Mathematical Institute

Magnus expansion based methods are an efficient class of integrators for solving Schrödinger equations that feature time dependent potentials such as lasers. These methods have been found to be highly effective in computational quantum chemistry since the pioneering work of Tal Ezer and Kosloff in the early 90s. The convergence of the Magnus expansion, however, is usually understood only for ODEs and traditional analysis suggests a much poorer performance of these methods than observed experimentally. It was not till the work of Hochbruck and Lubich in 2003 that a rigorous analysis justifying the application to PDEs with unbounded operators, such as the Schrödinger equation, was presented. In this talk we will extend this analysis to the semiclassical regime, where the highly oscillatory solution conventionally suggests large errors and a requirement for very small time steps.
 

Delsarte-Goethals frames are a popular choice for deterministic measurement matrices in compressive sensing. I will show that it is possible to construct extremely sparse matrices which share precisely the same row space as Delsarte-Goethals frames. I will also describe the combinatorial block design underlying the construction and make a connection to Steiner equiangular tight frames.
 

Each year Prof. Trefethen gives the Problem Solving Squad a sequence of problems with no hints, one a week, where the solution of each problem is a single real number to be computed by any method available.  We will present this year's three problems, involving (1) an S-shaped bifurcation curve, (2) shortest path around a random web, and (3) switching a time-varying system to maximize a matrix norm.

The 14 students this year are Simon Vary plus InFoMM cohort 2: Matteo Croci, Davin Lunz, Michael McPhail, Tori Pereira, Lindon Roberts, Caoimhe Rooney, Ian Roper, Thomas Roy, Tino Sulzer, Bogdan Toader, Florian Wechsung, Jess Williams, and Fabian Ying.  The presentations will be by (1) Lindon Roberts, (2) Florian Wechsung, and (3) Thomas Roy.

Derivative-free optimisation (DFO) algorithms are a category of optimisation methods for situations when one is unable to compute or estimate derivatives of the objective, such as when the objective has noise or is very expensive to evaluate. In this talk I will present a flexible DFO framework for unconstrained nonlinear least-squares problems, and in particular discuss its performance on noisy problems.

Part of the series "What do historians of mathematics do?"  

I will address this question by turning to another: "What is algebra?"  In answering this second question, and surveying the way that the answer changes as we move through the centuries, I will highlight some of the problems that face historians of mathematics when it comes to interpreting historical mathematics, and give a flavour of what it means to study the history of mathematics.