# Mathematics & StatisticsSeminars and colloquia

• 28
AUG
2024

### Prof Zhuoyi Song (Fudan University of China) - Stochastic Analysis

Title: Mathematical Analysis of Refractory Period Distribution and the Underlying Molecular Regulation Mechanisms for Signal Transduction Systems
Location: LT908
Time: 3.00pm
• 04
SEPT
2024

### Dr Jorge Carneiro (NOVA University of Lisbon) - Health Ecology Modelling

Title: TBC
Location: LT908
Time: 1.00pm

### Nonlinear evolutionary processes, operator theory for the study of differential and integral equations. Enumerative, bijective and algebraic combinatorics.

Title:   Mesh pattern occurrence in random permutations

Date:  3.00pm Tuesday 3rd October

Venue: LT907

Abstract:  We say that the likelihood of a mesh pattern is the asymptotic probability that a random permutation contains an occurrence of the pattern. In this talk we will investigate the likelihood of a variety of patterns, determining their values for every vincular pattern. For bivincular patterns, the Small Anchors Theorem distinguishes between those patterns whose likelihood equals zero, those whose likelihood is positive but less than 1, and those whose likelihood equals 1. We also know how to determine the (rational) likelihood of any bivincular pattern formed of what we call anchored trees. Other bivincular patterns, such as the small ascent and small descent, have irrational likelihoods, whose values can be established using the Chen–Stein method. This is joint work with Jason Smith.

Title: On semi-transitivity of (extended) Mycielski graphs

Date: 3.00pm Tuesday 17th October 2023

Venue: LT908

Abstract: An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v0 → v1 → · · · → vk either there is no arc between v0 and vk, or vi → vj is an arc for all 0 ≤ i < j ≤ k. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs, and they are precisely the class of word-representable graphs studied extensively in the literature.

The Mycielski graph of an undirected graph is a larger graph, constructed in a certain way, that maintains the property of being triangle-free but enlarges the chromatic number. These graphs are important as they allow to prove the existence of triangle-free graphs with arbitrarily large chromatic number. An extended Mycielski graph is a certain natural extension of Mycielski graphs.

In this talk, I will discuss a complete characterization of semi-transitive extended Mycielski graphs and comparability Mycielski graphs, as well as a conjectured complete characterization of semi-transitive Mycielski graphs. My results are a far-reaching extension of the result of Kitaev and Pyatkin on non-semi-transitive orientability of the Mycielski graph µ(C5) of the cycle graph C5. Also, I will mention how to use a recent result of Kitaev and Sun to shorten the length of the original proof of non-semi-transitive orientability of µ(C5) from 2 pages to a few lines.

Title: Eigenvalues of canonical systems

Date: 3.00pm Tuesday 31st October 2023

Venue: LT907

Abstract: In this talk I shall consider eigenvalues of 2x2 canonical systems, which cover one-dimensional Schrödinger equations, Sturm-Liouville equations, Jacobi operators, Dirac systems and (generalised) Krein strings as special cases.  I am particularly interested in the asymptotic behaviour and the density of eigenvalues.  This behaviour is connected with the growth and the smoothness of the coefficients.  It turns out that the results change substantially when one moves from dense to sparse spectrum.  I will also mention a trace formula for the eigenvalues.

Title: Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections

Date: 3.00pm Tuesday 7th November 2023

Venue: LT907

Abstract: Consider a single photon living in curved space. It is described by Maxwell’s equations. We seek solutions harmonic in time. This reduces to the spectral problem for the operator curl, whose spectrum is, in general, asymmetric about zero (think particle/antiparticle).

Spectral asymmetry is a classical subject in analysis and geometry, whose origins lie in the papers of Atiyah, Patodi and Singer. In this talk I will discuss a new approach to the study of spectral asymmetry based on the use of pseudodifferential techniques developed in a series of recent joint papers by Dmitri Vassiliev and myself.

Emphasis will be placed on ideas and motivation; the talk will include a brief historical overview of the development of the subject area.

Title: Travelling waves in the Rosenau-KdV equation

Date: 3.00pm Tuesday 28th November 2023

Venue: LT907

Abstract: We consider the existence of monotone travelling waves in the Rosenau-KdV equation, which exhibits phenomena that cannot be seen in the well-known Burgers-KdV equation. Many problems are still open. This is joint work with N. Bedjaoui and G. Maypaokha (UPJV).

Title: Mathematical analysis of fracture and related phenomena in atomistic modelling of crystalline materials

Date: 3.00pm, Tuesday 5th December 2023

Venue: LT907

Abstract: The atomistic modelling of fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial and exciting challenges, both on a theoretical and a practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms such as (i) crack propagation; (ii) near-crack tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results.

In this talk I will aim to summarise on-going efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will begin by introducing a mathematical framework, based in part on bifurcation theory, giving rise to well-defined models for which regularity and stability of solutions can be discussed, followed by describing how the theory can be used to set up numerical continuation-based simulations, e.g. for Mode I crack propagation in silicon on the (111) cleavage plane, using state-of-the-art interatomic potentials. Subsequently I will outline how this framework can be used to rigorously derive upscaled models of near-crack-tip plasticity.

Finally, I will also touch upon recent work on proposing a much more general framework in which a wide range of defect nucleation & migration phenomena in the atomistic modelling of crystalline materials can be treated as bifurcation events.

Title: Thresholds for patterns in random permutations

Date: 3.00pm Tuesday 12th December 2023

Venue: LT420

Abstract: In this talk we will investigate thresholds for the appearance and disappearance of consecutive patterns occurring within large random permutations as the number of inversions increases. We establish these lower and upper thresholds for any fixed consecutive pattern. We also consider thresholds for classical and vincular patterns. To do so, we work with inversion sequences, which we consider to be weak integer compositions. As a result, we introduce a model of random integer compositions which we call the geometric random composition. This talk will focus on how we transfer thresholds for patterns in the geometric random composition to get thresholds for patterns in the uniform random permutation.

Title: Distribution of mesh patterns

Date: 2.00pm Tuesday 23rd January 2024

Venue: LT907

Abstract: The notion of a mesh pattern, generalizing several classes of permutation patterns, was introduced in 2011 by Branden and Claesson to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. There is a long line of research papers dedicated to the study of mesh patterns and their generalizations.

In this talk, I will discuss a systematic study of avoidance and distribution of mesh patterns of short length, and also some other more general results

Title: Coagulation and Combinatorics

Date: 2.00pm on Tuesday 20th February 2024

Venue: LT907

Abstract: I will review the recent combinatorics approach to discrete coagulation equations due to Łepek and coworkers and the open problems that arise in that area.

Title: Vortex stretching in Navier-Stokes turbulence

Date: 3.00pm on Tuesday 27th February 2024

Venue: LT907

Abstract: Turbulence in the Navier-Stokes equations is a major nonlinear problem, that cannot be tackled with existing analytical methods. I discuss key characteristics of turbulence, explaining its physical content and the momentum balance it encodes. I  indicate the key analytical difficulty and past failures to tackle it in the velocity-pressure space. I introduce the vortex dynamical viewpoint and explain the association between the latter and key turbulent physics. By employing the vortex dynamical viewpoint, I compute key aspects of turbulent vortex stretching, including, among other, the Lyapunov exponents of the tangent system, and the correlations between strain rate eigendirections and coherent, filamentary vorticity.

Title: Coagulation, non-associative algebras and combinatorial trees

Date: 2.00pm Tuesday 12th March 2024

Venue: LT907

Abstract: We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by:

• grafting together the roots of all pairs of order-compatible trees at preceding orders, or
• attaching binary branches to all free branches of trees at the previous order.

We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.

Title: Asymptotic enumeration of monotone permutation grid classes

Date: 2.00pm Tuesday 26th March 2024

Venue: LT907

Abstract: A monotone grid class Grid(M) is a set of permutations whose shape satisfies constraints specifies by the matrix M, all of whose entries are in {1,−1, 0}. Each entry of the matrix corresponds to a cell in a gridding of a permutation. If the entry is 1, then the points in the cell must increase. If it is −1, they must decrease. If it is 0, the cell must be empty. To find the exact number of permutations of length n in Grid(M) is hard, because a permutation may have more than one gridding, so we only determine the asymptotic enumeration. To do this, we find the number of gridded permutations, the typical proportion of points in each cell, and the ways in which permutations can be gridded. Our focus is mainly on L-shaped, T-shaped, and X-shaped classes.

Title:  Non-interpenetration of matter in lower-dimensional structures

Date:  3pm Wednesday 19th June 2024

Venue: LT907

Abstract:  Non-interpenetration of matter is a well-known challenge for solid elastic materials combining analytical and geometrical aspects. In the bulk model, at least on the conceptual level, non-interpenetration is quite understood even if many challenges still remain open. In lower-dimensional structures (plates, rods), the situation seems to be even less clear. Focusing on rods in the plane, we will introduce a possible concept of noninterpenetration and show density and Γ-limit results in this case. This is a joint work in progress with B. Benešová, D. Campbell, and S. Hencl (all from Prague).

### Liquid crystals, Droplet evaporation, Thin-film flow, Complex fluids, Medical product design, Flows in porous & complex media, Non-linear waves.

Title: Resonant free-surface water waves in closed basins

Date:  1.00pm Thursday 25th January 2024

Title: Data-driven design optimisation of chemical reactors

Date:  1.00pm Thursday 1st February 2024

Title: Oscillatory bodily flows: the  eye and the brain

Date:  1.00pm Thursday 8th February 2024

Title: Determination of the index of refraction of anti-reflection coatings

Date:  1.00pm Thursday 22nd February 2024

Title: Electrostatics and variational perturbation theory

Date:  1.00pm Thursday 29th February 2024

Title: TBC

Date:  1.00pm Thursday 7th March 2024

Title: Interplay between zonal jets, waves and turbulence: application to gas giants

Date:  1.00pm Thursday 14th March 2024

Title: Growing in the wind - an interdisciplinary investigation of wind influence on plant growth

Date:  1.00pm Thursday 21st March 2024

### Departmental Colloquia

Title: tbc

Date: 3.00pm Wednesday 7th February 2024

Venue: tbc

Abstract:  tbc

Title: tbc

Date: 3.00pm Wednesday 28th February 2024

Venue: LT908

Abstract:  tbc

### Marine Population Modelling, Mathematical Biology, Epidemiology and Statistical Informatics

Title: The Role of Individuals in Determining the Impacts of Changing Environments on Population Level Dynamics

Date: Wednesday 7th February, 1.00-2.00pm

Venue: LT907

Abstract:

Climate change is having profound effects on the incidence of vector borne disease, such as dengue, chikungunya and West Nile virus. However, developing effective measures of disease risk on a global scale are challenged by the complex ways in which environmental variation acts in vector-host-pathogen systems. One way in which insect vectors, such as mosquitos, respond to environmental variation is to change their traits this can result in populations comprised of groups of individuals which differ in their traits (e.g. size, fecundity). For example, if food is scarce for juvenile mosquitos then when they become adults they are smaller, and lay fewer eggs to ensure there is less competition for food in the next generation. The environment of the juvenile determines the trait the individual has as an adult. In this way the individuals adapt to the environment as well as shape the environment for future generations.

Current models over-simplify the interaction between individuals, populations and the environment, so risk misestimating predictors of disease risk. Here, we derive a mathematical framework for capturing the interaction of individuals, their traits and the population dynamics. I will show how this new mathematical framework leads to both interesting mathematical questions and can be used to help explain the location, magnitude and timing of historical dengue outbreaks.

Title: Quantifying Performance and Resilience in Marine Assemblages with Complex Life-Histories

Date: Wednesday 21st February, 1.00-2.00pm

Venue: LT908

Abstract:  Ongoing global change challenges our ability to predict how natural populations will both respond to novel climatic regimes and utilise available habitat space. Corals are critical to the functioning of coastal reef ecosystems and, yet, in spite of their intrinsic and economic value, are threatened by an increasing plethora of abiotic and biotic disturbances. Preventing the ensuing loss of coral coverage and diversity calls for a mechanistic understanding of resilience across coral species and populations that is currently lacking in coral reef science. Meanwhile, changing coastal conditions and our ever-expanding exploitation of marine resources, has heralded a perceived increase in the abundance of coastal jellyfish assemblages. While jellyfish are also an important component of coastal marine communities, their public perception is often tainted by their proclivity for aggregating in vast numbers, known as jellyfish blooms, which can disrupt fishing and tourism activities. However, despite the socioeconomic ramifications associated with the formation of these jellyfish blooms, the complex and cryptic lifecycles exhibited by jellyfish species largely precludes accurate predictions into their timing and location, restricting our ability to manage and mitigate their ecological and economic impacts. Here, I will introduce research comprising state-structured population modelling, novel transient demographic approaches, and state-of-the-art hydrodynamic simulations, that offers valuable insight into the performance and resilience of these complex and cryptic marine assemblages under future climate scenarios; frameworks that represent key decision-support tools for informing both the conservation of global coral reefs and our management of the socioeconomic impacts of jellyfish bloom formation.

Title: Some Data-Driven Approaches to Surveillance of Covid-19 in the UK

Date: Wednesday 28th February, 1.00-2.00pm

Venue: LT908

Abstract: Whilst much of the work in modelling transmission of the pandemic was conducted using mathematical transmission models, the quantity of data made available through open data portals, such as the Covid Dashboard, provided alternatives to understanding and intervening to improve public health outcomes. In this talk I will outline some statistical approaches using surveillance data from varying spatial scales to study underlying dynamics of transmission of Covid-19 in the UK. In particular, multivariate flexible regression models and dimension spatial dimension reduction techniques will be used to estimate relative transmissibility of emerging variants of concern, as well as nowcasting current states of pandemic from noisy multivariate time series, respectively.

Title: Estimating the Size of Aedes Aegypti Populations from Dengue Incidence Data: Implications for the Risk of Yellow Fever, Zika Virus and Chikungunya Outbreaks

Date: Wednesday 17th April, 1.00-2.00pm

Venue: TBC

Abstract:  In this talk I present a model to estimate the density of aedes mosquitoes in a community affected by dengue. The model is based on the fitting of a continuous function to the incidence of dengue infections, from which the density of infected mosquitoes is derived straightforwardly. Further derivations allows the calculation of the latent and susceptible mosquitoes' densities, the sum of the three equals the total mosquitoes' density. The model is illustrated with the case of the risk of urban yellow fever resurgence  in dengue infested areas but the same methods apply for other aedes-transmitted infections like Zika and chikungunya viruses.

Title: Using Minimalistic Food-Web Models to Inform Fisheries Management

Date: Wednesday 1st May, 1.00-2.00pm

Venue: TBC

Abstract:  Chance and Necessity (CaN) modelling is a minimalistic food-web modelling framework integrating the existing knowledge about an entire or part of the ecosystem, the available data, and uncertainties. Unlike most of the large ecosystem models, the CaN framework does not aim at predicting the future state of commercial stocks, but rather to reconstruct possible past dynamics of a food-web. Reconstructions result from the delimitation of the possible “state-space” of the food-web based on ecological survey data and expert knowledge, and the exploration of this “state-space”. Here, I will present the underlying concepts of CaN modelling, present case studies of CaN modelling applications and provide examples on how the model outputs can be used to improve the management of commercial fisheries in the future.

Title: Modelling and Inference and Heterogeneity in Bacterial Growth

Date: Wednesday 22nd May, 1.00-2.00pm

Venue: LT908

Abstract:  E. coli is a common bacterium found in the intestines of humans and animals. While many strains are harmless, some can cause serious illness, such as diarrhoea, urinary tract infections, and in severe cases, even kidney failure or death. Understanding its behaviour and mechanisms of infection is therefore vital for public health.  Like many bacteria, E. coli can also develop resistance to antibiotics.  Persister cells are a small subpopulation of bacterial cells which are slow-growing, which allows them to survive in harsh conditions such as exposure to antibiotics or other stressful environments. These cells are distinct from regular bacterial cells because they are not killed by antibiotics. Studying how resistance emerges and spreads within bacterial populations helps in developing strategies to combat antibiotic-resistant strains. The aim of this talk is to discuss the range of modelling approaches to determine the impact non-inheritable variation between individual cells has on population growth and population response to stressful environments. Various models will be considered including systems of ODEs, age-structured PDEs, renewal equations, and stochastic branching process. Where possible, comparisons and connections will be made between the different modelling approaches. We also discuss some initial work on model inference, where the objective is to indirectly determine the possibility of heterogenous sub-populations from total population data. These techniques will be applied to a probabilistic model of data generated from a microfluidic dynamic cytometer.

Title: Heterogeneity and Identifiability in Mathematical Biology

Date: Wednesday 29th May, 1.00-2.00pm

Venue: LT908

Abstract:  Heterogeneity is a dominant factor in the behaviour of many biological processes and is often a significant source of the variation observed in biological data. Despite this, it is relatively rare for mathematical models of biological systems to incorporate variability in model parameters as a source of noise. In the first part of talk, I motivate and present a new computationally efficient method for inference and identifiability analysis of so-called random parameter models based on an approximate moment-matched solution constructed through a multivariate Taylor expansion.

Effective application of mathematical models to interpret biological data and make accurate predictions typically requires that model parameters are identifiable. Yet, there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture the spatial heterogeneities features inherent to many phenomena. In the second part of this talk, I provide an introduction to structural identifiability before presenting a new methodology applicable to a broad class of PDE models. I then conclude by discussing the future of identifiability analysis for the spatial, random parameter, and stochastic models that are fast becoming pervasive throughout mathematical biology.

Title: How Lifestyle Differences Affect Epidemic Spread: Heterogeneous Density Dependence in Infectious Contact Rates

Date: Wednesday 10th July, 1.00-2.00pm

Venue: LT908

Abstract:

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:

### Numerical solutions of PDEs, Stochastic computation, Numerical linear algebra, Computational physics & engineering

Title:  DEM applied in the mining and mineral extraction industry

Date:  1pm Tuesday 19th March 2024

Venue: LT907

Abstract:  As the world transitions towards net zero though electrification our reliance upon critical minerals and ores such as copper, iron, lithium, etc is ever increasing. The mining and  mineral extraction industry currently uses 4-7% of the worlds available energy supply.  Mining and mineral extraction companies are faced with decarbonising their impact on the planet while simultaneously increasing extraction from excavation sites to meet demand using traditional technology. Discrete Element Modelling (DEM) gives us insight into why products behave the way they do and process condition parameter exploration to ultimately drive towards new designs and a more sustainable future.  The presentation will cover who the Weir Advanced Research Centre (WARC) is at UoS, our journey so far in using DEM for Weir products and the future direction of coupling such tools with other mathematical tools.

Title:  Reducible networks of Prandtl-Ishlinskii operators with economic and financial applications

Date:  1pm Tuesday 26th March 2024

Venue: LT907

Abstract:  If the nodes in a network have input-output responses satisfying a certain property (ie. they are Prandtl-Ishlinskii (PI) operators) then remarkable simplifications are possible. For arbitrary network topologies (under mild additional conditions) the entire network can be rigorously reduced to a single aggregated PI operator. This is true even if cascading behaviour, such as bubbles and crashes, can occur in the network.

Two applications will be presented. One is a financial market model incorporating momentum traders. The other is a macroeconomic model with the aggregated PI operator representing inflation expectations in the economy.

Title:  Non-interpenetration of matter in lower-dimensional structures

Date:  3pm Wednesday 19th June 2024

Venue: LT907

Abstract:  Non-interpenetration of matter is a well-known challenge for solid elastic materials combining analytical and geometrical aspects. In the bulk model, at least on the conceptual level, non-interpenetration is quite understood even if many challenges still remain open. In lower-dimensional structures (plates, rods), the situation seems to be even less clear. Focusing on rods in the plane, we will introduce a possible concept of noninterpenetration and show density and Γ-limit results in this case. This is a joint work in progress with B. Benešová, D. Campbell, and S. Hencl (all from Prague).

### Stochastic Differential Equations, Stochastic Computation, Time Series, Probability, Image Analysis

Title:  Numerical Solutions of a Markov-Switching One-Factor Volatility Model with Non-Globally Lipschitz Continuous Coefficients

Friday 25th January, 2024, 4.00-5.00pm

Venue: LT907

Abstract: We extend the one-factor stochastic volatility model to incorporate coefficient terms of super-linear growth under the Markov-switching framework. Since the proposed model is intractable analytically, we develop various mathematical techniques to investigate convergence in probability of the numerical solutions under the local Lipschitz condition. Finally, we perform simulation examples to demonstrate the theoretical results and justify the theoretical results for the valuation of some financial options.

Title:  Explicit Convergence Rates for the M/G/1 Queue under Perturbation

Friday 19th April, 2024, 3.00-4.00pm

Venue: LT907

Abstract: Stochastically ordered Markov process is a topic of special concern to us. As is mentioned by Meyn and Tweedie, many Markov processes are stochastically ordered in their initial state. Thus, we established convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are calculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.

Title:  Truncated Euler-Maruyama Method for Time-Changed SDEs with Super-Linear State Variables and H/"older's Continuous Time Variables

Friday 3rd May, 2024, 3.00-4.00pm

Venue: LT907

Abstract: In this work, an explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey H\”older's continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Simulations are provided to demonstrate the theoretical results.

Title:  A New Criterion on Stability in Distribution for a Hybrid Stochastic Delay Differential Equation

Friday 17th May, 2024, 3.00-4.00pm

Venue: LT907

Abstract: A new sufficient condition for stability in distribution of a hybrid stochastic delay differential equation (SDDE) has been proposed in this work. Although the new criterion leads to stability for an SDDE, its main component only depends on the coefficients of a corresponding SDE without delay. The Lyapunov method is applied to find an upper bound, so that the SDDE is stable in distribution if the delay is less than the upper bound. Also, the criterion shows that delay terms can be an impetus toward the stability in distribution.

Title:  Limit Theorems for Weakly Dependent Random Fields with Applications to High-Dimensional Time Series

Friday 31st May, 2024, 2.00-3.00pm

Venue: LT908

Abstract:  We establish limit theorems, law of large numbers (LLN) and central limit theorem (CLT), for weakly dependent arrays of random fields which are not necessarily stationary and may have asymptotically unbounded moments. The weak dependence condition is proved to be inherited through transformation, and this makes our results applicable to statistical inference of high-dimensional time series models. Consistency and asymptotic normality of maximum likelihood estimation can be proved for high dimensional time series models which are checked to be weakly dependent as random fields, allowing for non-stationarity and unbounded trending moments, when sample size and/or dimension go to infinity. As an example for application of our general theory, asymptotic properties of estimation for network autoregression are obtained under mild conditions.

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.

### Edinburgh Mathematics Society (EMS)

Title: Can We Rely On AI?

Date: 3.00pm Friday 19th January 2024

Venue: LT908

Abstract: Over the last decade, adversarial attack algorithms have revealed instabilities in deep learning tools. These algorithms raise issues regarding safety, reliability and interpretability in artificial intelligence (AI); especially in high risk settings. Mathematics is at the heart of this landscape, with ideas from optimization, numerical analysis and high dimensional stochastic analysis playing key roles. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of successful attacks. I will present examples of attack algorithms in image classification and optical character recognition. I will also outline recent results on the overarching question of whether, under reasonable assumptions, it is inevitable that AI tools will be vulnerable to attack.

Tea/coffee will also be served in the staff common room (LT911) from 2:30pm.

For those of you who cannot attend in person, the talk will be streamed over zoom.

https://strath.zoom.us/j/87299195984

Meeting ID: 872 9919 5984

### Jointly Hosted Seminars

Title: Solution multiplicity and effects of data and eddy viscosity on Navier-Stokes solutions inferred by physics-informed neural networks

Date: 10am Wednesday 6th December 2023

Venue: Zoom ( Meeting ID: 976 1568 3533   Passcode: m1LJJ\$nU

Abstract: Physics-informed neural networks (PINNs) have emerged as a new simulation paradigm for fluid flows and are especially effective for inverse and hybrid problems. However, vanilla PINNs often fail in forward problems, especially at high Reynolds (Re) number flows. Herein, we study systematically the classical lid-driven cavity flow at Re=2,000, 3,000 and 5,000. We observe that vanilla PINNs obtain two classes of solutions, one class that agrees with direct numerical simulations (DNS), and another that is an unstable solution to the Navier-Stokes equations and not physically realizable. We attribute this solution multiplicity to singularities and unbounded vorticity, and we propose regularization methods that restore a unique solution within 1\% difference from the DNS solution. In particular, we introduce a parameterized entropy-viscosity method as artificial eddy viscosity and identify suitable parameters that drive the PINNs solution towards the DNS solution. Furthermore, we solve the inverse problem by subsampling the DNS solution, and identify a new eddy viscosity distribution that leads to velocity and pressure fields almost identical to their DNS counterparts. Surprisingly, a single measurement at a random point suffices to obtain a unique PINNs DNS-like solution even without artificial viscosity, which suggests possible pathways in simulating high Reynolds number turbulent flows using vanilla PINNs.

Speaker Bio:  Zhicheng Wang received Ph.D. degree in engineering thermal physics from University of Chinese Academy of Sciences in 2013. He is currently an associate professor at School of Energy and Power Engineering, Dalian University of Technology, Dalian, China. His main research interests include high order numerical methods and scientific machine learning for predicting turbulent flows and multiphase phase flows, as well as the fluid structure interactions. He has published more than 20 papers in Proc. Natl. Acad. Sci. U.S.A., J. Comput. Phys., J. Fluid Mech., Comput. Method Appl. M., J. Fluids Struct., Phys. Fluids.