Pure & Applied Mathematics Seminar

Strong components of random digraphs

We consider random digraphs in a directed version of the classical Erdős–Rényi model: given $n$ vertices, each possible directed edge is inserted with probability $p$, independently of the others. It turns out that these graphs undergo a phase transition when $p$ is about $1/n$, which can be seen in the answer to questions such as: what is the probability that there are no directed cycles (equivalently, that all strongly connected components are singletons)? Using methods from analytic combinatorics, we obtain very precise asymptotic answers to questions of this kind.

Mathematical Models for Evaporating Sessile Droplets

The evaporation of a sessile droplet is a multifaceted problem of enduring scientific interest that is key to a wide range of everyday and industrial situations, such as protein crystallography, surface patterning, ink-jet printing, and agrochemical spraying of plants. In this talk I shall review some of the recent developments in the study of evaporating droplets, focusing on situations in which relatively simple mathematical models can give new insights into this fascinating multidisciplinary problem, including the competitive evaporation of multiple droplets, the evaporation of a droplet on a non-planar substrate, the effect of the spatial distribution of the local evaporative flux on the final deposit left behind on the substrate after a particle-laden droplet has completely evaporated, the effect of gravity on the evaporation of a droplet, and (if time permits) the effect of particle-substrate adsorption on the final deposit. The results presented in this talk are the outcome of joint work with a large number of collaborators, including Drs Brian Duffy, David Pritchard and Alexander Wray (University of Strathclyde), Professor Khellil Sefiane (University of Edinburgh) and Professor Colin Bain (University of Durham), and past and present research students Gavin Dunn, Jutta Stauber, Feargus Schofield, Hannah-May D’Ambrosio, Laura Mills, David Craig and Henry Sharp, all of whose unique contributions are gratefully acknowledged.

Arithmetic questions in the "poor man's adèle ring"

The elements in the "poor man's adèle ring" are encoded by sequences $(t_p)_p$ indexed by prime numbers $p$, with each $t_p$ viewed as a residue modulo $p$. One can define rational and algebraic elements in this ring and then raise questions about transcendence. In the joint work with Florian Luca we prove the transcendence of the elements attached to Schur's $q$-Fibonacci numbers and to the Frobenius traces of non-CM elliptic curves. I will overview elementary aspects of these results and highlight some remaining arithmetic challenges.

Duality for Frames

The duality principle is a universal principle in frame theory which gives insight into many phenomena. Its original fiber matrix formulation for Gabor systems is the driving principle behind various seemingly different results and constructions. We discuss how the same duality principle—whose essence is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems—holds in general Hilbert spaces. An immediate consequence is that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems which are easier to control. For instance, in the arena of multiresolution-analysis (MRA) wavelet frames, the unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only requires the completion of an invertible constant matrix. Under minimal conditions on the MRA, this guarantees the existence and easy constructability of non-separable multivariate dual wavelet frames.

Tensor decompositions for multivariate function approximations

A ubiquitous task in much of (applied) mathematics is computations with functions of many variables. Particularly challenging is the integration, for example, the computation of posterior expectations in Bayesian inference. Standard quadratures may suffer from the curse of dimensionality, whereas Monte Carlo methods may suffer from a large variance. We consider low-rank decompositions of tensors of expansion coefficients of the sought function in order to approximate the function by a low-parametric representation, using a low number of function values. The unique strength of this approach is the multilinearity of the decomposition, enabling both efficient linear algebra construction methods surpassing a slow generic optimization often needed to compute other function approximations (such as neural networks), and efficient post-processing such as integration with the linear complexity in the number of variables. The fast integration enables also an efficient transformation of coordinates driven by an approximate function, which in turn simplifies the target function.

Variational approach to parabolic PDE

We show that parabolic PDE can be derived from the variational principle. This is particularly relevant to irreversible evolution of thermodynamic and fluid dynamic systems with friction. Unlike conservative systems, the evolution of which is governed by the principle of least action within the classical field theory, the systems with friction are dissipative and do not imply such principle. The dissipative nature of such systems is equivalent to the parabolic structure of the corresponding parabolic PDE, which is not symmetric with respect to inversion of time. We propose a Lagrangian that allows to formalize such dissipative systems within the variational framework. This Lagrangian is related to one of the key concepts of irreversible evolution, entropy production, which must be non-negative, according to the second law of thermodynamics. The proposed Lagrangian is both, symmetric in time (and therefore compatible with “microscopic reversibility” which underlines the evolution of all physical systems) and generates two complementary PDE that are not time-symmetric.

Apollonian gasket, old and new

Around 200 B.C. Greek geometer Apollonius of Perga in his work Tangencies stated the problem of drawing circles that are simultaneously tangent to three lines, or two lines and a circle, or two circles and a line, or three circles. As the years past, the topic attracted attention of Descartes, Princess Elizabeth of Bohemia, Leibniz, Nobel laureate Frederick Soddy, researches from AT&T lab working on an early GPS system, and connected number theory, geometry, and dynamics. Many captivating questions are still open.

Symmetrization for singular problems

We discuss Talenti-type symmetrization results in the form of mass concentration (i.e. integral comparison) for both local and nonlocal singular problems, whose prototype is

where $\mathrm{\Omega }$ is a bounded domain in ${ℝ}^n$ , $\gamma >0$ and $f\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ , $f\ge 0$ .

The different approaches will be compared, highlighting the differences in the selection of the basic ingredients and in the outcomes. Then, the results will be extended from the elliptic setting to the parabolic one.

The results are contained in some papers written in collaboration with F. Chiacchio, I. de Bonis, V. Ferone, C. Trombetti and B. Volzone.

Curve shortening flow with free boundary

I will present a comprehensive classification of the convex ancient solutions to the free boundary curve shortening flow in (bounded and unbounded) convex planar domains. This is joint work with Theodora Bourni and Nathan Burns.

Bridging exact and approximate solutions to Itô stochastic differential equations: a recursive series approach

This talk lays out a single, systematic framework for obtaining exact and approximate analytical solutions to Itô stochastic differential equations (SDEs). By linking each SDE to a linear first‑order partial differential equation for a carefully constructed operator, we build a recursive series. The series converges under a natural, checkable condition on the terms. Classical closed forms for linear models pop out as special cases, and we also derive exact solutions for nonlinear SDEs like the stochastic logistic and Gompertz models. When a closed form isn’t available, short series truncations give practical implicit approximations, validated by numerical simulations that show their accuracy. The method handles time‑dependent coefficients and nonlinearities, offering a flexible tool for stochastic modelling in finance, biology and engineering.

Modelling sea ice at different scales: from microstructure to effective properties

Sea ice is a crucial component of the Earth’s climate system, affecting the ocean circulation, the atmospheric temperature, and marine ecosystems. However, sea ice is not a simple solid material; it is a complex mixture of ice crystals, brine pockets, and air bubbles, that changes its structure and properties depending on the environmental conditions. In this talk, I will explore how we can model and understand the behaviour of sea ice at different scales, from the microscopic interactions of ice and salt to the macroscopic effects of heat transfer and fluid flow. I will present two mathematical models: a thermodynamically consistent model for the liquid-solid phase change in sea ice that incorporates the effects of salt, using multiscale analysis to derive a quasi-equilibrium Stefan-type problem. And a new rigorous derivation of bounds on the sea ice effective thermal conductivity obtained through Padé approximates and using Stieltjes integrals.

Inverse problems related to pattern formation on coupled oscillator networks, or where do chimeras live

Mathematical models describing the collective behavior of large populations of coupled phase oscillators can be found in various fields of physics, chemistry, and biology. In the thermodynamic limit, when the number of oscillators tends to infinity and the distribution of oscillator parameters converges to some probability density, it is often observed that after a sufficiently long transient, the state of the population approaches some statistical equilibrium. In this talk, we describe how the properties of this equilibrium can be used to reconstruct system parameters of the underlying network. The effectiveness of the approach is demonstrated by its application to so-called chimera states in networks of phase oscillators with nonlocal coupling.

When Contagions Collide: Mathematics of Interacting Epidemics and Beyond

Most mathematical models of contagion—whether for diseases, computer viruses, or ideas—focus on a single agent spreading in isolation. Reality is far more complex: pathogens interact within hosts, behaviors shape disease transmission, and multiple contagions often overlap in space and time. These interacting contagions can amplify, suppress, or fundamentally alter each other’s dynamics [1].

In this talk, I will introduce the idea of contagions as a unifying concept across biology and society [1], with examples from disease ecology and the One Health perspective, where human, animal, and environmental health are deeply interconnected. I will outline the first mathematical steps in modeling these interactions, beginning with extensions of classical epidemic models and network-based frameworks. Even simple models reveal surprising phenomena: shifts in epidemic thresholds, changes in persistence or/and the order of phase transitions, and unexpected outcomes when contagions couple together [1-15].

1. Hébert-Dufresne, Laurent, Yong-Yeol Ahn, Antoine Allard, Vittoria Colizza, Jessica W. Crothers, Peter Sheridan Dodds, Mirta Galesic et al. "One pathogen does not an epidemic make: a review of interacting contagions, diseases, beliefs, and stories." npj Complexity 2, no. 1 (2025): 26.
2. Roostaei, Arash, Hadi Barzegar, and F. Ghanbarnejad*. "Emergence of Hopf bifurcation in an extended SIR dynamic." Plos one 17.10 (2022): e0276969.
3. A. Khazaee and F. Ghanbarnejad*. "Effects of measures on phase transitions in two cooperative susceptible-infectious-recovered dynamics." Physical Review E 105.3 (2022): 034311.
4. F. Ghanbarnejad*, et al. "Emergence of synergistic and competitive pathogens in a coevolutionary spreading model." Physical Review E 105.3 (2022): 034308.
5. S. Sajjadi, M. R. Ejtehadi, F. Ghanbarnejad*, "Impact of temporal correlations on high risk outbreaks of independent and cooperative SIR dynamics" Plos one 16.7 (2021): e0253563.
6. F. Pinotti, F. Ghanbarnejad, P. Hoevel, C. Poletto, “Interplay between competitive and cooperative interactions in multi-pathogen systems” Royal Society open science 7.1 (2020): 190305.
7. D. Soriano-Paños, F. Ghanbarnejad, S. Meloni, J. Gómez-Gardeñes, “Markovian approach to tackle the interaction of simultaneous diseases” Physical Review E100.6 (2019): 062308.
8. F. Zarei, S. Moghimi-Araghi, F. Ghanbarnejad*,“Exact solution of generalized cooperative SIR dynamics” Physical Review E 100.1 (2019): 012307.
9. J. P. Rodríguez, F. Ghanbarnejad*, V. M. Eguíluz, "Particle velocity controls phase transitions in contagion dynamics” Scientific reports 9.1 (2019): 1-9.
10. R. Goel, A. Singh, F Ghanbarnejad*, “Competitive marketing strategies in social networks”Physica A: Statistical Mechanics and its Applications 518 (2019): 50-70.
11. Jorge P. Rodríguez, F. Ghanbarnejad*, Víctor M. Eguíluz, "Risk of coinfection outbreaks in temporal networks: a case study of a hospital contact network" Frontiers in Physics5 (2017): 46.
12. L. Chen, F. Ghanbarnejad*, D. Brockmann "Fundamental properties of cooperative contagion processes" New Journal of Physics 19.10 (2017): 103041.
13. P. Grassberger, L. Chen, F. Ghanbarnejad, W. Cai "Phase Transitions in Cooperative Coinfections: Simulation Results for Networks and Lattices"Physical Review E 93.4 (2016): 042316.
14. W. Cai, L. Chen, F. Ghanbarnejad, P. Grassberger "Avalanche-Outbreaks Emerging in Cooperative Contagions" Nature physics 11.11 (2015): 936-940.
15. L. Chen, F Ghanbarnejad*, W. Cai, P. Grassberger “Outbreaks of coinfections: the critical role of cooperativity” EPL (Europhysics Letters) 104.5 (2013): 50001.

Ravi Pethiyagoda: Diagrammatic approach to dynamical systems

In this talk I will introduce a way to construct and represent dynamical systems using a diagram (based on the causal loop diagram). I will then demo a program where we can construct these diagrams graphically and output the associated governing equations as well as solution realisations. The goal of this program is to give MATH1800 students a greater ability to construct and test mathematical models while not having the requisite knowledge to solve systems of ODEs.

Dave Smith: Collaborative peer feedback

Feedback on assessed work is invaluable to student learning, but there is a limit to the amount of feedback an instructor may provide. Peer feedback increases the volume of feedback possible, but typically reduces the quality of the feedback. We describe a model of collaborative peer feedback designed to increase quality of feedback, and describe its implementation in an undergraduate Mathematics module at a small liberal arts college and reimplementation in MATH2350 at UoN. The implementations include bespoke software through which administrative tasks are automated.

Traveling waves and breathers in the nonlocal NLS models

A nonlocal derivative nonlinear Schrodinger equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero-Moser-Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background.

Function field arithmetic and irreducibility of division polynomials associated to Drinfeld modules.

In 1993, Masser and W\"ustholz proved a famous result on existence of isogeny, with degree bounded by an explicit formula,  between two isogenous Elliptic curves. Building upon this achievement, they subsequently employed the isogeny estimation to establish an explicit bound on the irreducibility of division polynomials associated to elliptic curves over a number field without complex multiplication (CM).  This bound is then used to deduce a bound on the surjectivity of mod l Galois representation associated to elliptic curves over a number field without CM. Inspired by elliptic curve theory, David and Denis introduced an isogeny estimate applicable to Anderson t-modules, specifically deriving an isogeny estimate for Drinfeld modules over global function fields. This raises the question of whether the Masser-Wüstholz strategy can be adapted to obtain a similar bound on the irreducibility of division polynomials associated with Drinfeld modules without CM. However, the Masser-W\"ustholz strategy can not be applied directly to the context of Drinfeld modules. Thus we develop an alternative strategy, which involves heights of isogenous Drinfeld modules studied by Breuer, Pazuki, and Razafinjatovo, to deduce a function field analogue of Masser-W\"ustholz irreducibility bound. In this talk, I will begin by reviewing the theory of elliptic curves and the Masser-Wüstholz theorem. Then, I will compare classical elliptic curve theory with function field arithmetic and Drinfeld modules. Finally, I will explain how we adapt the Masser-Wüstholz strategy to the context of Drinfeld modules.

Curvature measures and volume preserving curvature flows

Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class of 1-homogeneous curvature functions. In this talk, we discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k=1,...,n. As a key step, the curvature measure theory will be used to prove the Hausdorff convergence to a sphere. We also discuss some generalizations including the flows in the hyperbolic setting and the flows in the anisotropic setting.

Adventures in the Parallel Worlds of Number Theory

I will explain the parallel worlds of number fields and function fields, and my recent work on modular polynomials, modular forms as well as parity of quadratic units and the Cohen-Lenstra-Martinet heuristics.

Flag Hardy space theory—an answer to a question by E.M. Stein.

The theory of multi-parameter flag singular integral originates from the study of ¯∂-problem on the Heisenberg group by D. Phong and E.M. Stein. In 1999, E. M. Stein asked “What is the Hardy space theory in the flag setting?” in the conference at Washington University in Saint Louis to celebrate the 70th birthday of G. Weiss. In our recent work, we established a complete flag Hardy space theory on the Heisenberg group, including characterisations via Littlewood–Paley area function, square function, non-tangential and radial maximal functions, atoms, and the flag Riesz transforms. It provided a unified approach for proving the Lp boundedness of different types of singular integrals, and led to the endpoint L log(L) → L^{1,∞} estimates. The representations of flag BMO functions are also provided.

Quasi-Monte Carlo methods for PDEs with random coefficients

Mathematical models often contain uncertainty in parameters and measurements. In this talk we focus on partial differential equations where some parameters are modelled by random variables. The main example comes for the diffusion equation where the diffusion parameters is modelled as a random field which randomly fluctuates around a given mean. To sample the random fluctuations we use quasi-Monte Carlo methods. We provide an elementary introduction to PDEs using the diffusion equation with random coefficients and discuss some of the newer methods towards the end.

A neural-network surrogate Bayesian algorithm for the Helmholtz inverse-shape problem

We present a novel approach to the classical inverse problem of reconstructing the shape of scatterers from noisy far-field data. The far-field data in our model is generated by multiple incident waves striking an impenetrable scatterer. This data can be mathematically modeled using the Helmholtz equation in the unbounded region outside the scatterer. For reconstruction, our method employs a Bayesian framework that incorporates data and utilizes Markov Chain Monte Carlo (MCMC) sampling. To address the computational challenges posed by the high-dimensional prior space, we introduce a physics-property informed (PPI) neural network (NN) surrogate model for the forward problem. The PPI component of the algorithm facilitates the training of the NN using far-field data from just one incident wave. Subsequently, the PPINN-surrogate model takes advantage of rotational symmetries in the prior space, allowing for efficient evaluation of the forward model across several incident wave directions. We showcase our method’s effectiveness by demonstrating excellent reconstruction for a range of test scatterer shapes.

Stability of breather solutions to the discrete Nonlinear Schrodinger equation

In this talk, I will show how ideas from exponential asymptotics, resurgence, and complex analysis can be used to explain the stability properties of breathers (or stationary waves that are periodic in time) in discrete systems. I first introduce the history and ideas of exponential asymptotics and Stokes' phenomenon. I will then introduce the discrete analogue of the famous nonlinear Schrodinger equation explain why the only breather solutions that can exist are site-centered (on-site) or midpoint-centered (inter-site) solutions, before determining the asymptotic behaviour of the (exponentially) small eigenvalues to determine the stability of these breathers. Finally, I will explain why these methods break down once long-range interactions are introduced to the system, and how this can be resolved by using Borel transform theory and conformal mapping.