L1

When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is  algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.

Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.

Building on the work of Schöberl, Olshanskii and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier--Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines a novel tailored finite element discretisation, discrete augmented Lagrangian stabilisation, a custom prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each
level. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000.

Employing the usual multilevel Monte Carlo estimator, we introduce a framework for estimating the solutions of SDEs by an Euler-Maruyama scheme. By considering the expected value of such solutions, we produce simulations using approximately normal random variables, and recover the estimate from the exact normal distribution by use of a multilevel correction, leading to faster simulations without loss of accuracy. We will also highlight this concept in the framework of reduced precision and vectorised computations.

Robert Timms

Title: Multiscale modelling of lithium-ion batteries

Lithium-ion batteries are one of the most widely used technologies for energy storage, with applications ranging from portable electronics to electric vehicles. Due to their popularity, there is a continued interest in the development of mathematical models of lithium-ion batteries. These models encompass various levels of complexity, which may be suitable to aid with design, or for real-time monitoring of performance. After a brief introduction to lithium-ion batteries, I will discuss some of the modelling efforts undertaken here at Oxford and within the wider battery modelling community.
 

Jan Vonk

Title: Singular moduli for real quadratic fields

At the 1900 ICM, David Hilbert posed a series of problems, of which the 12th remains completely open today. I will discuss how to solve this problem in the simplest open case, by considering certain exotic (so called p-adic) metrics on the set of numbers, and using its concomitant theories of analysis and geometry.
 

Are you teaching intercollegiate classes or tutorials this term? Would you like to explore inclusive teaching strategies that could help all students make the most of your sessions? In this interactive workshop, we'll explore strategies that have been found effective. This will be a self-contained session, but will also be a good introduction to the "Developing Learning and Teaching" course offered by MPLS for graduate students and early career researchers. The session will be led by Vicky Neale (Mathematics) and Delia O'Rourke (Oxford Learning Institute). 
 

How much do you know actually about the research that is going on across the department? The SIAM Student Chapter brings you a 3 minute thesis competition challenging a group of DPhil students to go head to head to explain their research in just 3 minutes with the aid of a single slide. This is the perfect opportunity to hear about a wide range of topics within applied mathematics, and to gain insight into the impact that mathematical research can have. The winner will be decided by a judging panel comprising Professors Helen Byrne, Jon Chapman, Patrick Farrell, and Christina Goldschmidt.
 

Cristina Palmer-Anghel: Quantum invariants via topological intersection pairings
The world of quantum invariants for knots started in 1984 with the discovery of a strong link invariant, namely the Jones polynomial. Then, Reshetikhin and Turaev developed a conceptual algebraic method that, starting with any quantum group, produces invariants for knots. The question that we have in mind is to find topological models for certain types of quantum invariants. On the topological side, in 2000, Bigelow, building on earlier work of Lawrence,
interpreted the original Jones polynomial in a homological manner- as a graded intersection pairing in a covering of a configuration space of the punctured disc. On the quantum side of the story, the coloured Jones polynomials are a sequence of quantum invariants constructed through the Reshetikhin-Turaev recipe from the quantum group Uq(sl(2)). The first invariant of this sequence is the original Jones polynomial. In this talk we will present how one can use topological intersection pairings in order to describe a topological model for all coloured Jones polynomials.

Francis Woodhouse: Autonomous mechanisms inspired by biology

Unlike the air around us, biological systems are not in equilibrium: cells consume chemical energy to keep growing and moving, forming a clear arrow of time. The recent creation of artificial versions of these ‘active’ materials suggests that these concepts can be harnessed to power new soft robotic systems fuelled by as simple a source as oxygen. After an introduction to the physics of natural and artificial active systems, we will see how endowing a mechanical network with activity can create an intricate self-actuating mechanism.

In this session we discuss various different routes for promoting your research through a panel discussion with Dawn Gordon (Project Manager, Oxford University Innovation), Dyrol Lumbard (External Relations Manager, Mathematical Institute), James Maynard (Academic Faculty, Mathematical Institute) and Ian Griffiths, and chaired by Frances Kirwan. The panel discussion will include the topics of outreach, impact, and strategies for promoting aspects of mathematics that are less amenable to public engagement. 

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the  workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

The Goldbach conjecture is a famous unsolved problem in mathematics. It asks whether every even number greater than or equal to 4 is the sum of two primes. I will discuss some of the history of the problem, explaining among other things why the answer is surely yes, and also why this appears to be very hard to prove.