Oxford

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

In this talk we will discuss when one right-angled Artin group is a subgroup of another one and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.

Ice sheets are among the key controls on global climate and sea-level change. A detailed understanding of ice sheet dynamics is crucial so to make accurate predictions of their mass balance into the future. Ice streams are the dominant negative component in this balance, accounting for up to 90$\%$ of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete.

A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localization feedback between ice flow, basal melting and water pressure in the underlying sediments. I will present a model of subglacial water flow below ice sheets, and particularly below ice streams. This hydrologic model is coupled to a model for ice flow. I show that under some conditions this coupled system gives rise to ice streams by instability of the internal dynamics.

Seismology currently undergoes rapid and exciting advances fueled by a simultaneous surge in recorded data (in both quality and quantity), realistic wave-propagation algorithms, and supercomputing capabilities. This enables us to sample parameter spaces of relevance for imaging the Earth's interior 3D structure with fully numerical techniques. Seismic imaging is the prime approach to illuminate and understand global processes such as mantle convection, plate tectonics, geodynamo, the vigorous interior of the Sun, and delivers crucial constraints on our grasp of volcanism, the carbon cycle and seismicity. At local scales, seismic Earth models are inevitable for hydrocarbon exploration, monitoring of flow processes, and natural hazard assessment.

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With a slight focus on global-scale applications, I will present the underlying physical model of realistic wave propagation, its numerical discretization and link such forward modeling to updating Earth models by means of inverse modeling. The associated computational burden to solve high-resolution statistical inverse problems with precise numerical techniques is however entirely out of reach for decades to come. Consequently, seismologists need to take approximations in resolution, physics, data and/or inverse methodology. I will scan a number of such end-member approximations, and focus on our own approach to simultaneously treat wave physics realistically across the frequency band while maximizing data usage and allow for uncertainty quantification. This approach is motivated by decisive approximations on the model space for typical Earth structures and linearized inverse theory.

This is joint work with Angus Macintyre.

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

To prove this we prove universal definability of the valuation rings of such fields using work of Cluckers-Derakhshan-Leenknegt-Macintyre on existential

definability, quantifier elimination of Basarab-Kuhlmann for valued fields in a many-sorted language involving higher residue rings and groups,

a model completeness theorem for certain pre-ordered abelian groups which generalize Presburger arithmetic (we call finite-by-Presburger groups),

and an interpretation of higher residue rings of such fields in the higher residue groups.

Seismic exploration in the oil industry is one example where wave equations are used as models. When the wave speed is spatially varying one is naturally concerned with questions of homogenisation or upscaling, where one would like to calculate an effective or average wave speed. As a canonical problem this short workshop will introduce the one-dimensional acoustic wave equation with a rapidly varying wave speed, perhaps even a periodic variation. Three questions will be asked: (i) how do you calculate a sensible average wave speed (ii) does the wave equation suffice or is there a change of form after averaging and (iii) if one can induce any particular excitation at one end of a finite one-dimensional medium, and make any observations that we like at that end, what - if anything - can be inferred about the spatial variability of the wave speed?