2023 - 2024

Mathematical Sciences Seminar Series

Workshop on "Frontiers in Applied Dynamical Systems"

2022 - 2023

Mathematical Sciences Seminar Series

Workshop on "Nonlinear Dispersive Waves"

2021 - 2022

Mathematical Sciences Seminar Series

Applied Mathematics

Mathematics [Pending]

Statistics & Actuarial [Pending]

2020 - 2021

“G​oing all online - teaching practice, learning design and assessment" (National Seminar Series)

Applied Mathematics

Mathematics [Pending]

Statistics & Actuarial [Pending]

2019 - 2020

Applied Mathematics

Mathematics

Statistics & Actuarial [Pending]

Past seminars

• 2018 - 2019
  
  • Applied Mathematics
  • Mathematics
  • Statistics / Actuarial
• 2017 -2018
  • Mathematics

Mathematics Seminars 2017 - 2018

Ralf Peeters (Maastricht University, 16 Jan)

Department of Data Science and Knowledge Engineering, Maastricht University, The Netherlands

(this is joint work with my colleague Joel Karel)

Title: Data-driven design of orthogonal wavelets with compact support and vanishing moments

Abstract: We present a framework to design an orthogonal wavelet with compact support and vanishing moments, matched to a given application in a data-driven way. This is achieved by optimizing a criterion, such that a prototype signal, which is characteristic for the application and must be supplied by the user, becomes sparse in the wavelet domain. Such a sparsity approach is claimed to have benefits for compression and detection purposes.

Starting from a filter bank approach with lossless polyphase matrices, a parameterization is developed for which wavelet orthogonality and compact support are built in, and in terms of which we can express the vanishing moment conditions relatively conveniently. For filters of order 2n-1, the orthogonality and compact support requirements leave n degrees of freedom. Instead of using all this freedom to obtain as many vanishing moments as possible (which would lead to Daubechies wavelets), the idea is to impose p vanishing moments, where 0<p<n, still leaving n-p degrees of freedom for matching the wavelet to a prototype signal by optimization. For low values of p, these vanishing moment conditions can in fact be built into the parameterization; we show the topological structure this induces. For high values of p (and for high orders of the filters) this becomes hard; we then may have to resort to constrained optimization. The approach is developed for critically sampled wavelet transforms as well as for the stationary wavelet transform. Examples are given to illustrate the wavelets generated by these methods. 

Reference:

Joël Karel, Ralf Peeters, (2017), Orthogonal Matched Wavelets with Vanishing Moments: A Sparsity Design Approach, Circuits, Systems & Signal Processing (CSSP), published online Nov. 2017. https://doi.org/10.1007/s00034-017-0716-1.

Steve Wills (University College Cork, 23 Jan)

Title: Construction of quantum stochastic cocycles

Abstract: Quantum stochastic cocycles generalise (semi)groups of automorphisms of operator algebras in two ways - the automorphism condition is weakened to a positivity preservation condition, and the time evolution law of a semigroup is suitably modified. These objects can be used to describe the evolution of an open quantum system interacting with its environment.

In this talk I will survey known results that show the correspondence between cocycles and solutions of the Evans-Hudson quantum stochastic differential equation for bounded generators. I will then outline the challenges of constructing such cocycles when the generator is unbounded, giving some techniques that can be used in a variety of settings to solve the QSDE/construct the cocycle.

David Wraith (Maynooth University, 30 Jan)

Title: When is a product not a product?

Abstract: The answer to the question in the title is: when the product is a Hatcher bundle. The first part of this talk will explore the construction and curious properties of Hatcher bundles. In the second part we will discuss the space of Ricci positive metrics on odd dimensional spheres, and indicate how Hatcher bundles can be used to uncover subtle aspects of the topology of this space. This is joint work with Boris Botvinnik and Mark Walsh.

Andreea Nicoara (Trinity College Dublin, 5 Feb)

Title: The Global Theory of Denjoy-Carleman Functions: Nullstellensatz Results and Beyond

Abstract: Denjoy-Carleman classes are intermediate classes between real-analytic and smooth defined by imposing certain bounds on the growth of their derivatives. Classically studied by analysts, they have more recently attracted the attention of model theorists and algebraic geometers due to their rather unique properties. I will describe recent progress on global Nullstellensatz results for the Denjoy-Carleman classes as well as challenges to figuring out their global algebraic geometric behaviour.

Clifford Nolan (University of Limerick, 13 Feb)

Title: Microlocal Analysis in Radar and Seismic Imaging 

Abstract: In this talk we consider data acquisition geometries which are applicable to both radar and seismic imaging. What seems like a cosmetic change in how one acquires (or synthetically pre-processes) the data can have a dramatic effect on the image one obtains when performing standard back-projection, or almost any other imaging method that one cares to consider. Microlocal analysis is the perfect tool to analyse the (back-projected) image and identify whether or not one should expect artefacts to appear. We will use this approach to analyse the images obtained using some standard acquisition geometries used in radar and seismic data processing.  No prior knowledge of microlocal analysis will be assumed.

Paolo Guasoni (Dublin City University, 20 Feb)

Title: Asset Prices in Segmented and Integrated Markets

Abstract: Agents with equal preferences live in two regions that yield two respective dividend streams, cointegrated with each other, but with uncorrelated fluctuations. We find equilibrium asset prices and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is negligible before integration, but significantly positive afterwards, partially explaining financialization effects.

Madalin Guta (University of Nottingham, 27 Feb)

Title: An introduction to quantum statistics

Abstract: Statistical inference plays an increasingly important role in quantum information and technology. In this talk I will give an overview of some of tools and research topics in “quantum statistics” centred around the problem of quantum state tomography and system identification. In particular I will show how key statistical concepts such as Fisher information and local asymptotic normality can be extended to the quantum domain, and how thresholding and compressed sensing techniques can be used for estimating high dimensional, low rank quantum states.

James Wright (University of Edinburgh, 6 Mar)

Title: A Sharp L2 Fourier Restriction Theorem

Abstract: We give a variant of the classical Stein-Tomas argument proving an L2 restriction

estimate. Interestingly the result is sharp for a class of homogeneous varieties of any dimension.

Hiroaki Aikawa (Hokkaido University, 20 Mar)

Title: Global integrability of superharmonic functions and supertemperatures

Abstract: Ever since Armitage showed, in 1972, that a nonnegative superharmonic function on a bounded smooth domain in $R^n$ is $L^p$-integrable up to the boundary, provided $0<p<n/(n-1)$, the global integrability of nonnegative supersolutions has attracted many mathematicians.

In this talk we consider a parabolic counterpart.  We study the global integrability of nonnegative supertemperatures on the cylinder $D\times(0,T)$.  We show that the integrability depends on the lower estimate of the Green function for the Dirichlet Laplacian on $D$.

In particular, if $D$ is a bounded $C^1$-domain, then every nonnegative supertemperature on $D\times(0,T)$ is $L^p$-integrable over $D\times(0,T')$ for any $0<T'<T$, provided $0<p<(n+2)/(n+1)$. The bound $(n+2)/(n+1)$ is sharp.

Joint work with Hara and Hirata.

Dan Petersen (Stockholm University, 10 Apr)

Title: Moduli spaces and operads

Abstract: The moduli space of Riemann surfaces parametrizes all complex structures on a two-dimensional surface. After compactifying the moduli space one finds an interesting structure corresponding to how surfaces can be glued together. Specifically, these gluings give rise to something called an operad, and this leads to an interesting interplay between algebra and geometry.

Neil O’Connell (University College Dublin, 17 Apr)

Title: Birational RSK correspondence and Whittaker functions

Abstract: The Robinson-Schensted-Knuth (RSK) correspondence is a combinatorial bijection which plays an important role in the theory of Young tableaux and symmetric functions, particularly in understanding combinatorial aspects of Schur polynomials (Cauchy-Littlewood identity, Littlewood-Richardson rule, etc.). I will give some background on this and then explain how a birational version of the RSK correspondence provides a similar `combinatorial’ framework for the study of GL(n,R)-Whittaker functions.  These functions arise in the context of automorphic forms associated with GL(n,R), and reduce to the classical Whittaker functions in the case n=2.

Eoin Ó Colgáin (Asia Pacific Center for Theoretical Physics, 24 Apr)

Title: Classical Yang-Baxter Equation from Gravity 

Abstract: Solving the equations of motion of a gravitational system, one expects to encounter differential equations. We present a theory where given a solution, there exists a deformation where the equations of motion are equivalent to the Classical Yang-Baxter Equation, a hallmark of integrability (exact solvability) across many subjects in physics. I will introduce the theory and present some examples and time permitting explain initial attempts at a proof. 

Mathematics Seminars 2018 - 2019

Seminar Titles and Abstracts 2018-2019

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 9 April 2019, WGB G04

Speaker:  Padraig Ó Catháin (Worcester Polytechnic Institute)

Title: Morphisms of Complex Hadamard matrices

Abstract:

Let $M$ be a matrix with complex entries of unit norm. A well-known theorem of Hadamard bounds the magnitude of the determinant of $M$ as a function of its dimension, and $M$ is a complex Hadamard matrix if $M$ meets Hadamard's bound with equality. In this talk I will survey some known results on existence of special types of complex Hadamard matrices, in particular matrices with entries in the $k^{\textrm{th}}$ roots of unity. I will report on recent joint work with Ronan Egan and Eric Swartz on the existence of tensor-product-like maps which reduce the number of entries in a complex Hadamard matrix at the cost of increasing the dimension. This work generalises previous constructions of Turyn and Compton-Craigen-de Launey of real Hadamard matrices from certain complex Hadamard matrices with entries in the fourth and sixth roots of unity respectively.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 2 April 2019, WGB G04

Speaker:  Vladimir Dotsenko (Trinity College Dublin) 

Title: Homotopy type of the moduli space of stable rational curves

Abstract:

The moduli space of stable rational curves, also known as the Deligne-Mumford compactification \bar{M}_{0,n} of the moduli space of rational curves with marked points, has been studied in many different areas of mathematics for decades, but some questions about it have remained open until now. An instance of such questions is the rational homotopy type of this space. I shall show that the rational cohomology of this space is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 19 March 2019, WGB G04

Speaker:  Oliver Mason (Maynooth University)

Title: The Joint Spectral Radius and Extremal Norms in Nonnegative and Max Algebra

Abstract:

The joint spectral radius (JSR) helps to characterise the asymptotic behaviour of non-homogeneous matrix products, and plays an important role in the stability analysis of difference and differential inclusions.  In this talk, I will outline some classical results concerning the JSR before going on to discuss recent work on the existence of extremal and Barabanov norms for semigroups of nonnegative matrices and matrices over the max algebra.  Time-permitting, I will also describe some applications to persistence theory for switched models in epidemiology.  

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 5 March 2019, WGB G04

Speaker:  Eduardo Mota Sánchez (University College Cork)

Title: Constant Mean Curvature Surfaces and Heun's Differential Equations

Abstract:

The generalised Weierstrass representation for surfaces with constant mean curvature allows to describe any conformal constant mean curvature immersion in R3, H3 or S3 with four ingredients: a Riemann surface, a base point, a meromorphic loop Lie algebra valued 1-form and finally the initial condition for a linear system of ODE's.

Associating to the linear system a second order differential equation from the class of Heun's Differential Equations, we prescribe certain singularities in the linear system that appear in the resulting surface. Regular singularities produce asymptotically Delaunay ends and irregular singularities produce irregular ends. We discuss global issues such as period problems and asymptotic behavior involved in the construction of this kind of surfaces. Finally, using the generalised Weierstrass representation, we construct new parametric families of constant mean curvature surfaces in R3 with genus zero that possess at least one irregular end.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 19 February 2019, WGB G04

Speaker:  Alexandru Nica (University of Waterloo)

Title: A free probabilistic approach to meandric systems

Abstract:

I will consider a family of diagrammatic objects (well-known to mathematical physicists and to combinatorialists) which go under the name of ``meandric systems''.  These objects offer some very appealing, yet difficult problems -- in particular, denoting by $E_n$ the expected number of components of a random meandric system of order $n$, there are no precise results concerning the asymptotic behaviour of $E_n$ for large values of $n$. Numerical experiments suggest the conjecture that the limit of $E_n / n$ should exist, with a value around 0.23.  In this talk I will present a result obtained in joint work with Ian Goulden and Doron Puder, giving some evidence in favour of the above conjecture.  Quite interestingly, our result is intimately related to the combinatorial side of an area of research called free probability, in particular to a very basic notion of ``free additive convolution'' which is used in free probability (and which will be reviewed as part of the talk)

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 12 February 2019, WGB G17

Speaker:  Jonathan Hickman (University of St. Andrew’s)

Title: On convergence of Fourier integrals

Abstract:

In the first half of the 20th century great advances were made in understanding convergence of Fourier series and integrals in one dimension. Many natural convergence problems in higher dimensions are still poorly understood, however, despite great attention by many prominent mathematicians over the last five decades. In this talk I will introduce the basic questions, describe their rich underlying geometry, and explain some recent developments in joint works with L. Guth (MIT) and M. Iliopoulou (UC Berkeley) and K. Rogers (ICMAT) which have applied tools from incidence and algebraic geometry to these problems.

Time and location: 4-5pm, Tuesday 29 January 2019, WGB G17

Speaker:  Clifford Gilmore (University College Cork)

Title: The Dynamics of Linear Operators

Abstract:

Linear dynamics has been a rapidly evolving area since the early 1990s.  It lies at the intersection of operator theory and topological dynamics, and its central property is hypercyclicity.  In this talk I will give a general introduction to hypercyclicity and the stronger property of frequent hypercyclicity, whence I will demonstrate that many natural continuous linear maps turn out to possess these properties.

I will finish by outlining some recent joint work with Eero Saksman and Hans-Olav Tylli (University of Helsinki) on the growth rates of harmonic functions that are frequently hypercyclic with respect to the partial differentiation operator.

Time and location: 4-5pm, Tuesday 22 January 2019, WGB 402

Speaker:  Arundhathi Krishnan (University College Cork)

Title: On the continuity of the pseudospectrum

Abstract: The pseudospectrum of an element of a unital Banach algebra is a particular subset of the complex plane which contains properly the spectrum, and is determined by the norm of the resolvent function. Under certain conditions, the pseudospectrum has the important property of being stable under perturbations.

This property is not shared by the spectrum. We discuss some of these continuity properties.

Speaker: Donnacha Oisin Kidney (UCC)

Time and location: Tuesday 27 November 4-5pm WGB G02

Title: TBC

Speaker:  Michiel van den Berg (University of Bristol)

Time and location: 4-5pm, Tuesday 6 November 2018, WGB G02 

Title: Sign changing solutions of Poisson's equation

Abstract: Michiel van den Berg

Claus Köstler (University College Cork, 11 September)

Title: Markovian fun with F

Abstract: Markovianity is a stochastic phenomenon which does not care about the past - the presence 'dictates' the future. Unexpectedly this phenomenon is closely linked to representations of the Thompson group F. I will playfully introduce you to this new connection between randomness and symmetry. Toy examples are given by moving marbles on a two-dimensional grid. I will explain why this entails the general result that every stationary Markov chain induces a representation of the Thompson group F. Furthermore I will briefly address that, conversely, a large class of representations of F yields stationary Markov chains. Finally I will introduce 'partial spreadability' as a new distributional symmetry, aiming at a de Finetti type characterization of Markovianity. The presented results are based on ongoing research with Rajarama Bhat, Gwion Evans, Rolf Gohm, Arundhathi Krishnan, Vijaya Kumar, and Stephen Wills. My talk should be accessible to a general mathematical audience.

Fran Burstall (University of Bath, 4 September)

Title: Conformal submanifold geometry for beginners

Abstract: I shall describe those aspects of the geometry of surfaces in R^3 which are invariant under angle preserving transformations.  I shall eschew the technical machinery of the subject so as to (try to) make the talk accessible to all.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 21 January 2020, WGB G17

Speaker: Tuomas Sahlsten (University of Manchester)

Title:

Quantum Chaos on Random Surfaces

Abstract:

The study of Laplacian eigenfunctions on Riemannian manifolds has a long history and is motivated by many applications in physics. For instance, an interest from the quantum mechanics perspective is the behaviour of the eigenfunctions in the large eigenvalue limit; an example of which is the quantum unique ergodicity conjecture of Zeev Rudnick and Peter Sarnak and the subsequent related work by Elon Lindenstrauss. More recently, there has been growing interest in the study of eigenfunctions and their connection to the underlying geometric aspects of the manifold. In this talk I will present an approach to quantum chaos by trying to randomise the space (surface) and describe some recent results that have been developed jointly with Cliff Gilmore (Cork), Etienne Le Masson (Cergy) and Joe Thomas (Manchester). Here we establish probabilistic relationships between L^p norms of Laplacian eigenfunctions and the genus of hyperbolic surfaces using random surface techniques for Weil-Petersson probabilities in the moduli space of Riemann surfaces developed by the late Maryam Mirzakhani. This is motivated by the recent works of Bauerschmidt et al. on the spectral theory of large random regular graphs and level aspect delocalisation of holomorphic cusp forms.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 14 January 2020, WGB G17

Speaker: Nina Snigireva (University College Dublin)

Title:

Order continuous multilinear maps and polynomials

Abstract:

In this talk we will first review the notions of order convergence and order continuity in Banach lattices. Then we will address the question of what properties are preserved when a multilinear map is extended to the bidual and the role order continuity plays in this process. (This is joint work with C. Boyd and R. Ryan.)

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 07 January 2020, WGB G.02

Speaker: Rolf Gohm (Aberystwyth University)

Title:

The probabilistic motivation to study semi-cosimplicial Hilbert spaces

Abstract:

De-Finetti-type theorems in probability theory can be analysed from the point of view of cosimplicial identities. What happens if we do similar arguments in the category of Hilbert spaces and isometries?

UCC Mathematics Seminar

Time and location: 5-6 pm, Tuesday 19 November 2019, WGB, G02

Speaker: Dmitri Zaitsev (Trinity College Dublin)

Title:

Effectiveness of multiplier ideal sheaves and triangular resolutions of singularities of holomorphic maps 

Abstract:

Multiplier ideals have been invented by Kohn as one of very few known bridges connecting singularities in PDE systems with those in geometry. However, effective control of the parameters in the PDE estimates via geometric invariants remains a major open problem. 

In my recent work with Sung Yeon Kim, we propose a new purely geometric tool of triangular resolutions that allows to control  effectiveness by means of a new set of geometric complexity invariants in terms of intersection multiplicities arising in our resolutions. 

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 12 November 2019, WGB 304

Speaker: Eberhard Mayerhofer (University of Limerick)

Title:

Three essays on stopping

Abstract:

This talk is on my article "Three essays on stopping "(https://arxiv.org/abs/1909.13050), where the exponential distribution plays a crucial role, and where the focus is on particularly simple proofs of partly known facts. The audience is only required to have a basic idea about scalar diffusion processes and compound Poisson processes.

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This modifies a formula by Perry et al (2004). Second, we characterize all diffusion processes, where the maximum before a fixed draw-down is exponentially distributed. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give a particularly simple proof for the fact that the maximum at a fixed drawdown threshold is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatovic and Pistorius (2012).

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 29 October 2019, WGB 405

Speaker: Gabriel Lord (Radboud University Nijmegen)

Title:

Numerics and a model for the stochastically forced vorticity equation

Abstract:

This talk will introduce the stochastically forced vorticity equation and a reduced stochastic differential equation model that we show captures the infinite dimensional behaviour. We start by introducing the background to stochastic differential equations before explaining the space-time forcing. The talk will discuss some of the issues in performing numerical simulations for these systems.

UCC Mathematics Seminar

Time and location: 4-5pm, Tuesday 22 October 2019, WGB G17

Speaker: Michel Schellekens (University College Cork)

Title:

Diyatropic Algorithms: Viewing entropy through the looking glass
(On Modular Analysis & Entropy Conservation)

Abstract:

Timing-modularity, the capacity to determine the time of an algorithm from the times of its parts, is not guaranteed. In general, the time of a part intricately depends on inputs passed on by other parts. Modularity, when it holds, drastically simplifies timing, for instance by enabling the derivation of recurrence equations for worst-case time, average case time or second moments. Little is known about the intrinsic properties of algorithms for which the time analysis is guaranteed to be modular. We study the question for comparison-based algorithms.

Standard algorithm analysis assumes a uniform input distribution. In the same vein, a modular analysis, deriving information of the whole from information on its parts, should assume that each of the algorithm's parts must, once again, operate over inputs that are uniformly distributed. This proves too strict a requirement. Fortunately modular analysis extends to algorithms that preserve uniform distributions ``locally". In this case, uniform distributions merely hold on parts of an input partition referred to as a ``global state". Algorithms preserving global states exhibit the tight distribution control which admits an elegant modular analysis. Those that do not, lie at the root of deep open problems in algorithm analysis.

Algorithms preserving global states can be designed based on the Modular Quantitative Analysis framework MOQA, which consists of: (a) modelling data structures as partially-ordered finite sets; (b) modelling data on these by topological sorts; (c) considering computation states as finite multisets of such data; (d) analysing algorithms by their induced transformations on states. In this view, an abstract specification of a sorting algorithm has input state given by any possible permutation of a finite set of elements (represented, according to (a) and (b), by a discrete partially-ordered set together with its topological sorts given by all permutations) and output state a sorted list of elements (represented, again according to (a) and (b), by a linearly-ordered finite set with its unique topological sort).

Series-parallel (SP-)orders form an important, computationally tractable class of data structures that support the model's computations. We introduce Mod-SP, the least MOQA-fragment sufficient to construct all topological sorts on SP orders via computations guaranteed to support modular timing. Mod-SP-computations, when made reversible through history-keeping, act as closed systems in which entropy is conserved, linking modularity of timing to entropy conservation of data. This sharpens traditional entropy preservation guaranteed by the second law of thermodynamics for reversible systems. For Mod-SP-computations entropy is neither created nor destroyed, merely transferred in modular fashion from one form (quantitative entropy) to another (positional entropy).

We establish an entropic duality theorem coupling each Mod-SP computation with a dual computation. The latter effects an increase in positional entropy proportional to the decrease in quantitative entropy effected by the original computation. We refer to algorithms satisfying this type of ``entropic coupling" as ``diyatropic" and show that all Mod-SP computations are diyatropic. This includes Insertion sort and the Heapify algorithm but not standard Heapsort's Selection Phase. Finally we show that Entropic Duality implies an Entropic Correspondence Theorem, conjectured by M. Fiore.

This work was completed during a Fulbright Scholarship (April 5 - Aug 31, 2019) at Stanford's Computer Science theory group, research host Don Knuth. The author is grateful for discussions with Don Knuth, Vaughan Pratt and Marcelo Fiore.

Time and location: 12pm, Thursday 25 October, WGB 304

All are welcome at the following seminar:

UCC Applied Mathematics Seminar

Speaker: Adrián Rodriguez-Sanjurjo (University College Cork)

Title: Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions

Abstract: In this talk I outline recent work providing a rigorous mathematical analysis of some nonlinear surface waves in the presence of a depth-invariant zonal current constituting a generalisation of Pollard's waves. These oceanic water waves are remarkable due to the fact that they represent exact, explicit nonlinear solutions of the geophysical f-plane equations accounting for rotational effects. By way of applying a mixture of analytical and degree-theoretical arguments, it is shown that the three-dimensional Lagrangian flow-map prescribing these solutions is a global diffeomorphism and the fluid motion is dynamically possible. This is achieved by imposing certain conditions on the physical and Lagrangian labelling parameters. Furthermore, it is also proven that parameter specifications not meeting those conditions can produce solutions that fail to be globally valid; thereby demonstrating the necessity of subjecting these Lagrangian solutions to such rigorous analytical considerations.    

Rodríguez-Sanjurjo, A. Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions. Annali di Matematica (2018) 197:1787. 

https://doi.org/10.1007/s10231-018-0749-5

Time and location: 12noon, Thursday 18 October, WGB 405

Speaker: Eoin Clerkin, TU Darmstadt

Title: Problems of stability determination for large time-periodic mechanical systems.

All are welcome to attend

Time and location: 12pm, Thursday 13 September, WGB 304

Speaker: Philipp Hövel (University College Cork)

Title: A safari tour guide for chimeras

Abstract: Systems of nonlocally coupled oscillators can exhibit surprisingly complex spatio-temporal patterns, called chimera states, that consist of coexisting domains of spatially coherent (synchronized) and incoherent (desynchronized) dynamics. First observed in systems of identical elements with symmetric coupling topology, these hybrid states have been intensively studied during the last decade.

In my talk, I will first present a number of examples of these peculiar states of partial synchrony including time-discrete maps and time-continuous models. Then, I will discuss an approach to control chimeras. The control scheme is based on targeted modifications of system parameters of a few elements. In detail, I will explore the influence of a block of excitable units on the existence and behavior of chimera states in a nonlocally coupled ring-network of FitzHugh-Nagumo elements. The FitzHugh-Nagumo system, a paradigmatic model in many fields from neuroscience to chemical pattern formation and nonlinear electronics, exhibits oscillatory or excitable behavior depending on the values of its parameters. In previous studies, chimera states have been realized in networks of coupled oscillatory FitzHugh-Nagumo elements. I will show that introducing a block of excitable units into the network may lead to several interesting effects. It allows for controlling the position of a chimera state as well as for generating a chimera state directly from the synchronous state.

Time and location: 12pm, Thursday 20 September, WGB 304

Speaker: Sajjad Bakrani (Imperial College London)

Title: Invariant Manifolds of a homoclinic orbit in a 4D system with a first integral and a Z_2 symmetry

Abstract: A homoclinic orbit is an orbit of a dynamical system which joins an equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable invariant manifold and the unstable invariant manifold of an equilibrium. These orbits play an important role in understanding the global behavior of a dynamical system. In this talk we describe the dynamics in a small neighborhood of a homoclinic orbit of a particular family of 4D systems of differential equations. In particular we provide necessary and sufficient conditions for the existence of the stable and the unstable invariant manifolds of the homoclinic orbit. We then pay a special attention to the case where these two invariant manifolds intersect.

Time and location: 12pm, Thursday 27 September, WGB 304

Speaker: Alan Compelli (UCC)

Title: Soliton propagation in a fluid of non-uniform depth

Abstract: A surface water wave over a bed of non-uniform depth is considered. The fluid is incompressible, inviscid and irrotational. The Hamiltonian is determined in terms of wave-only quantities using a Dirichlet-Neumann operator. By introducing an appropriate scaling regime, and considering the bottom to vary `slowly', a KdV equation with variable coefficients is derived. A one-soliton solution approaching a ramp on the seabed is then considered and numerical results demonstrate the effect the ramp shape has on the birth of new solitons as the soliton passes over it.

Time and location: 12pm, Thursday 4 October, WGB 304

Speaker: Bernd Krauskopf (The University of Auckland)

Title: Dynamic complexity of two coupled photonic nanocavities

Abstract: This is joint work with Andrus Giraldo, Neil Broderick and Alejandro Giacomotti.

We consider two coupled passive optical resonators in the form of photonic crystal nanocavities, which operate with only a few hundred photons. This low-photon system is described mathematically by the Bose–Hubbard model consisting of two complex ordinary differential equations for the slowly varying amplitudes of two electric fields. It was shown experimentally that this system exhibits spontaneous symmetry breaking and bistable behaviour, which is of particular interest for optical memories and logical switching. This type of dynamics has also been found in the Bose–Hubbard model, and previous theoretical work has concentrated on finding parameter regions where stable symmetric and asymmetric continuous-wave solutions exist.

We consider here the overall dynamics of the system as described by the Bose–Hubbard model. In particular, we focus on the existence of complex self-pulsations. As more energy is pumped into the system self-pulsations arise from Hopf bifurcations. These periodic solutions then change or disappear in sequences of homoclinic bifurcations. In particular, we find chaotic Shilnikov bifurcations and the appearance of chaotic attractors. Such complicated dynamics can take place in either an individual cavity or in both of them simultaneously. We present how local and global bifurcations bound regions of different dynamics in relevant parameter planes. Our global bifurcation analysis of the two coupled photonic crystal nanocavities predicts types of (chaotic) dynamics well within the range of future experiments.

Time and location: 12pm, Thursday 11 October, WGB 304

Speaker: Martin Stynes (Beijing Computational Science Research Center)

Title: Fractional-order derivatives and the numerical solution of time-fractional problems

Abstract: An introduction to fractional derivatives and some of their properties is presented. The regularity of solutions to Caputo fractional initial-value problems in one dimension is discussed; it is shown that typical solutions have a weak singularity at the initial time t=0. This singularity has to be taken into account when designing and analysing numerical methods for the solution of such problems. To address this difficulty we use graded meshes, which cluster mesh points near t=0, and answer the question: how exactly should the mesh grading be chosen? Next, initial-boundary value problems in one space dimension are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative generalisation of the classical parabolic heat equation.) Once again a weak singularity appears at t=0, and the mesh in the time coordinate should be graded to compute satisfactory numerical solutions.

The seminar is meant to highlight major topics of recent interest to both senior and junior researchers in the field.

For questions and inquiries, please contact the seminar coordinator Dr. Philipp Hoevel (philipp.hoevel@ucc.ie)

Semester 1

Location: Western Gateway Building WGB G08 

Time: 16.00 - 17.00 on Wednesdays

Semester 2

Location: Western Gateway Building WGB G02

Time: 12:00 - 13:00 on Thursdays

The seminar is meant to highlight major topics of recent interest to both senior and junior researchers in the field.

For questions and inquiries, please contact the seminar coordinators Dr. Philipp Hoevel (philipp.hoevel@ucc.ie, semester 1) and Dr. Andrew Keane (andrew.keane@ucc.ie, semester 2).

Semester 1

Location: Western Gateway Building WGB G08 (unless the presentation is given online)

Time: 16.00 - 17.00 on Wednesdays

COVID-19 regulations: Full information can be found here.

Please note that face coverings are:

• Required to be worn by all students whilst in teaching spaces and lab spaces where less than 2m Social Distancing is not achievable 
• Recommended to be worn indoors when moving through corridors and where it is difficult to maintain 2m distance

Semester 2

Location: Western Gateway Building WGB G02 (unless the presentation is given online)

Time: 12.00 - 13.00 on Thursdays

COVID-19 regulations: Full information can be found here. 

The seminar is meant to highlight major topics of recent interest to both senior and junior researchers in the field.

For questions and inquiries, please contact the seminar coordinators Dr. Philipp Hoevel (philipp.hoevel@ucc.ie, semester 1) and Dr. Andrew Keane (andrew.keane@ucc.ie, semester 2).

Semester 1

Location: Western Gateway Building WGB G08 (unless the presentation is given online)

Time: 16.00 - 17.00 on Wednesdays

COVID-19 regulations: Full information can be found here.

Semester 2

No Applied Mathematics seminar scheduled.

The seminar is to be replaced by a school-wide seminar.

The seminar series will highlight major topics of recent interest to both senior and junior researchers in the field.

For questions and inquiries, please contact the seminar coordinators Spyridon Dendrinos (sd@ucc.ie), Tony Fitzgerald (t.fitzgerald@ucc.ie) and Andrew Keane (andrew.keane@ucc.ie).

Semester 2

Location: Western Gateway Building (WGB)

The seminar series will highlight major topics of recent interest to both senior and junior researchers in the field.

For questions and inquiries, please contact the seminar coordinators Spyridon Dendrinos (sd@ucc.ie), Tony Fitzgerald (t.fitzgerald@ucc.ie) and Andrew Keane (andrew.keane@ucc.ie).

Semester 1

Location: Western Gateway Building WGB G16 (unless otherwise stated below)

Semester 2

Location: Western Gateway Building WGB

**The schedule is still being prepared and will be updated regularly.**