C5

I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an L^p norm of the potential.

These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

Group theoretic Dehn filling, motivated by Dehn filling in the theory of 3- manifolds, is a process of constructing quotients of a given group. This technique is usually applied to groups with certain negative curvature feature, for example word-hyperbolic groups, to construct exotic and useful examples of groups. In this talk, I will start by recalling the notion of word-hyperbolic groups, and then show that how group theoretic Dehn filling can be used to answer the Burnside Problem and questions about mapping class groups of surfaces.

A big mapping class group is the mapping class group (MCG) of a surface of infinite type. Although several aspects of big MCGs remain mysterious, their geometric definition allows some simple, interesting arguments. In this talk, we will use big MCGs as an excuse to survey some (more or less) classical results in geometric group theory: we will present a quick introduction to infinite type surfaces, highlight differences between standard and large MCGs, and use Higman’s embedding theorem to deduce that there exists a big MCG that contains every finitely presented group as a subgroup.

TBA

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.