Several problems of great importance in the study of glaciers and ice sheets involve processes of attachment and reattachment of the ice from the bedrock. Consider, for example, an ice sheet sliding from the continent into the ocean, where it goes afloat. Another example is that of subglacial cavitation, a fundamental mechanism in glacial sliding where the ice detaches from the bedrock along the downstream area of an obstacle. Such problems are generally modelled as a viscous Stokes flow with a free boundary and contact boundary conditions. In this talk, I will present a framework for solving such problems numerically. I will start by introducing the mathematical formulation of these viscous contact problems and the challenges that arise when trying to approximate them numerically. I will then show how, given a stable scheme for the free boundary equation, one can build a penalty formulation for the viscous contact problem in such a way that the resulting algorithm remains stable and robust.

Organoids are three–dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low–cost and efficient methods of producing them in a reproducible manner and at scale. We are working in collaboration with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, which utilises flow of culture media to enhance nutrient delivery to the organoids and facilitate the removal of waste metabolites. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on the bioreactor design and operating conditions. A complete understanding of the system requires knowledge of the spatial and temporal information regarding flow and the resulting oxygen and metabolite concentrations throughout the bioreactor. However, it is impractical to obtain this data empirically, due to the highly–controlled environment of the bioreactor posing difficulties for online real–time monitoring of the system. Thus, we exploit a mathematical modelling approach, to provide spatial as well as temporal information.

In the bioreactor, organoids are seeded as single cells in a layer of hydrogel. We present a general model for the nutrient and waste metabolite concentrations in the hydrogel and organoid regions of the bioreactor. Resolving for the millions of organoids within the hydrogel is computationally expensive and infeasible. Hence, we take a mathematical homogenisation approach to understand how the behaviour of the organoids on the microscale influences the macroscale behaviour in the hydrogel layer. We consider the case of growing organoids, with a temporally and spatially dependent radii, and exploit the separation of scales to systematically derive an effective macroscale model for metabolite transport. We explore some canonical problems to understand our homogenised system.

A complex plane curve singularity gives rise to two objects: (1) a moduli space that representation theorists call an affine Springer fiber, and (2) a topological link up to isotopy. Roughly a decade ago, Oblomkov–Rasmussen–Shende conjectured a striking identity relating the homology of the affine Springer fiber to the so-called HOMFLYPT homology of the link. In unpublished writing, Shende speculated that it would follow from advances in nonabelian Hodge theory: the study of transcendental diffeomorphisms relating “Hitchin” and “Betti” moduli spaces. We make this dream precise by expressing HOMFLYPT homology in terms of the homology of a “Betti”-type space, which, we conjecture, deformation-retracts onto the affine Springer fiber. In doing so, we recast the whole story in terms of an arbitrary semisimple group. We give evidence for the nonabelian Hodge conjecture at the numerical level, using a mysterious formula that involves rational Cherednik algebras and the degrees of unipotent principal-series representations.

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.

A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.

 I will review some well-established relationships between four manifolds and vertex algebras that can be deduced from studying the M5-brane worldvolume theory, and outline some of the corresponding results in mathematics which have been understood so far. I will then describe a proposal of Gaiotto-Rapcak to generalize these ideas to the setting of multiple M5 branes wrapping divisors in toric Calabi-Yau threefolds, and explain work in progress on understanding the mathematical implications of this proposal as a complex network of relationships between the enumerative geometry of sheaves on threefolds and the representation theory of affine Lie algebras.