A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.

We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.

We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.

Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.

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

Both as a canonical mathematical text and as a representative of ancient thought, Euclid's Elements of Geometry has been a subject of study since its creation c. 300 BCE. It has been read as a practical and a theoretical text; it has been studied for its philosophical ramifications and for its perceived potential to inculcate logical thought. For the historian, it is where the history of mathematics meets the history of ideas; where the history of the book meets the history of practice. The study of the Elements enjoyed a particular resurgence during the Early Modern period, when around 200 editions of the text appeared between 1482 and 1700.  Depending on their theoretical and practical functions, they ranged between elaborate folios and pocket-size compendia, and were widely studied by scholars, natural philosophers, mathematical practitioners, and schoolchildren alike.

In this talk, I will present some of the preliminary results of the research we have been conducting for the AHRC-funded project based at the History Faculty 'Reading Euclid: Euclid's Elements of Geometry in Early Modern Britain', paying particular attention to how the books were printed, collected, and annotated. I will concentrate on our methodologies and introduce the database we have been building of all the early modern copies of the text in the British Isles, as well as the 'catalogue of book catalogues'.

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

In 1873 the Personal Recollections from Early Life to Old Age of Mary Somerville were published, containing detailed descriptions of her life as a 19th century philosopher, mathematician and advocate of women's rights. In an early draft of this work, Somerville reiterated the widely held view that a fundamental difference between men and women was the latter's lack of originality, or 'genius'.

In my talk I will examine how Somerville's view was influenced by the historic treatment of women, both within scientific research, scientific institutions and wider society. By building on my doctoral research I will also suggest an alternative viewpoint in which her work in the differential calculus can be seen as original, with a focus on her 1834 treatise On the Theory of Differences.

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

In this talk, we will survey the movement of mathematical ideas in the 17th century. We will explore, in particular, the mathematical cultures of Paris, Amsterdam, Rome, Cape Town, Goa, Kyoto, Beijing, and London, as well as the journey of mathematical knowledge on a global scale. As it will be an ambitious task to complete a round-the-world history tour in an hour, the focus will be on East Asia. By employing the digital humanities technique, this presentation will use digital media to effectively show historical sources and help the audience imagine the world as a “round” entity when we discuss a global history of mathematics.