$$\Large \qquad \rho\frac{D\mathbf{u}}{Dt} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{g}, \qquad \varphi_X(t) = \exp \left\{ it\mu - \frac{1}{2} \sigma^2 t^2 \right\}, \qquad e^{A+B} = \lim_{N \rightarrow \infty} \left(e^{A/N}e^{B/N}\right)^N$$

# Upcoming webinars:¶

## Speaker: Cyprien Soulaine¶

### Micro-continuum modelling: A hybrid-scale approach for solving coupled processes in porous media¶

Effective use of the Earth subsurface for resources and energy applications requires very good control of time- and space-dependent fluid flow regimes. Such applications include the exploitation of groundwater reservoirs, the injection and sequestration of CO2 into deep saline aquifers, the storage of nuclear waste repositories, the development of geothermal energy, and the oil and gas recovery. Subsurface acidizing can lead to a reorganization of the pore-space through mineral dissolution and precipitation. These modifications of the rocks topology locally change the streamlines and, therefore, the hydraulic properties of rocks. The feedback between flow and geochemical reactions are complex and highly coupled. A good understanding of these non-linear phenomena is crucial to assess the long-term effectiveness and the environmental impact of such processes.

Pore-scale analysis of the involved physico-chemical processes is the elementary step in a modeling strategy in a cascade of scales nested in each other. Using devoted numerical modeling based on the micro-continuum approach and microfluidic experiments, we highlighted and characterized several regimes of dissolution instability at the scale of individual grains. We then demonstrated that the presence of a gas phase produced from the dissolution of a calcite crystal changes the stability diagrams. These results, upscaled to a continuum-scale formulation, inform the hydraulic and transport properties such as effective surface area, and shed new light on the differences observed between lab and field measurements.

## Speaker: Karel Devriendt¶

### Soules vectors: applications in graph theory and the inverse eigenvalue problem¶

George W. Soules introduced a set of vectors $r_1,\dots,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n\lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, discuss some basic properties and their relevance to the symmetric nonnegative inverse eigenvalue problem. I will also cover a number an important applications to (spectral) graph theory: Soules vectors allow to realize any positive spectrum as the spectrum of a Laplacian matrix.

## Speaker: Derek Cook¶

### The Open Ball Topology¶

The open ball topology is a well known construction that induces a topology on a metric space. By saying a subset of a metric space is open if every point in it has a little ball around it we can turn any metric space into a topological space. But this leads to the question: can all topological spaces be obtained from metric spaces in this way? If not can we suitably extend the concept of a metric space sufficiently to do so? In finding some answers to these questions we will also examine whether or not the open ball topology is the only reasonable way to induce a topology on a metric space.

## Speaker: James Burridge¶

### Understanding human vowel systems as random walking words and flipping spins¶

Vowels are the primary carriers of information in human speech, and are therefore important sounds. Acoustically, they are resonances which depend on the shape of the mouth cavity, and may be reliably identified by the first two peaks of their frequency spectrum. We can think of them as points in an approximately triangular subset of $\mathbb{R}^2$, bounded by the constraints of the human vocal apparatus. The number and arrangement of vowel sounds within this triangle varies widely between languages, but there are some particularly common arrangements and repeating patterns. In this talk I will explore the general typology of vowel systems, derived from a database of 3020 languages and dialects, and present two different evolutionary models which attempt to explain them. In the first model, I will view the words of a language as a diffusing soup of Brownian particles, interacting via the vowel sounds they contain. Using a simple model of language learning I will derive inter-particle forces which cause the words to form globules, each representing a different vowel. In the second model, I will draw on ideas from Markov Chain Monte Carlo and Ising spin dynamics to construct a simple evolutionary dynamics, capable of matching the statistics of real sound inventories almost perfectly. I will then show how Lasso regularization may be used to generate a version of this model which is simple enough for humans to understand. I will explain how these models shed light on some long standing debates in linguistics as to the origin of observed vowel patterns.

The work presented in the talk was carried out in collaboration with Bert Vaux, a phonologist (studies the organization of sounds) at the Faculty of Modern and Medieval Languages and Linguistics, University of Cambridge.

## Speaker: David Greenhalgh¶

### Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model¶

In this talk we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for $R_0 <1$, where $R_0$ is the basic reproductive number, and a unique supercritical endemic equilibrium for $R_0>1$. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The talk starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small $b$, where $b$ is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the talk, both for $b$ small and for larger values of $b$.

## Speaker: Yuliya Kyrychko¶

### Dynamics of systems with distributed delays¶

Many physical, biological and engineering processes can be represented mathematically by models of coupled systems with time delays. Time delays in such systems are often either hard to measure accurately, or they are changing over time, so it is more realistic to take time delays from a particular distribution rather than to assume them to be constant. In this talk, I will show how distributed time delays affect the stability of solutions in systems of coupled oscillators. Furthermore, I will present a system with distributed delays and Gaussian noise, and illustrate how to calculate the optimal path to escape from the basin of attraction of the stable steady state, as well as how the distribution of time delays influences the rate of escape away from the stable steady state. Throughout the talk, analytical calculations will be supported by numerical simulations to illustrate possible dynamical regimes and processes.

## Speaker: Jorge Bruno¶

### Locally finite trees and the topological minor relation.¶

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1. What is the number $\lambda$ of topological types of locally finite trees?

2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, $\omega_0 \leq \lambda \leq c$ and Matthiesen refined it to $\omega_1 \leq \lambda \leq c$. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

• $\lambda = \omega_1$, and

• the size of an equivalence class can only be either $1$ or $c$.

## Speaker: Hipolito Treffinger¶

### Keep calm and build your objects in order¶

In the sixties, when Mumford introduced in algebraic geometry the so-called geometric invariant theory, he showed that, under the right (stability) conditions, some times we do not have to prove something for all objects, but only for those that are stables. Some years later, Harder and Narasimhan showed that every vector bundle of an algebraic curve can be built using the stable objects following the order induced by the slope function. Nowadays, the stability conditions have been adapted to multitude of branches of mathematics and often one can find a theorem which is an adaptation of the result of Harder and Narasimhan to each particular environment.

In this talk we will recall the definition of a torsion pair and we will introduce the indexed chains of torsion classes. Our main theorem is that every indexed chain of torsion classes induce a Harder-Narasimhan filtration. The result for stability conditions becomes then a particular case of our theorem.

After that, we will follow ideas of Bridgeland to show the existence of a (pseudo)metric in the set of indexed chain of torsion classes. Implying that all indexed chains of torsion classes form a topological space.

No previous knowledge on abelian categories nor stability condition will be assumed.

## Speaker: Patricia Verrier¶

### Mathematical Modelling at Quintessa¶

Quintessa is an employee-owned scientific consultancy of 25 staff based in Henley-on-Thames and Warrington. We work in the areas of decision support, geosciences, materials modelling, risk assessment and software development, and mathematical modelling is an important part of our day-to-day work. With a focus on the right level of detail for the problem, the methods we use can range from high-level statistical analysis to system-level compartment-based models to detailed physics-based materials modelling. This talk aims to provide an overview of the role of mathematical modelling at Quintessa and how it supports our work.

## Speaker: Vincenzo Tamma¶

### The quantum information supremacy of linear optics quantum interference¶

Multiphoton quantum interference underpins fundamental tests of quantum mechanics and quantum technologies, including applications in quantum computing, quantum sensing and quantum communication. Standard quantum information processing schemes rely on the challenging need of generating a large number of identical photons. In this talk, we show how the difference in the photonic spectral properties, instead of being a drawback to overcome in experimental realisations, can be exploited as a remarkable quantum resource. Interestingly, we demonstrate how harnessing the full multiphoton quantum information stored in the photonic spectra by frequency and time resolved correlation measurements in linear interferometers enables the characterisation of multiphoton networks and states, produces a wide variety of multipartite entanglement, and scales-up experimental demonstrations of boson sampling quantum computational supremacy.  These results are therefore of profound interest for future applications of universal spectrally resolved linear optics across fundamental science and quantum technologies with photons with experimentally different spectral properties.

### References:¶

• [1] S. Laibacher and V. Tamma, Phys. Rev. A 98, 053829 (2018)
• [2] S. Laibacher and V. Tamma arXiv:1801.03832 (2018)
• [3] V. Tamma and S. Laibacher, Phys. Rev. Lett. 114, 243601 (2015)
• [4] S. Laibacher and V. Tamma, Phys. Rev. Lett. 115, 243605 (2015)
• [5] V. Tamma and S. Laibacher,  Quantum Inf. Process. 15(3), 1241-1262 (2015)

## Speaker: Marianna Cerasuolo¶

### Mathematical model of induced drug resistance in the TRAMP model of prostate cancer¶

Androgen deprivation therapy ability to reduce tumour growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistance prostate cancer (CRPC). Enzalutamide is a second-generation drug recently approved for the treatment of CRPC. However, cases of tumour resistance to enzalutamide have now been reported. In this study the multistage model of TRAMP mice and TRAMP-derived cells have been used to extensively characterise in vitro and in vivo response and resistance to second-generation androgen receptor antagonist enzalutamide.

Within the study a multiscale mathematical model has been developed. The set of six stochastic differential equations describes the interaction between heterogeneous cancer cell populations, the pharmacokinetics of the administered drug and the dynamics of the tumour microenvironment. The integration of experimental data with a mathematical framework of tumour growth allowed to gain insights into the dynamics of intratumoral evolution. The response of prostate cancer to different therapy strategies was explored with in silico experiments. The model clearly shows that the use of enzalutamide does always cause the onset of resistance and that no strategy allows to successfully control the disease, which suggests that drug combination therapies are needed.

## Speaker: Michal Gnacik¶

### On PCA, fitting an elipse into data, and shapes of memorised random walks¶

In the first part of the talk, we discuss the principle component analysis (PCA, a procedure used for data dimensionality reduction) and describe a simple method of fitting an error elipse into data via the convariance matrix resulting from the data. In the second part of the talk we consider a random walk with sites obtained via evaluating two dimensional Brownian motion at the arrival times of a non-homogeneous Poisson process. We view this random walk as the memory (mental map) of a foraging animal, perhaps searching for food while forming a mental map of its surroundings. We will work with the corresponding covariance matrix (similarly as in the first part of the talk) and obtain a formula that measures how spherical this walk is on average with respect to the initial (or current) location of the walker; in particular, we show how this formula depends on the intensity function of a non-homogeneous Poisson process used to construct the walk. We give examples of intensity functions that lead to both spherical and aspherical shapes.

## Speaker: James Burridge¶

### Language Evolution in Spatial Domains, and Statistical Physics¶

Language is evolving everywhere, all the time. As a result, people from different parts of a language area may use their language in quite different ways. This geographical variation has often been visualized using “isoglosses”: lines marking the approximate geographical boundaries of different linguistic features. In this talk I will introduce a simple mathematical model in which domains of distinctive language use emerge spontaneously, with transition zones in between. I will show that domain boundaries (isoglosses) feel a form of surface tension and are also warped and moved by variations in population density, allowing us to predict the shapes of distinctive linguistic zones in different countries. Much of this simplicity arises from a connection between linguistics and physics: isoglosses behave much like domain walls between different atomic orderings in certain magnetic or crystalline materials, first studied by Ilya Lifshitz (a Russian Physicist) in 1962. I will then describe recent developments, in which continuous space is replaced with an embedded network, and more realistic processes describing language dynamics are investigated.

## Speaker: Jamie Foster¶

### Ion migration in perovskite solar cells¶

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding $22\%$; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around $33\%$ (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the 'anomalous' hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

## Speaker: Ittay Weiss¶

### An invitation to category theory¶

Category Theory was created by Eilenberg and Mac Lane more than 50 years ago. In this relatively short period of time Category Theory had transformed several areas of mathematics, primarily Algebraic Topology, and, perhaps unexpectedly, infiltrated nearly all areas of mathematics and even extending to physics, computer science, linguistics, and biology. Recent developments are strongly indicating that Category Theory reached a level of maturity allowing a line of applications attacking problems at the forefront of scientific and industry-fuelled inquiry. In particular, the new journal "Compositionality", the book "Seven Sketches in Compositionality: An Invitation to Applied Category Theory" by Spivak and Fong, and the book "What is Applied Category Theory?" by Bradley are some of the indications of the expected utility in the subject.

In this introductory talk I will offer an invitation to category theory assuming no prior knowledge of categories. Employing the perfect vision of hindsight I will offer a new perspective on the notion of injective, surjective, and bijective functions which will naturally lead to basic concepts of category theory and illustrate the power inherent in a categorical perspective. The lecture is accessible to second year undergraduate students. Anybody interested in accepting the invitation may be interested to attend a series of lectures I will give on Friday afternoons which will do more than just scratch the surface.