Integrating lab experiments into fluid dynamics models
The join button will be published 30 minutes before the seminar starts (login required).
Ashleigh Hutchinson is an applied mathematician with a strong research focus on fluid mechanics problems rooted in nature and industry. Her work centres on low-Reynolds number flows and non-Newtonian fluids, where she adopts a multidisciplinary approach that combines theoretical models, laboratory experiments, and numerical simulations.
Her other research interests include applying mathematical modelling to solve problems in industries such as finance, sugar, fishing, mining, and energy conservation.
Abstract
In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.
Master Stability for Traveling Waves on Networks
The join button will be published 30 minutes before the seminar starts (login required).
Stefan Ruschel’s research focuses on dynamical systems theory and its applications to nonlinear optics and mathematical biology, among others. He specialises in analytical and numerical methods for delay differential and functional differential equations when the delay is large compared to other time scales of the system. His specific contributions include work on the fixed point spectrum for large delay, as well as the characterisation of slowly oscillating solutions such as travelling pulses and waves.
His future research is dedicated to applying these techniques to delay and lattice dynamical systems arising from coupled excitable and coupled bi-stable systems in laser dynamics and neuroscience, where such solutions play an important role in data transmission and neural signal propagation.
He is currently a research fellow at the University of Leeds (UK), funded by UKRI in recognition of a Horizon Europe MSCA award post-Brexit.
Abstract
I will present a new framework for determining effectively the spectrum and stability of traveling waves on networks with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike traditional methods, MSCs are independent of system size and can be readily used to assess wave destabilization and multi-stability in small and large networks.
17:00
Feferman's Completeness Theorem
Abstract
Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of uniform reflections. This result is commonly known as Feferman's completeness theorem. The talk aims to give one or two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked result.
Moreover, one of the proofs furnishes sharp bounds on the order types of well-orders necessary to attain completeness.
(This is joint work with Fedor Pakhomov and Dino Rossegger.)
17:00
The open core of NTP2 topological structures
Abstract
The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are often inter-definable with their open core. Structures such as M = (ℝ,<, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk, we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination (every definable set is constructible), and its definable functions are generically continuous.
17:00
The Index of Constant Mean Curvature Surfaces in Three-Manifolds
Abstract
Path integral formulation of stochastic processes
Abstract
Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).
In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].
[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)
[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).
[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)
[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)
15:30
Scaling limits for planar aggregation with subcritical fluctuations
Abstract
Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.
Complex crystallographic groups and Seiberg--Witten integrable systems
Abstract
For any smooth complex variety Y with an action of a finite group W, Etingof defines the global Cherednik algebra H_c and its spherical subalgebra B_c as certain sheaves of algebras over Y/W. When Y is an n-dimensional abelian variety, the algebra of global sections of B_c is a polynomial algebra on n generators, as shown by Etingof, Felder, Ma, and Veselov. This defines an integrable system on Y. In the case of Y being a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero--Moser system. Recently, together with P. Argyres and Y. Lu, we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg--Witten integrable systems of certain supersymmetric quantum field theories. I will describe our progress in understanding this connection for groups W=G(m, 1, n), corresponding to the case Y=E^n where E is an elliptic curves with Z_m symmetry, m=2,3,4,6.
15:30
Strong regularization of differential equations with integrable drifts by fractional noise
Abstract
We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.