C5

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.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.

Orbifolds are a generalisation of manifolds which allow group actions to enter the picture. The most basic examples of orbifolds are quotients of manifolds by (non-free) finite group actions.
I will give an introduction to orbifolds, recalling a number of philosophically different but mathematically equivalent definitions. For starters, I will try to convince you that "a space locally modelled on a quotient of R^n by a finite group" is misleading. I will draw many pictures of orbifolds, make the connection to complexes of groups, and explain the definition of a map of orbifolds. In the process, I hope to demystify the definition of the orbifold fundamental group, the orbifold Euler characteristic and orbifold cohomology.

In this talk, I will discuss the important Grothendieck conjecture which originated Grothendieck-Teichmuller Theory, a bridge between Topology and Number Theory. On the geometric side, there is the study of automorphisms of mapping class groups that satisfy compatibility conditions with respect to subsurface inclusions. On the other side, there is the study of the absolute Galois group of the rationals, one of the most important objects in Number Theory today.
In my talk, I will introduce these objects and discuss the recent progress that has been made in understanding such automorphisms of mapping class groups. No background in Number Theory or Galois Theory is required.

Finiteness properties of groups provide various generalisations of the properties "finitely generated" and "finitely presented." We will define different types of finiteness properties and discuss Bestvina-Brady groups as they provide examples of groups with interesting combinations of finiteness properties.

Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G. We discuss some recent work on local properties that X may possess which allow us to answer these questions - in many cases we can in fact deduce that the group is a linear group over Z.

Given an arbitrary group presentation, often very little can be deduced about the underlying group. It is thus something of a miracle that many properties of one-relator groups can be simply read-off from the defining relator. In this talk, I will discuss some of the classical results in the theory of one-relator groups, as well as the key trick used in many of their proofs. Time-permitting, I'll also discuss more recent work on this subject, including some open problems.

“The Royal Swedish Academy of Sciences has today decided to award the 2021 Nobel Prize in Physics for ground-breaking contributions to our understanding of complex physical systems”

Last Tuesday this announcement got many in our community very excited: never before had the Nobel prize been awarded to a topic so closely related to Network Science. We will try to understand the contributions that have led to this Nobel Prize announcement and their ties with networks science. The presentation will be held by Erik Hörmann, who has been lucky enough to have had the honour and pleasure of studying and working with one of the awardees, Professor Giorgio Parisi, before joining the Mathematical Institute.

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.