Schedule

Welcome

Sara Jabbari (University of Birmingham)

Title:  Mathematical tools to aid with the development of novel antimicrobials

Abstract:  The ability of bacteria to become resistant to previously successful antibiotic treatments is an urgent and increasing worldwide problem. Solutions can be sought via a number of methods including, for example, identifying novel antibiotics, re-engineering existing antibiotics or developing alternative treatment methods. The nonlinear interactions involved in infection and treatment render it difficult to predict the success of any of these methods without the use of computational tools in addition to more traditional experimental work. We use mathematical modelling to aid in the development of anti-virulence treatments which, unlike conventional antibiotics that directly target a bacterium's survival, may instead attenuate bacteria and prevent them from being able to cause infection or evade antibiotics. Many of these approaches, however, are only partially successful when tested in infection models. Our group are studying a variety of potential targets, including preventing bacteria from binding to host cells, inhibiting the formation of persister cells (these can tolerate the presence of antibiotics) and blocking efflux pump action (a key mechanism of antimicrobial resistance). I will present results that illustrate how mathematical modelling can suggest ways in which to improve the efficacy of these approaches.

Coffee Break

David Versluis  (Universiteit Leiden)

Title:  Multiscale mathematical modelling of the metabolism and metabolic interactions of the infant gut microbiota

Abstract:  Nearly immediately after birth, a complex and dynamic ecosystem forms in the human infant gut that influences the infant's health in both the short and long term. Oligosaccharides in nutrition strongly affect the composition of the microbiota by nurturing some species at the expense of others and influencing bacterial interactions. We use a multiscale mathematical model of a community of cross-feeding gut bacteria to investigate how oligosaccharides in nutrition may steer the infant gut microbiota. 

To cover spatial variation and interactions over time the model represents the microbiota as a grid of bacterial populations that exchange metabolites. At each timepoint the metabolism  of each population is calculated using flux balance analysis with an enzymatic constraint on a set of genome-scale metabolic models. There is an additional set of equations for modelling the extracellular production of public goods. Taken together, this allows the model to predict species-specific metabolisms, including metabolic switches, nutrient preferences, and cross-feeding. 

The model provides predictions for a wide variety of questions around the composition of the infant gut microbiota. For example, the model predicts that the prebiotic milk oligosaccharide 2’-FL improves butyrate production in the infant gut by stimulating the production of the cross-feeding substrate propane-1,2-diol, which allows butyrate producers to outcompete other cross-feeding species. The model furthermore predicts that specific competitors can outcompete the butyrate producers in the presence of different oligosaccharides. Our overall aim is to provide mechanistic explanations of the influence that nutrition can have on the development of the infant microbiota.

Richard Howat  (University of Birmingham)

Title: Intersection of Thick Fractal Sets

Abstract: Typically, fractal sets do not behave well under intersections and it can be a challenge to verify said intersection is non-empty, let alone has non-zero Hausdorff dimension. Inspired by work from Alexia Yavicoli we introduce a variation on the potential game to show that, in the square metric, thickness can be used to verify that a countable collection of self-similar sets satisfying the open set condition with feasible open set B(0,1), have positive Hausdorff dimension if they have sufficiently large thickness.

Lunch

Tony Samuel  (University of Birmingham & University of Exeter)

Title: Diffraction of return time measures

Abstract: The diffraction spectrum of Dirac combs supported on point sets such as regular model sets arising from cut and project schemes and deterministic incommensurate structures have been extensively studied and shown to be pure point. Diffraction of translation bounded weighted Dirac combs supported on locally compact Abelian groups have also been studied.  In this talk we introduce a new class of weighted Dirac combs where the weights are determined by the dynamics of a given interval map or circle rotation.  We will discuss the diffraction spectrum of this new class of Dirac combs and highlight some interesting limiting behaviours.

Alice Peng  (Universiteit Leiden)

Title: Model upscaling of cellular forces and diffusive compounds in the living tissue

Abstract:  In biomedical applications, there are many interactions between cells and their direct environment, for instance, mechanical interaction via cellular stress and chemical interaction via diffusive compounds such as signalling molecules. For example, in wound healing, fibroblasts (skin cells) apply pulling forces on the extracellular matrix (ECM) to contract the wound, and these skin cells are attracted from the uninjured skin to wound by the signalling molecules secreted by the immune cells.

Generally speaking, agent-based modelling and continuum-based modelling are the main categories used to describe the interactions between cells and their direct environment. While continuum-based modelling is more computationally efficient, agent-based modelling provides a more precise physical description of the biological phenomena but with higher computational cost. Hence, it is important to search for a "bridge" to upscale the agent-based models to the continuum-based models, with controlling the information loss. 

In this talk, we will present how to upscale the linear momentum balance model (describing the cellular forces) and the diffusion model (describing the diffusion of the compounds), with the use of Dirac delta distributions as a representation of point forces and point sources, respectively. Both analytical and numerical results will be shown.

Break

Ale Jan Homburg (Universiteit Leiden & Universiteit Amsterdam)

Title: Iterated function systems of affine expanding and contracting maps on the unit interval

Abstract: I will discuss iterated function systems on the unit interval generated by expanding and contracting affine maps, and its connection to heterochaos baker maps.  I will focus on a study of two-point motions. This is joint work with Charlene Kalle.

Conference Dinner

Francesca Arici  (Universiteit Leiden)

Title: Noncommutative spheres from quantum trees

Abstract: In the paradigm of noncommutative geometry, C*-algebras play the role of generalized topological spaces, often encoding interesting dynamics. In this talk, I will introduce a class of C*-algebras constructed from a special class of polynomials in non-commuting variables and I will discuss some of their properties, including how and why we can interpret them both as algebras of functions on algebraic subsets of noncommutative spheres and as algebras of functions on the boundary of a quantum tree.

Coffee Break

Yuezhao Li (Universiteit Leiden)

Title: A hitchhiker's guide to topological insulators in noncommutative geometry

Abstract: Topological insulators are remarkable materials that, while being insulating in the bulk (i.e. they do not conduct electric current), permit a current to flow on their boundaries without dissipation. This intriguing property arises from the spectral theory of the physical Hamiltonian of such a material, and has surprising relation with the topology of the material's Brillouin zone. It has been challenging to describe this in a mathematically precise fashion, until Jean Bellissard introduced noncommutative geometry to this field in the 1980s. In this talk, I will give a gentle introduction and some historical remarks to this exciting story.

Andrew Mitchell  (University of Birmingham) 

Title: Pisot substitutions, Rauzy fractals, and their generalisations

Abstract: Rauzy fractals are geometric objects that can be associated with substitution sequences in a natural way. Questions concerning dynamical properties of substitution sequences can often be reframed in terms of the geometry of the associated Rauzy fractal. In this talk, I will provide an introduction to substitutions and present how their associated Rauzy fractal is constructed. I will then discuss how this construction can be extended to random substitutions, which are a generalisation of substitutions where the substituted image of a letter is determined by a Markov process. This talk is based on joint work with Philipp Gohlke (University of Lund), Dan Rust (Open University) and Tony Samuel (University of Birmingham and University of Exeter).

Lunch

Ayreena Bakhtawar (Scuola Normale Superiore Pisa)

Title: Continued fractions and Dirichlet’s Improvability

Abstract: Diophantine Approximation is a branch of Number theory in which the central theme is understanding how well real numbers can be approximated by rationals. Dirichlet's theorem (1842) is a fundamental result that gives an optimal approximation rate of any irrational number. The set of real numbers for which Dirichlet's theorem admits an improvement was originally studied by Davenport and Schmidt. After them, Kleinbock and Wadleigh in 2018  proved that the improvements to Dirichlet's theorem are related to the growth of the products of consecutive partial quotients. In this talk I will discuss some new metrical results for the set of Dirichlet non-improvable numbers in connection with the theory of continued fractions.

Sabrina Kombrink  (University of Birmingham)

Title: Weyl asymptotics, renewal theorems and fractal Steiner-formulae

Abstract: In this talk an overview of links between Weyl asymptotics, renewal theorems and fractal Steiner-formulae will be presented. Moreover, we will investigate potential applications of the involved methods in epigenetics.

Weyl asymptotics relate the spectrum of the Dirichlet Laplacian to geometric properties of the underlying domain and relate to the question: Can you hear the shape of a drum? by Kac 1966. Steiner-formulae were first obtained for convex non-empty subsets A of d-dimensional Euclidean space. For such sets A, the Steiner-formula says that the volume of the set of points that are distance at most t away from A, can be written as a polynomial of degree d in t. The coefficients of this polynomial give important geometric characteristics of the set A, such as the volume, surface area, curvatures and Euler characteristic. Asymptotic expansions of renewal functions provide tools to obtain Weyl asymptotics and Seiner-formulae in highly irregular (fractal) settings.