Gerard Killeen ()

Title: 

Abstract: 

Jerome Moloney ( University of Arizona, US)

Title: Ultrafast intense laser pulse propagation: Dispersion vs nonlinearity dominated physics.

Abstract: In this talk, I will briefly review the early development of the theory of ultrashort pulse (USP) filamentation. The Unidirectional Pulse Propagation equation (UPPE), a pseudo-spectral solver, will be introduced as a robust Maxwell-like full wave propagator that allows long distance propagation of arbitrary vectorial optical waveforms while resolving the underlying optical carrier wave. In limiting cases, propagating USPs, whether dispersion or nonlinearity dominated, can exhibit very different singular behaviors. In the former limit, the respective mathematical descriptions are typically the envelope NLSE-types which exhibit a blow-up singularity at a finite distance. Nonlinearity dominated interactions are best captured by the optical carrier resolved modified Kadomtsev-Petviashili (mKP)) equation, describing optical carrier shock-like singularities. Applications to be discussed will include both near-IR and long wave atmospheric propagation, multifilament generation seeded by a modulational instability, pulse splitting and X-wave generation, exotic Bessel and Airy beams, extended plasma string generation proposed for guiding high voltage discharges, RF and Microwave guiding and lightning deflection.

Dana Contreras Julio (University of Auckland,  New Zealand) 

Title: Close Enough for Continuation: A Numerical Method for Computing Pseudo-Orbits on Invariant Manifolds as Seeds for Boundary Value Problems in Maps

Abstract: In the study of discrete dynamical systems, identifying special orbits, specifically connecting orbits between saddle fixed or periodic points, is essential for understanding complex dynamics. Such special orbits are closely associated with stable or unstable manifolds and can be found numerically and then continued in parameters as solutions of suitably formulated boundary value problems (BVPs). However, obtaining the initial "seed" solution required for Newton's method to converge to the BVP remains a significant numerical challenge.

Pseudo-orbits are guaranteed to have a true finite orbit of the diffeomorphism nearby, and hence, they serve as robust and natural seeds for BVPs. In this talk, we present a numerical method for computing one-dimensional invariant manifolds such that a pseudo-orbit is readily available for every computed point. We illustrate the efficiency of this method using two examples of robust non-hyperbolic dynamics, where we identify pseudo-orbits to systematically find many orbits in a chosen section, as well as numerous homoclinic and heteroclinic connections. 

Laura Byrne (UCC, Ireland) 

Title: The Design & Analysis of Biodiversity Experiments using Diversity-Interactions Models

Abstract: Biodiversity and ecosystem functioning (BEF) relationships define the ways in which the diversity of species in an ecosystem drive the quantity and quality of the goods and services provided. Species diversity may be defined in various ways, such as richness (species number) and/or evenness (relative proportions of species), and may or may not include species identity. Designing BEF experiments, where species diversity is manipulated across experimental units, can present unique challenges given that species diversity has multiple dimensions. The development of the Diversity-Interactions (DI) modelling framework, a regression-based approach, builds upon the traditional usage of ‘richness-only’ models in the BEF literature; richness-only models equate species diversity to solely the number of species, which reduces the true dimensionality of species diversity and therefore may cause loss of information and confounding within an analysis.

A vast number of BEF studies record multiple different responses, either to study the provision of simultaneous functions by an ecosystem (referred to as multifunctionality) or the stability of an ecosystem across time. In such studies, the data that arises from each experimental unit potentially has multiple sources of correlation which must be accounted for in analysis of the data. A series of DI models have been developed for such analyses. In this session, I will talk about the advancements of DI modelling framework, specifically for use in grassland BEF relationship studies, and the associated challenges.

Alannah Neff (University of Edinburgh,  UK) 

Title: Solute transport in the cranial subarachnoid space

Abstract: Cerebrospinal fluid (CSF) is a clear, Newtonian fluid that fills the subarachnoid space (SAS)  surrounding the brain and spinal cord and plays a central role in maintaining brain health. It is responsible for the delivery of nutrients and the clearance of metabolic waste and neurotoxic proteins. Disruptions to CSF transport are thought to contribute to the accumulation of proteins such as amyloid-β, which is implicated in neurological diseases including Alzheimer’s disease. CSF motion within the SAS is driven by brain pulsations associated with the cardiac and respiratory cycles, leading to predominantly oscillatory flow. Although this motion is largely periodic, it can 
generate secondary transport mechanisms that produce a net movement of solutes over long  timescales. Understanding how these pulsations give rise to effective solute transport remains an open question. In this talk, I present simplified mathematical models of oscillatory CSF flow in the cranial SAS 
aimed at understanding the link between brain pulsations and solute transport. Using lubrication  theory to reduce the governing fluid equations, and exploiting the separation of timescales between fast oscillatory motion and slow net transport, we derive an effective long-time transport equation 
for solutes. We then investigate how both temporally and spatially varying oscillations influence solute transport within the SAS.

Sergei Petrovskii (University of Leicester, UK) 

Title: Interplay between climate forcing and evolutionary rescue may explain mass extinctions in the Earth history 

 Abstract: Species get extinct all the time with a certain background extinction rate; this is a normal course of macroevolution. However, several times through the 540 Ma of the recorded history of life on Earth, the extinction rates exceeded the average background rate by more than an order of magnitude, resulting in 50-90% loss in the global biodiversity. Apart from the “Big Five”, there were many smaller mass extinctions with the global biodiversity loss ranging between 10-50%. Mass extinctions came into the focus of scientific community in early 1980s and significant progress has been made over the last few decades. However, given the inherent deficiency of the fossil data, statistical analysis alone (which is the main research tool used in paleontology) does not always allow to distinguish between the effect of different processes, which hampers further progress. Process-based mathematical models are needed.

In my talk, I introduce a novel modelling approach that counterpoise the effect of a fast climate change on the population dynamics with species evolutionary response. Different variants of the model may or may not also take into account species’ active feedback on the global energy balance and/or the dependence of population growth rate on the ambient temperature, which is a generic property of many plant and animal species. The model shows that species extinction or survival following a climate change depends on a subtle interplay between the magnitude of the climate change and the rate of species’s adaptive evolution. The model predicts a distribution of extinction frequencies which is generally consistent with the fossil data. Our study therefore suggests that mass extinctions in the Earth history occurred due to r-tipping on the global scale. 

 Matthias Wolfrum (Weierstrass Institute, Germany)  

Title: Triplet synchronization in higher-order networks

 Abstract:  The locking of coupled oscillators is a fundamental phenomenon in nonlinear science. In its simplest form it describes the frequency entrainment of two coupled oscillators close to a resonance of their natural frequencies. The classical work of Kuramoto shows how pairwise interaction of oscillators can induce global synchronization. We study here the case of networks where more than two oscillators, all close to a resonance, interact. It turns out that in such higher-order networks there may arise new types of locking phenomena. Based on normal form transformations and phase reduction methods we show how they depend on an interplay between the coupling structure and the nonlinear interaction functions.

Bruce Sutherland (University of Alberta, Canada) 

Title: Instabilities of Waves Inside the Ocean

 Abstract: The greatest unknown in predicting the climate of the Earth's oceans is understanding how energy, either from surface winds or gravity-driven tides, transforms from large (100km) scale motion to small (mm) scales where it drives mixing and dissipation.   This occurs through a myriad of processes.   The talk will focus on two energy cascade processes involving weakly nonlinear wave-wave interactions.   Specifically, we examine the evolution of a horizontally periodic ``parent'' internal wave that moves within the ocean under the influence of buoyancy forces where the ocean density increases nonlinearly with depth.  The parent wave has the vertical structure of a low mode, having half a wavelength from the surface to bottom.  Due to the nonuniform stratification, the parent wave excites superharmonics with double the horizontal wavenumber  and nearly double the frequency.  If close to resonance, the superharmonics can grow to large amplitude, themselves exciting superharmonics, resulting in a superharmonic energy cascade.  This process can be described by coupled ordinary differential equations.  They predict that the superharmonic cascade leads to the formation of a solitary wave-train. 

A distinct energy cascade mechanism occurs through triadic resonant instability, whereby a pair of sibling waves in the background noise interact with a parent internal tide so that they grow in amplitude. In this work we develop a theory for the growth of sibling waves in near resonance due to a frequency mismatch between the forced frequency and natural frequency of each sibling wave.  The predictions are tested against fully nonlinear numerical simulations and laboratory experiments.  Theory, simulations and experiments show that near-resonance can occur if the parent wave has frequency close to the maximum allowable frequency.  However, the growth rate is negligibly small if the parent wave has frequency more representative of internal tides in the ocean.

Andrew Simpkin (University of Galway, Ireland) 

Title: Modelling the derivatives of functional data

 Abstract: P-splines provide a flexible and computationally efficient smoothing framework and are commonly used for derivative estimation in functional data. Including an additive penalty in P-splines improves derivative estimation. We propose a new method which incorporates an additive penalty in P-splines with a fast covariance estimation (FACE) algorithm. The proposed method is used to estimate the derivative of the covariance for functional data, which plays an important role in derivative-based functional principal component analysis (FPCA). Following this, we provide an algorithm for estimating the eigenfunctions and their corresponding scores in derivative-based FPCA. We extend the algorithm to multivariate cases, referred to as derivative multivariate functional principal component analysis (DMFPCA). The method will be presented and applied to several contexts involving sensor data. 

Tom Carroll (UCC, Ireland) 

Title: Frequent Hypercyclicity of Bounded Linear Operators

A bounded linear operator T on a separable Fréchet space X is said to be hypercyclic if the orbit under T of some element in the space is dense in the space. This is the same as topological transitivity, which means that the orbit of any open set under T will meet any other open set. 

An element with dense orbit under T is called a hypercyclic vector for T. In the case of any open subset U of X, the orbit of a hypercyclic vector will enter U infinitely often. The question addressed in this talk is to quantify how much time the orbit of a hypercyclic vector spends in U, hence various densities of sets of natural numbers arise.  

We will present some of the extensive literature on this topic and outline our own approach. A good portion of the time will be devoted to weighted backward shifts on sequence spaces. This is joint work with Clifford Gilmore at Université Clermont-Ferrand.