Applied Mathematics Seminars Autumn 2023
Autumn 2023
Seminars take place online on Tuesdays at 3.00pm on Zoom via the link https://ucl.zoom.us/j/99614222402. Some of the seminars will be 'hybrid' (i.e. in person +zoom). If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).
Tuesday 3 October 2023
Speaker: Ben Binder (University of Adelaide, Australia)
TITLE: QUANTIFYING AND MODELLING PATTERN FORMATION IN YEAST COLONIES
Abstract:
Yeasts have been used for biotechnology throughout recorded history. They are important human pathogens, and major experimental models of eukaryotic cells. Although yeasts are some of the most studied organisms in biology, their modes of colony pattern formation are not fully understood. In this talk, we examine macroscale spatial patterns in experimental images of Saccharomyces cerevisiae yeast, using our purpose-built app TAMMiCoL to process images into binary data which is subsequently quantified with spatial statistics. Both continuum and discrete modelling approaches are implemented to investigate the mechanisms that produce spatially non-uniform colony formation. We show the continuum approach can model regular spatial pattering observed in floral biofilm colonies and demonstrate the necessity to use a discrete model to capture highly non-uniform patterning in filamentous colonies. The basic research challenges the pioneering work of Pirt 1967 that filamentous yeast patterns are the result of diffusion limited growth. Thus, our ongoing aim is to provide a deeper understanding of the fundamental mechanisms for colonial morphology in the different modes of growth of Saccharomyces cerevisiae, with implications for this and other biofilm-forming yeasts of biotechnological or medical importance.
Tuesday 10 October 2023
Speaker: Xin Guan (UCL)
TITLE: SIMULATION OF INTERFACIAL FLOWS: LIMITING FORMS, SYMMETRY BREAKING AND A NEW TIME-EVOLUTION ALGORITHM
Abstract:
Interfacial flows, characterized by their complex nonlinear evolution and various instabilities, are a fascinating subject of fluid dynamics. Related phenomena include internal waves in oceans, lee waves in the atmosphere behind mountains, and wind-driven ripples. This talk explores the mathematical modeling of interfacial flows, where a simple but effective assumption of constant density for each fluid phase is employed. We further assume flows are inviscid and irrotational, allowing the application of potential flow theory for simulation. In the first part of this talk, we delve into the early research directions of interfacial waves dating back to the 1980s, focusing on the bifurcations of traveling waves and their limiting forms. We discuss the singular limiting solution proposed by Pullin and Grimshaw, characterized by a mushroom-shaped profile and a 120-degree angle, reminiscent of the well-known highest Stokes wave in surface gravity waves. Recent numerical confirmations by various authors using highly accurate methods highlight the significance of this discovery. We explore a unified bifurcation mechanism arising from symmetry breaking. In the second part of the talk, we introduce a novel time-evolution algorithm based on arclength-parameterization. This innovative approach enables the simulation of various interfacial flow scenarios, including wave breakings, solitary wave collisions, drop splashes, Kelvin-Helmholtz instability-induced roll-up structures, etc. What sets our method apart from existing algorithms based on particle-tracking techniques is its ability to ensure a uniform distribution of sample points on the interface, effectively mitigating numerical stiffness issues caused by the accumulation of sample particles in capillary flows.
Tuesday 17 October 2023 - hybrid on Zoom and in Maths Room 706
Speaker: Tomas Alarcon (Centre de Recerca Matematica, Barcelona)
TITLE: COARSE-GRAINING OF COMPLEX BIOLOGICAL MODELS
Abstract:
The mathematical study of biological systems often involves the use of complex mathematical models. Their analysis often becomes cumbersome and even their efficient numerical simulation becomes a problem of significant difficulty. This, in turn, hinders further study regarding mathematical analysis, parameter fitting and identifiability, etc. To overcome these issues, we have been developing a number of approaches to coarse-graining such models. In this talk, I will address three such approaches applied to different models: a multiscale stochastic model of tumour growth, an individual-based model with multistable agents, and a model for large-scale transitions in epigenetic landscapes.
Tuesday 24 October 2023 - No seminar
Tuesday 31 October 2023
Speaker: Luca Giuggioli (University of Bristol)
TITLE: RESOLUTION OF A HUNDRED YEAR OLD PROBLEM ON LATTICE RANDOM WALKS AND ITS IMPACT ON MODELLING MOVEMENT AND INTERACTIONS IN HETEROGENEOUS SPACE
Abstract:
In many complex systems the emergence of spatio-temporal patterns depends on the movement and interaction between pairs of individuals, agents or subunits comprising the whole system as well as interactions with the spatial heterogeneities of the environment. Theoretical predictions of such patterns rely upon quantifying when and where interaction events might occur. Even in simple scenarios when the dynamics are Markovian, it has been challenging to obtain estimates of interaction statistics due to the lack of a mathematical formalism that accounts for the co-occurrence of multiple random processes. In this talk I will present such formalism, that is a general theory that allows to quantify the spatio-temporal dynamics of interactions when a token of information is transferred upon co-location or proximity, the so-called reactive interactions, e.g. an infection or an encounter, and when interactions result from the spatial heterogeneities of the environment, the so-called inert interactions, e.g. when movement is modified by the presence of areas with different diffusivity, partially permeable or impenetrable barriers or long-range connections between distant sites. I will present examples when the underlying lattice topology is Cartesian, hexagonal and triangular. Spatial discretisation is key to develop such a theory bypassing the need to solve unwieldy boundary value problems, giving predictions that are either fully analytical or semi-analytical. For the case of permeable barriers I also show that the spatial continuous limit of the discrete lattice dynamics allows to derive a new fundamental equation that go beyond the diffusion and the Smoluchowski equation in the presence of an arbitrary number of thin permeable barriers. If time allows I will also present results on the spatio-temporal dynamics when the dynamics is non-Markov, specifically when the lattice walker steps are correlated.
Tuesday 7 November 2023 - Reading Week - No Seminar
Tuesday 14 November 2023
Speaker: Matthew Butler (UCL)
TITLE: STICK OR TWIST: CONTROLLING MICRO-SYSTEMS WITH FLUID-INDUCED DEFORMATION
Abstract:
Although deformation of solids is often seen as a negative trait (e.g. failure through buckling, twisting and warping), recent work has shown that knowledge of elastic behaviour can be exploited to achieve positive outcomes. One area where this is becoming increasingly prevalent is in micro-systems. At these small scales, the dominant physics is different to everyday experience and new mechanisms are needed to perform simple tasks such as swimming, gripping, sorting and transporting. Designed structures or components that distort in pre-specified ways are a promising mechanism to achieve some of these goals.
In this talk, I will explore two distinct, useful ways that fluids are able to deform compliant solid structures: elasto-capillary adhesion and hydrogel swelling. I will first discuss how adhesion forces due to surface tension can be enhanced and controlled by the addition of deformable structures, before considering the poro-elastic dynamics of hydrogel swelling and shrinking, including how control of this can be exploited to make shape-changing micro-structures. Comprehension of these fundamental model systems can ultimately help us develop new micro-technologies, as well as improve existing devices.
Tuesday 21 November 2023 - No Seminar
Tuesday 28 November 2023
Speaker: Alexandria Volkening (Purdue University)
TITLE: MODELING AND TOPOLOGICAL DATA ANALYSIS OF ZEBRAFISH PATTERNS
Abstract:
Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that appear on their bodies and fins. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to better link genes, cell behavior, and visible animal characteristics — I seek to identify the alterations to cell interactions that lead to mutant patterns. Toward this goal, I will overview our work using agent-based models to simulate pattern formation and make experimentally testable predictions. However, stochastic, microscopic models are not analytically tractable using traditional techniques, and comparing model output and in vivo patterns is often a qualitative process, both for zebrafish skin and other biological patterns. To help address this broader challenge, I will also discuss the topological methods that we have developed to quantify variability in cell-based patterns at large scale.
Tuesday 5 December 2023
Speaker: Giulia Celora (UCL)
TITLE: SELF-ORGANIZED PATTERNING IN COMPLEX FLUIDS
Abstract:
Understanding how multicomponent systems self-organise to control their emergent dynamics across spatial and temporal scales is a fundamental problem with important applications in many areas; from the design of soft materials to the study of developmental biology. In this talk, I will discuss how we can use mathematical modelling to understand the role of microscale physical interactions in the self-organisation of complex fluids. I will illustrate this by presenting two examples. Firstly, I will discuss self-organization in stimuli-responsive polyelectrolyte gels surrounded by an ionic solution; secondly, I will discuss self-organization during collective migration of multicellular communities. Our results reveal hidden connections between these two initially disconnected applications hinting at the existence of general principles controlling self-organisation of both inanimate and living matter.
Tuesday 12 December 2023
Speaker: Triantaphyllos Akylas (MIT,USA)
TITLE: STABILITY OF INTERNAL GRAVITY WAVE MODES: FROM TRIAD RESONANCE TO BROADBAND INSTABILITY
Abstract:
A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter μ, compares to the basic-state amplitude parameter, ε ≪ 1. For μ = 0(1), the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that μ ≪ 1 but α = μ/ε ≫ 1, this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as μ is further decreased holding ε fixed, higher-frequency perturbations than these two subharmonics come into play, and when α = 0(1) Floquet modes feature broadband spectrum. In this regime, PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilizing effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.