (Imperial)

Title: *Solving PDEs on triangles with multivariate orthogonal polynomials*

Abstract: Univariate orthogonal polynomials have a long history in applied and computational mathematics, playing a fundamental role in quadrature, spectral theory and solving differential equations with spectral methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials exist, they have an unfair reputation of being unwieldy on non-tensor product domains, and their use in applications has been limited. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators, fast evaluation of expansions using Clenshaw’s algorithm and the ability to construct sparse partial differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast solution of general partial differential equations.

Matt Thorpe (Cambridge)

Title: Analysis of p-Laplacian Regularization in Semi-Supervised Learning

Abstract: This talk concerns a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of n sample points, provided a small training subset of N labelled points. A goal of semi-supervised learning is to take advantage of the (geometric) structure provided by the large number of unlabelled data when assigning labels. In this talk the geometry is represented by the random geometric graph model with connection radius r(n). The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian regularization.

The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of r(n) for the asymptotic consistency to hold. For standard approaches used thus far there is a restrictive upper bound on how quickly r(n) must converge to zero as n goes to infinity. I will present a new model which overcomes this restriction. It is as simple as the standard models, but converges as soon as r(n) -> 0. This is joint work with Dejan Slepcev (CMU).

Tobias Grafke (Warwick)

Title: Non-equilibrium self-organization of motile bacteria

Abstract: Active materials can self-organize in many more ways than their equilibrium counterparts. For example, self-propelled particles with density dependend motility can display motility-induced phase separation (MIPS), resulting in novel routes to pattern formation. In this talk it is shown how internal fluctuations in the population size and swimming speed of motile bacteria have a significant impact on the way they self-organize. Two nontrivial regimes are identified, depending on the population carrying capacity. Below a certain threshold, the fluctuations make bacteria clusters appear and disappear periodically in time at random locations in space, with a period that is roughly independent of the noise amplitude. Above the threshold, bacteria organize in metastable clusters, and fluctuations lead to transitions between those at random times that are exponentially long in the noise amplitude, following specific out-of-equilibrium pathways. Both in the quasi-periodic and the metastable regimes, these findings can be explained by combining tools from large deviation theory with a bifurcation analysis in which the mean bacteria density, assumed to vary slowly via birth and death, plays the role of control parameter.