Title: A storm in a tea-cup(?)

Date:  2pm Tuesday 12th November 2024

Venue: LT907

Abstract: This talk has two parts. In the first part I will talk about the amazing results obtained recently by using techniques that originate with Nash and Gromov, and in the second part I will talk about implications of these results for admissibility criteria, which is ongoing work with H. Gimperlein, R. J. Knops and M. Slemrod.  Some audience participation will be required in the first part.

Title: The 1-11-representability of graphs 

Date: 2.00pm Tuesday 19th November 2024

Venue: LT907

Abstract: 

A graph is word-representable if it can be represented by rules concerning the alternation of letters in words. While many well-studied classes of graphs are word-representable, not all graphs can be represented in this way. By relaxing the conditions that define an edge, we can represent all graphs in a so-called 2-11 manner. In the new terminology, word-representable graphs are precisely 0-11-representable graphs. Whether all graphs can be 1-11-represented remains unknown, and the 1-11-representation of graphs has gained attention in the literature. In this talk, I will discuss methods and tools for proving the word-representability and 1-11-representability of graphs.

Title: Ribbon Tilings

Date: 2.00pm Tuesday 26th November 2024

Venue: LT907

Abstract: A ribbon tile consists of a finite edge-connected sequence of unit squares in the plane, each square either directly above or directly to the right of the previous square in the sequence. An n-ribbon tile is a ribbon tile made up of n unit squares. Which finite regions of the plane can be tiled with n-ribbon tiles? How many n-ribbon tilings can a region have? In this talk I will give an introduction to n-ribbon tilings, focussing on these questions. The new material in the talk is based on joint work with Yinsong Chen and Vladislav Kargin (arXiv:2408.09272).

Title: Topology and Transcription in 3D Chromatin Loop Networks

Date: 2.00pm Thursday 28th November 2024

Venue: LT908

Abstract: The 3D folding of a mammalian gene can be studied by polymer models, where the chromatin fiber is represented by a semiflexible polymer which interacts with multivalent proteins, representing complexes of DNA-binding transcription factors and RNA polymerases. This physical model leads to microphase separation and the natural emergence of clusters of proteins and binding sites, which can be understood via a mean field theory. The clustering is  accompanied by the folding of chromatin into a set of topologies, each associated with a different network of loops. By combining Brownian dynamic simulations, statistical mechanics and combinatorics, we first classify these networks, and then find their relative importance or statistical weight, when the properties of the underlying polymer are those relevant to chromatin. Unlike polymer networks previously studied, our chromatin networks have finite average distances between successive binding sites, and we show that this feature leads to giant differences between the weights of topologies with the same number of edges and nodes but different wiring. These weights strongly favor rosette-like structures with a local cloud of loops with respect to more complicated nonlocal topologies. Our results suggest that genes should overwhelmingly fold into a small fraction of all possible 3D topologies, which can be robustly characterized by the framework we propose here.

Title: Irrational Enumeration

Date: Tuesday 11th February 2025, 2.00pm

Venue: LT907

Abstract: It is well known that the asymptotic behaviour of the number of objects in a combinatorial class can be determined from the singularities on the circle of convergence of its generating function. In this talk we extend the scope of analytic combinatorics to classes whose objects have irrational sizes by establishing an analogous result when it is no longer required that the size of an object must be an integer.

The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. The technique will be illustrated with a variety of applications, including tilings with tiles of irrational area, lattice walks enumerated by Euclidean length, and trees with vertices of irrational size. We will also explore phase transitions in the asymptotics of families of irrational combinatorial classes.

Title: A Variational Perspective on Auxetic Metamaterials of Checkerboard Type

Date: Tuesday 18th March 2025, 2.00pm

Venue: LT907

Abstract:  Auxetic metamaterials have the counterintuitive property that they expand perpendicular to applied forces under stretching. In this talk, we discuss homogenization via Gamma-convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumption of non-self-interpenetration. Our result rigorously confirms these structures as auxetic. The challenging part of the proof is determining the admissible macroscopic deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components. To this end, we establish an asymptotic rigidity result showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. Our strategy is to tackle first an idealized model with full rigidity on the stiff tiles and then transfer the findings to a model with diverging elastic constants. The latter requires a new quantitative geometric rigidity estimate for non-connected touching squares and a tailored Poincaré-type inequality for checkerboard structures.

This is joint work with Wolf-Patrick Düll (University of Stuttgart) and Carolin Kreisbeck (KU Eichstätt-Ingolstadt).

Title: On the Connection of the Prandtl Equations and the Harmonic Oscillator

Date: Tuesday 25th March 2025, 2.00pm

Venue: LT907

Abstract:  A deep connection  is established between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schrödinger operator. The first result concerns a spectral condition associated with unstable quasimodes of the Prandtl equations. We give all solutions of the governing ODE in terms of Kummer's functions. By classifying their asymptotic behaviour, we determine that the spectral condition has unique, determined eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions.

Moreover, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schrödinger operator with a quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.

Title: The effect of imbibition on the deposition from an evaporating droplet

Date:  1.00pm Wednesday 25th September 2024

Title: Recent progress on modelling of interfacial flows with soluble surfactants including micelle formation, contact-line, and interfacial viscosity effects

Date:  1.00pm Wednesday 2nd October 2024

Title: Droplet dynamics: evaporation, hydrodynamics and particle transport

Date:  1.00pm Wednesday 9th October 2024

Title: Droplet Dynamics and Self-Cleaning Mechanisms on Liquid-Repellent Surfaces

Date:  1.00pm Wednesday 16th October 2024

Title: TBC

Date:  1.00pm Wednesday 23rd October 2024

Title: The Ouzo effect: adding a splash of dynamics to the water-ethanol-oil phase diagram

Date:  1.00pm Wednesday 30th October 2024

Title: TBC

Date:  1.00pm Wednesday 6th November 2024

Title: Shear-induced dispersion in reactive Hele-Shaw flows

Date:  1.00pm Wednesday 20th November 2024

Title: New in-orbit self-assembly principles: experimental results and high-fidelity simulations

Date:  1.00pm Wednesday 4th December 2024

Title: A conservative finite element method for the Navier-Stokes equation with free surface

Date:  1.00pm Wednesday 22nd January 2025

Title: Granular Material Mechanics and Rheology - From Discrete Simulations to Continuum Models

Date:  1.00pm Wednesday 5th February 2025

Title: Fluid Mechanics of Biological Cells

Date:  1.00pm Tuesday 11th February 2025

Title: Active nematic multipoles: a threefold application of nematic harmonics

Date:  1.00pm Wednesday 19th February 2025

Title: A Multicompartment Darcy Flow Model for Myocardial Perfusion Incorporating Patient-Specific Microvasculature

Date:  1.00pm Wednesday 5th March 2025

Title: Homogenization of a Coupled Electrical and Mechanical Bidomain Model for the Myocardium

Date: 3.00pm Wednesday 29th January 2025

Venue: LT908

Abstract:  We propose a coupled electrical and mechanical bidomain model for the myocardium tissue. The heart muscle structure that we investigate possesses an elastic matrix with embedded cardiac myocytes. We are able to apply the asymptotic homogenization technique by exploiting the length scale separation that exists between the microscale where we see the individual myocytes and the overall size of the heart muscle. We derive the macroscale model which describes the electrical conductivity and elastic deformation of the myocardium driven by the existence of a Lorentz body force. The model comprises balance equations for the current densities and for the stresses, with the novel coefficients accounting for the difference in the electric potentials and elastic properties at different points in the microstructure. The novel coefficients of the model are to be computed by solving the periodic cell differential problems arising from application of the asymptotic homogenization technique. By combining both the mechanical and electrical behaviours, we obtain a macroscale model that highlights how the elastic deformation of the heart tissue is influenced and driven by the application of a body force that varies with the difference in electric potentials at different points in the microstructure. We numerically investigate the novel model as well as various extensions and consider properties such as the effective electrical conductivities and elastic parameters and find behaviours that replicate physiological observations.

Title: Plankton thermal diversity modulates the responses of primary production to ocean warming

Date: 3.00pm Wednesday 12th February 2025

Venue: LT908

Abstract:  Marine phytoplankton plays a pivotal role in taking up CO2 from the atmosphere and supporting marine fisheries. As our ocean is warming at an unprecedented rate due to anthropogenic activities, it is imperative to understand how the phytoplankton community and primary production will respond to warming. However, a fixed rate of increase of phytoplankton photosynthetic rate to increasing temperature is assumed in most Earth System Models used for projecting how the Earth system will respond to climate warming, which ignores phytoplankton diversity and leads to great uncertainties in future projection. Here, I explore how the differences in the thermal responses of phytoplankton species affect our perception of how to parameterize phytoplankton thermal responses in plankton models and the implications for how primary production will change in a warming ocean.

Title: A Riemannian View on PDE-constrained Shape Optimisation

Date: 3.00pm Wednesday 26th February 2025

Venue: LT908

Abstract:  Shape optimisation involves finding the optimal shape of a domain (e.g., a boundary or region in space) to minimise or maximise an objective function. This process is often subject to constraints, many of which are governed by partial differential equations (PDEs) since physical phenomena—such as fluid flow, heat conduction, or elasticity—are typically described by these equations. Common examples include minimising drag in fluid dynamics, maximising structural strength, and optimising material distribution within a design domain.

A significant challenge in shape optimisation is modelling the shapes themselves, as no universal consensus exists on the best approach. Current methods include level set techniques, the method of mappings, and boundary parametrisation, each with its own advantages and limitations. In this talk, we focus on the Riemannian view of shape optimisation. Here, Riemannian geometry provides a mathematical framework for understanding and navigating the space of shapes. The key idea is to treat the space of admissible shapes as a Riemannian manifold, where each shape (e.g., a curve in 2D or a surface in 3D) represents a point in either a finite- or infinite-dimensional manifold. Additionally, Riemannian metrics define distances and gradients on this manifold, enabling the rigorous application of optimisation techniques such as gradient descent.

Title: Cascading and Multi-Stressor Effects in Coastal Ecosystems

Date: Monday 19th August, 3.00-4.00pm

Venue: LT511

Abstract:  Coastal ecosystems are simultaneously exposed to a plethora of human-induced stressors, such as climate warming, eutrophication, pollution, overfishing, and pathogens. These stressors interact with each other, driving counter-intuitive responses and undesirable results in management efforts. For example, the reduction of nutrient loads is widely believed to be the solution for eutrophication problems in coastal ecosystems. However, we show that these efforts have not always resulted in, and may not in the future result in, the desired reduction of phytoplankton biomass. Instead, the effects of de-eutrophication are overridden by climate warming, which intensifies temperature-dependent grazing of zooplankton by small carnivores, such as juvenile fish, leading to reduced herbivory (by zooplankton on phytoplankton) and thus increased standing stock of algae. This effect is especially strong in the shallow and turbid waters of coastal seas worldwide. High turbidity persistently limits the rates of photosynthesis, shifting bottom-up control towards top-down control and a stronger influence of higher trophic levels. In another case study, we show the compounding effects of climate warming and marine viruses on food web dynamics in a coastal environment, leading to a decline in primary production and carbon export, and higher retention of nutrients in the upper water column. Our results highlight the importance of stressor interactions and cascading effects in understanding responses in coastal ecosystems, a benchmark for ecosystem modelling and the effective development of management and conservation strategies.

Title: TBC

Date: Wednesday 4th September, 1.00-2.00pm

Venue: LT908

Abstract:  

Date: 3.00pm Thursday 13th March 2025

Venue: LT907

Title:  Uniform in time approximations of stochastic

Abstract:   

Complicated models, for which a detailed analysis is too far out of reach, are routinely approximated via a variety of procedures; this is the case when we use multiscale methods, when we take many particle limits and obtained a simplified, coarse-grained dynamics, or, simply, when we use numerical methods, which will be the focus of this talk.  While approximating,  we make an error which is small over small time-intervals but it typically compounds over longer time-horizons. Hence, in general, the approximation error grows in time so that the results of our "predictions" are less reliable when we look at longer time-horizons.

However this is not necessarily the case and one may be able to find dynamics and corresponding approximation procedures for which the error remains bounded, uniformly in time.  We will discuss a very general approach to understand when this is possible. I will show how the approach we take is very broad and show how it can be used for all of the approximation procedures mentioned above – however the focus of the talk will be on numerical discretizations for SDEs. This is based on a series of joint works with a number of people

Date: 3.00pm Thursday 27th March 2025

Venue: LT907

Title:  GMRES upper bound based on epsilon-pseudospectra for preconditioned Toeplitz systems

Abstract:   PDE models typically require numerical discretisation with sufficiently fine meshes to capture solution features, transforming the problem into solving one or more systems of linear equations, Ax=b. Due to several factors, these systems are often large and nonsymmetric and highly nonnormal. Krylov subspace methods, such as GMRES, are powerful tools for solving these sparse systems, but preconditioning is usually required to increase efficiency. While descriptive convergence theory exists to guide preconditioner choice when A is normal, it is less informative for nonnormal ones. Due to the nonnormality, GMRES bounds based on eigenvalues, the field of values and pseudospectra can fail to be descriptive.

Our work aims to also utilise the underlying PDE and discretisation properties to better understand the GMRES convergence. We show that in many instances the preconditioned matrix can be decomposed into the identity plus a low-rank perturbation and a small-norm perturbation, resulting in an eigenvalue clustering near 1 with a few outliers.  We characterise the pseudospectra in this case, and present a new GMRES bound that takes into account the outlier in a way that overcomes overestimations seen in other bounds.

Title:  Mathematical Analysis of Refractory Period Distribution and the Underlying Molecular Regulation Mechanisms for Signal Transduction Systems

Friday 28th August, 2024, 3.00-4.00pm

Venue: LT908

Abstract:  Cellular decisions are governed by signal transduction pathways involving a series of chemical reactions. The refractory period (RP) represents the time it takes for the reaction system to regain responsiveness after a stimulus, making it a crucial factor in signal transduction pathways. Analytical expressions for RP distributions are essential for understanding its molecular regulation mechanisms. However, it depends on solving CME for systems with second or higher-order reactions, which remain open problems with traditional methods. We are developing new theories and methodologies to solve RP distributions for general time-variant signal transduction systems with second-order reactions. Our recent research shows that using path-wise representations can bypass solving CMEs analytically. Using this method, we solved the RP distribution for a class of nonlinear time-variant systems with A+A-C type of second-order reactions. We will extend to more complicated systems with A+B — C type of systems.

Title:  Stochastic Approximation and Applications

Friday 6th February, 2025, 3.00-4.00pm

Venue: LT907

Abstract:  Because of the demand and current interests in using machine learning to solve optimization problems, stochastic gradient algorithms have gained resurgent interests. These, however, are rooted in stochastic approximation (SA). In this talk, we will give a brief introduction to SA, mention the issues involved, main results, and applications in image processing, filtering, system identification, manufacturing, and financial engineering. [The main reference is H.J. Kushner and G. Yin, 2nd Ed., Springer, 2013 together with some of our more recent work that are in the preprint form.]

Title:  Ergodicity of Euler-Maruyama Schemes by the Coupling Method

Wednesday 12th March, 2025, 3.00-4.00pm

Venue: LT907

Abstract: In this talk I will explain how to use the probabilistic technique known as the coupling method in order to analyse ergodicity of Euler-Maruyama (EM) schemes for a broad class of stochastic differential equations (SDEs). After a general introduction to the coupling method, I will discuss how to use it to show that the Markov transition kernel of an EM scheme is contractive in an appropriately designed Wasserstein distance, and then I will present several useful consequences of that contractivity, such as obtaining precise convergence rates of the EM scheme to stationarity, or the Poincare inequality for its invariant measure. The talk is based on the following series of papers:

[4] J. Bao, M. B. Majka and J. Wang, Geometric ergodicity of modified Euler schemes for SDEs with super-linearity, submitted, 2024

[3] L. Liu, M. B. Majka and P. Monmarché, L^2-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems, Stochastic Process. Appl. 179 (2025), 104504

[2] L.-J. Huang, M. B. Majka and J. Wang, Strict Kantorovich contractions for Markov chains and Euler schemes with general noise, Stochastic Process. Appl. 151 (2022), 307-341,

[1] A. Eberle and M. B. Majka, Quantitative contraction rates for Markov chains on general state spaces, Electron. J. Probab. 24 (2019), paper no. 26, 36 pp.