In this talk, a novel discontinuous Galerkin (DG) method is introduced by utilising the principle of discrete least squares. The key idea is to build polynomial approximations by the method of (weighted) discrete least squares instead of usual interpolation or (discrete) $L^2$ projections. The resulting method hence uses more information of the underlying function and provides a more robust alternative to common DG methods. As a result, we are able to construct high-order schemes which are conservative as well as linear stable on *any* set of collocation points. Several numerical tests highlight the new *discontinuous Galerkin discrete least squares (DG-DLS)* method to significantly outperform present-day DG methods.

# Past Forthcoming Seminars

Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs. However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework.

However, if generalised SBP operators are used, the treatment of boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely —interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain.

In this talk, the concept of generalised SBP operators and their application to entropy stable semidiscretisations will be presented. Several recent ideas extending the range of possible methods are discussed, presenting both advantages and several shortcomings.

In this talk I will first briefly introduce 1-parameter persistent homology, and discuss some applications and the theoretical challenges in the multiparameter case. If time remains I will explain how tools from commutative algebra give invariants suitable for the study of data. This last part is based on the preprint https://arxiv.org/abs/1708.07390.

Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.

In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this

talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

Inverting the signature of a path with ideas from linear algebra with implementations.

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole.