Fridays@4

Thomas Prince The double life of the number 24.

The number 24 appears in a somewhat surprising result in the study of polyhedra with integer lattice points. In a different setting, the number 24 is the Euler characteristic of a K3 surface: a four (real) dimensional object which plays a central role in algebraic geometry. We will hint at why both instances of 24 are in fact the same, and suggest that integral affine geometry can be used to interpolate between the realm of integral polytopes and the world of complex algebraic geometry.

Mohit Dalwadi A multiscale mathematical model of bacterial nutrient uptake

In mathematical models that include nutrient delivery to bacteria, it is prohibitively expensive to include many small bacterial regions acting as volumetric nutrient sinks. To combat this problem, such models often impose an effective uptake instead. However, it is not immediately clear how to relate properties on the bacterial scale with this effective result. For example, one may intuitively expect the effective uptake to scale with bacterial volume for weak first-order uptake, and with bacterial surface area for strong first-order uptake. I will present a general model for bacterial nutrient uptake, and upscale the system using homogenization theory to determine how the effective uptake depends on the microscale bacterial properties. This will show us when the intuitive volume and surface area scalings are each valid, as well as the correct form of the effective uptake when neither of these scalings is appropriate.
 

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. 

Claudia Scheimbauer

Title: Quantum field theory meets higher categories

Abstract: Studying physics has always been a driving force in the development of many beautiful pieces of mathematics in many different areas. In the last century, quantum field theory has been a central such force and there have been several fundamentally different approaches using and developing vastly different mathematical tools. One of them, Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories, provides a beautiful link between the geometry of "spacetimes” (mathematically described as cobordisms) and algebraic structures. Combining this approach with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. The Cobordism Hypothesis classifies "fully local" topological field theories and gives us a recipe to construct examples thereof by checking certain algebraic conditions generalizing the existence of the dual of a vector space. I will give an introduction to the topic and very briefly mention on my own work on these "extended" topological field theories.

Alberto Paganini

Title: Shape Optimization with Finite Elements

Abstract: Shape optimization means looking for a domain that minimizes a target cost functional. Such problems are commonly solved iteratively by constructing a minimizing sequence of domains. Often, the target cost functional depends on the solution to a boundary value problem stated on the domain to be optimized. This introduces the difficulty of solving a boundary value problem on a domain that changes at each iteration. I will suggest how to address this issue using finite elements and conclude with an application from optics.

Research suggests that students with a 'growth mindset' may do better than those with a 'fixed mindset'.

• What does that mean for our teaching?
• How can we support students to develop a growth mindset?
• What sorts of mindsets do we ourselves have?
• And how does that affect our teaching and indeed the rest of our work?