Thu, 07 Mar 2024
12:00
L6

Well-posedness of nonlocal aggregation-diffusion equations and systems with irregular kernels

Yurij Salmaniw
(Mathematical Institute, University of Oxford)
Abstract

Aggregation-diffusion equations and systems have garnered much attention in the last few decades. More recently, models featuring nonlocal interactions through spatial convolution have been applied to several areas, including the physical, chemical, and biological sciences. Typically, one can establish the well-posedness of such models via regularity assumptions on the kernels themselves; however, more effort is required for many scenarios of interest as the nonlocal kernel is often discontinuous.

 

In this talk, I will present recent progress in establishing a robust well-posedness theory for a class of nonlocal aggregation-diffusion models with minimal regularity requirements on the interaction kernel in any spatial dimension on either the whole space or the torus. Starting with the scalar equation, we first establish the existence of a global weak solution in a small mass regime for merely bounded kernels. Under some additional hypotheses, we show the existence of a global weak solution for any initial mass. In typical cases of interest, these solutions are unique and classical. I will then discuss the generalisation to the $n$-species system for the regimes of small mass and arbitrary mass. We will conclude with some consequences of these theorems for several models typically found in ecological applications.

 

This is joint work with Dr. Jakub Skrzeczkowski and Prof. Jose Carrillo.

Wed, 28 Feb 2024
12:00
L6

Non-regular spacetime geometry, curvature and analysis

Clemens Saemann
(Mathematical Institute, University of Oxford)
Abstract

I present an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor
on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study
curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective
proved extremely fruitful in the Riemannian case (Alexandrov- and CAT(k)-spaces). After introducing the basics of our
approach, we report on recent progress in developing a Sobolev calculus for time functions on such non-smooth
Lorentzian spaces. This seminar talk can also be viewed as a primer and advertisement for my mini course in
May: Current topics in Lorentzian geometric analysis: Non-regular spacetimes

Fri, 20 Oct 2023

15:00 - 16:00
L5

Euler characteristic in topological persistence

Vadim Lebovici
(Mathematical Institute, University of Oxford)
Further Information

Vadim Lebovici is a post-doc in the Centre for Topological Data Anslysis. His research interests include: 

  • Multi-parameter persistent homology
  • Constructible functions and Euler calculus
  • Sheaf theory
  • Persistent magnitude
Abstract

In topological data analysis, persistence barcodes record the
persistence of homological generators in a one-parameter filtration
built on the data at hand. In contrast, computing the pointwise Euler
characteristic (EC) of the filtration merely records the alternating sum
of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical
"signal/noise" dichotomy, EC tools are powerful descriptors, especially
when combined with new integral transforms mixing EC techniques with
Lebesgue integration. Our motivation is fourfold: their applicability to
multi-parameter filtrations and time-varying data, their remarkable
performance in supervised and unsupervised tasks at a low computational
cost, their satisfactory properties as integral transforms (e.g.,
regularity and invertibility properties) and the expectation results on
the EC in random settings. Along the way, we will give an insight into
the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and
the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].

 

 

Fri, 13 Oct 2023

15:00 - 16:00
L5

What do we want from invariants of multiparameter persistence modules?

Luis Scoccola
(Mathematical Institute, University of Oxford)
Further Information

Luis Scoccola is a post-doc in the Centre for Topological Data Analysis, Mathematical Institute. He is a mathematician and computer scientist working in computational topology and geometry, and applications to machine learning and data science.

Abstract

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

 

Thu, 12 May 2022

12:00 - 13:00
L1

Averaged interface conditions: evaporation fronts in porous media (Ellen Luckins) & Macroscopic Transport in Heterogeneous Porous Materials (Lucy Auton)

Lucy Auton & Ellen Luckins
(Mathematical Institute, University of Oxford)
Abstract

Macroscopic Transport in Heterogeneous Porous Materials

Lucy Auton

Solute transport in porous materials is a key physical process in a wide variety of situations, including contaminant transport, filtration, lithium-ion batteries, hydrogeological systems, biofilms, bones and soils. Despite the prevalence of solute transport in porous materials, the effect of microstructure on flow and transport remains poorly understood and improving our understanding of this remains a major challenge.  In this presentation, I consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium, allowing for anisotropy.  I use a nontrivial extension to classical homogenisation theory via the method of multiple scales to rigorously upscale the novel problem involving cells of varying area. This results in simple effective continuum equations for macroscale flow and transport where the effect of the microscale geometry on the macroscopic transport and removal is encoded within these simple macroscale equations via effective parameters such as an effective local anisotropic diffusivity and an effective local adsorption rate.  For a simple example geometry I exploit the two degrees of microstructural freedom in this problem, obstacle size and obstacle spacing, to investigate scenarios of uniform porosity but heterogenous microstructure, noting the impact this heterogeneity has on filter efficiency. 

This model constitutes the development of the core framework required to consider other crucial problems such as solute transport within soft porous materials for which there does not currently exist a simple macroscale model where the effective diffusivity and removal depend on the microstructure. Further, via this methodology I will  derive a  bespoke model for fluoride and arsenic removal filters. With this model I will be able to optimise the design of fluoride-removal filters which are being deployed across rural India. The design optimisation will both increase filter lifespan and reduce filter cost, enabling more people to access safe drinking water

 

Averaged interface conditions: evaporation fronts in porous media

Ellen Luckins

Homogenisation methods are powerful tools for deriving effective PDE models for processes incorporating multiple length-scales. For physical systems in which interface processes are crucial to the overall system, we might ask how the microstructure impacts the effective interface conditions, in addition to the PDEs in the bulk. In this talk we derive an effective model for the motion of an evaporation front through porous media, combining homogenisation and boundary layer analysis to derive averaged interface conditions at the evaporation front. Our analysis results in a new effective parameter in the boundary conditions, which encodes how the shape and speed of the porescale evaporating interfaces impact the overall drying process.

Tue, 06 Mar 2018

12:45 - 13:30
C5

Modelling the collective migration of neural crest cells

Rasa Giniunaite
(Mathematical Institute, University of Oxford)
Abstract

Collective neural crest (NC) cell migration determines the formation of peripheral tissues during vertebrate development. If NC cells fail to reach a target or populate an incorrect location, improper cell differentiation or uncontrolled cell proliferation can occur. Therefore, knowledge of embryonic cell migration is important for understanding birth defects and tumour formation. However, the response of NC cells to different stimuli, and their ability to migrate to distant targets, are still poorly understood. Recently, experimental and computational studies have provided evidence that there are at least two subpopulations of NC cells, namely “leading” and “trailing” cells, with potential further differentiation between the cells in these subpopulations [1,2]. The main difference between these two cell types is the mechanism driving motility and invasion: the leaders follow the gradient of a chemoattractant, while the trailing cells follow “gradients” of the leaders. The precise mechanisms underlying these leader-follower interactions are still unclear.

We develop and apply innovative multi-scale modelling frameworks to analyse signalling effects on NC cell dynamics. We consider different potential scenarios and investigate them using an individual-based model for the cell motility and reaction-diffusion model to describe chemoattractant dynamics. More specifically, we use a discrete self-propelled particle model [3] to capture the interactions between the cells and incorporate volume exclusion. Streaming migration is represented using an off-lattice model to generate realistic cell arrangements and incorporate nonlinear behaviour of the system, for example the coattraction between cells at various distances. The simulations are performed using Aboria, which is a C++ library for the implementation of particle-based numerical methods [4]. The source of chemoattractant, the characteristics of domain growth, and types of boundary conditions are some other important factors that affect migration. We present results on how robust/sensitive cells invasion is to these key biological processes and suggest further avenues of experimental research.

 

[1] R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R.E. Baker, P. K. Maini and P. M. Kulesa. (2012). Multiscale mechanisms of cell migration during development: theory and experiment, Development, 139, 2935-2944.

[2] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, P. K. Maini, R. E. Baker and P. M. Kulesa. (2015). Neural crest migration is driven by a few trailblazer cells with a unique molecular signature narrowly confined to the invasive front, Development, 142, 2014-2025.

[3] G. Grégoire, H. Chaté and Y Tu. (2003). Moving and staying together without a leader, Physica D: Nonlinear Phenomena, 181, 157-170.

[4] M. Robinson and M. Bruna. (2017). Particle-based and meshless methods with Aboria, SoftwareX, 6, 172-178. Online documentation https://github.com/martinjrobins/Aboria.

Tue, 20 Feb 2018

12:45 - 13:30
C5

Modular Structure in Temporal Protein Interaction Networks

Florian Klimm
(Mathematical Institute, University of Oxford)
Abstract

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.

Tue, 14 Nov 2017

12:45 - 13:30
C5

A Bio-inspired Design for a Switchable Elastocapillary Adhesive

Matt Butler
(Mathematical Institute, University of Oxford)
Abstract

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation significantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

Tue, 17 Oct 2017
12:45
C5

Analysis of small contacts between particles in a furnace

Caoimhe Rooney
(Mathematical Institute, University of Oxford)
Abstract

Many metallurgical processes involve the heat treatment of granular material due to large alternating currents. To understand how the current propagates through the material, one must understand the bulk resistivity, that is, the resistivity of the granular material as a whole. The literature suggests that the resistance due to contacts between particles contributes significantly to the bulk resistivity, therefore one must pay particular attention to these contacts. 

My work is focused on understanding the precise impact of small contacts on the current propagation. The scale of the contacts is several order of magnitude smaller than that of the furnace itself, therefore we apply matched asymptotics methods to study how the current varies with the size of the contact.

Fri, 26 May 2017

16:00 - 17:00
L1

North meets South Colloquium

Erik Panzer + Yuji Nakatsukasa
(Mathematical Institute, University of Oxford)
Abstract

Erik

Erik Panzer

Feynman integrals, graph polynomials and zeta values

Where do particle physicists, algebraic geometers and number theorists meet?

Feynman integrals compute how elementary particles interact and they are fundamental for our understanding of collider experiments. At the same time, they provide a rich family of special functions that are defined as period integrals, including special values of certain L functions.

In the talk I will give the definition of Feynman integrals via graph polynomials and discuss some examples that evaluate to values of the Riemann zeta function. Then I will discuss some of the interesting questions in this field and mention some of the techniques that are used to study these.

 

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Yuji Nakatsukasa

Computing matrix eigenvalues

The numerical linear algebra community solves two main problems: linear systems, and eigenvalue problems. They are both vastly important; it would not be too far-fetched to say that most (continuous) problems in scientific computing eventually boil down to one or both of these.

This talk focuses on eigenvalue problems. I will first describe some of their applications, such as Google's PageRank, PCA, finding zeros and poles of functions, and global optimization. I will then turn to algorithms for computing eigenvalues, namely the classical QR algorithm---which is still the basis for state-of-the-art. I will emphasize that the underlying mathematics is (together with the power method and numerical stability analysis) rational approximation theory.

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