Conall Kelly (University College Cork, Ireland)

Title: Simulating stochastic systems subject to shocks

Abstract: Stochastic differential equations (SDEs) are used to model the evolution of real-world phenomena subject to random noise and uncertainty. Consider, for example, asset prices or stochastic interest rates in finance, models of ecological systems with complex interaction between species or models of chemical reactions in biological cells. The random noise may act as a diffusion, for example reflecting market volatility, or as a jump process, for example when an ecosystem is influenced by a random external event. 

Daniel Devine (Trinity College Dublin, Ireland) 

Title: Convergence results for an elliptic system of PDE

Abstract: In this talk, we will discuss a nonlinear elliptic system of PDE which has its origins in the study of the dynamics of viscous, heat-conducting fluids. To model viscous heating effects, the system of interest contains source terms with a nonlinear gradient dependence, which presents considerable theoretical challenges. By restricting our attention to solutions which are radially symmetric, the problem becomes far more mathematically tractable. To begin, we will outline some of the progress made since the early 2000’s, and then move onto some more recent results. In particular we will see that all solutions converge monotonically to an explicit solution which we can easily calculate. This talk is based on results jointly obtained with Paschalis Karageorgis, and Gurpreet Singh.

Eoin Hurley (University of Oxford, UK) 

Tom Carroll  (UCC, Ireland) 

Title: An uncertainty principle for the Vaserstein distance

Abstract: This is joint work with Xavier Massaneda and Joaquim Ortega-Cerdà (Barcelona). I will discuss an uncertainty principle of the following form: for a function f with mean zero, the size of the zero set of the function and the cost of transporting the mass of the positive part of f to its negative part cannot both be small at the same time. The result in two dimensions is due to Steinerberger. A partial result in higher dimensions, which we improve upon, is due to Sagiv and Steinerberger.  Related to this is a sharp upper estimate of the cost of transporting the positive part of an eigenfunction of the Laplacian to its negative part. This proves a conjecture of Steinerberger and provides a lower bound on the size of a nodal set of the eigenfunction.

Andrew D Smith (University College Dublin, Ireland)

Title: Spirals in spaces of holomorphic functions

Abstract: Functions W(t,z) of real time t and complex z satisfy the spiral relation:

W(2t, z) = (1+e^z) W(t, z)

For fixed t, these are holomorphic functions of z in the region:

  | Im(z) |   < arccos [ - e^{ - | Re(z) |} / 2 ]

Viewed as functions of t, for fixed z, the functions W(t,z) are Holder continuous and nowhere differentiable. They have a time-homogeneity property if Re(z) = 0, while for Im(z) = +/- pi/2 the paths have finite quadratic variation; a property also associated with semi-martingale paths in the theory of stochastic processes.

The W functions can produce beautiful images. Familiar fractal sets: Levy's C-curve, Heighway's dragon curve and van Roy's unicorn curve arise as the loci of W(t,z) when t is between 0 and 1 while and $z = +/- i.pi/2, that is, functions of t that satisfy both the time-homogeneity and quadratic variation criteria

Marvin Anas Hahn (Trinity College Dublin, Ireland)

Title: A tropical perspective on twisted Hurwitz numbers

Abstract: Hurwitz numbers count branched morphisms between Riemann surfaces with fixed numerical data. While a classical invariant, having been introduced in the 19th century, Hurwitz numbers are an active topic of study, among others due to their interplay with Gromov-Witten theory and their role in mirror symmetry. In recent work of Chapuy and Dołęga a non-orientable generalisation of Hurwitz numbers was introduced, so-called $b$-Hurwitz numbers. These invariants are a weighted enumeration of maps between non-orientable surfaces weighted by a power of a parameter $b$. This parameter should be viewed as measuring the non-orientability of the involved covers. For $b=0$, one recovers classical Hurwitz numbers, while $b=1$ represents a non-weighted count of non-orientable maps yielding so-called twisted Hurwitz numbers. In this talk, we derive a combinatorial model of twisted Hurwitz numbers via tropical geometry and employ it to derive a wide array of new structural properties. This talk is based on joint work with Hannah Markwig.

Luigi Roberti (University of Vienna, Austria) 

Title: Well-posedness of a semilinear parabolic equation arising from the modelling of atmospheric flows

Abstract: So-called “morning glory clouds” are a spectacular atmospheric phenomenon, observed in coastal regions at several locations around the globe, consisting of elongated tubular clouds travelling perpendicularly to the cloud line, in an essentially two-dimensional motion. Recently, a new mathematical model for the dynamics of such clouds has been derived by Constantin and Johnson. Exploiting mass conservation to eliminate the vertical velocity component, the problem can be reduced to studying a semilinear parabolic equation with nonlocal nonlinearity.

The aim of this talk is to illustrate the key features of the model and some recent results concerning the well-posedness of the problem, including local strong well-posedness and existence of global weak solutions as well as global strong well-posedness for small initial data. This is joint work with Bogdan Matioc (U Regensburg) and Christoph Walker (Leibniz U Hannover).

Kevin Burke (University of Limerick, Ireland) 

Title: Automating variable selection in distributional regression 

Abstract: Variable selection is an important scientific endeavour as it identifies important associations. Of course, this is more challenging than simply fitting a model for a given pre-specified set of covariates. From a scientific perspective, “distributional regression” models allow us to better understand the phenomenon under study compared to the classical mean-view of the world; for example, we can discover how covariates impact both the mean and variance of the response. However, variable selection is even more challenging in this setting since there is a regression equation for each of the distributional parameters. Stepwise regression procedures are quite computationally intensive in general, but so too are penalised regression procedures due to the need to select the penalty tuning parameter(s); the issue is compounded in distributional regression models due to the fact that there are multiple regression equations. Therefore, we introduce a tuning-parameter-free (and, hence, automated) procedure for selecting variables based on a differentiable approximation to an information criterion that we optimise directly. This method is especially advantageous in the distributional-regression setting, but is also useful in classical regression settings. For further details, see https://doi.org/10.1007/s11222-023-10204-8

Muhammed Fadera (University of Exeter,  UK) 

Title: Arbitrary Sensitive Transitions in Recurrent Neural Networks

Abstract: An Excitable Network Attractor (ENA) is a forward-invariant set in phase space composed of two or more attractors and parts of their basins that allow transitions between them under input or noise perturbations.  ENAs have recently been used to explain the input-driven dynamics of RNNs trained on sequence-to-sequence classification problems. For example, errors in performance of such trained RNNs are related to transitions to attractors outside the associated ENA. Typically an ENA is extracted from a trained RNN by finding fixed points in the autonomous dynamics of the RNN and using input-driven trajectories to infer transitions between these fixed points. While successful, this approach is computationally expensive. An alternative approach is to train a model which can realise arbitrary ENAs, and tune the number and sensitivity of attractors to inputs to match the dynamics of the trained RNN. Previous work has demonstrated that ENAs of arbitrary sensitivity can be realised in a RNN by suitable choice of connection weights and nonlinear activation function. The issue with this approach is that RNNs are trained by adapting the weight matrix while the activation function is held fixed. In this talk, I will show that ENAs of arbitrary sensitivity and structure can be realised even using a suitable fixed nonlinear activation function, i.e. by suitable choice of weights only. Furthermore, the weight matrix can be chosen so that the probability of transitions that could result in errors is 0.

Emily Gleeson (Met Éireann, Ireland) 

Title: Weather forecasting – crystal balls or cutting edge science

Abstract: In the talk, I will talk about the model we use in Met Éireann for short range weather forecasting – the HARMONIE-AROME model. I will talk about some of the recent developments we are doing in the model and also show some of the ways in which machine learning is being employed. 

Peter Ashwin (University of Exeter,  UK) 

Title: The quest for robust early warnings of bifurcations in noisy systems

Abstract: There is a lot of interest in finding "early warning signals" or "precursors" of tipping points due to bifurcations in cases where a system parameter is drifting. This is particularly relevant for tipping points where a fold bifurcation leaves no nearby solutions as stable. The idea of an early warning of a tipping point is compelling and potentially useful in a wide range of applications. In general, one examine the behaviour of a nonlinear system with weak noise, and finds properties of the noise response (critical slowing down) that indicates a tendency towards less stable solutions. The challenge is to understand when one can expect such a signal to be reliable and robust indicator of a future tipping event; this includes problems related to timescales, estimation and extrapolation, and begs the question of when a warning might be "too early" or "too late". I will outline some ongoing work that aims to better understand when such warnings are reliable.

Qi Wu (UCC,  Ireland) 

Title: Statistical methods for mapping kinetics in Long Field of View Dynamic PET Studies

Abstract: Positron Emission Tomography (PET) plays a critical role in medical imaging, providing quantitative insights into biological and pathological processes. However, conventional static PET imaging struggles to capture dynamic metabolic changes, while traditional dynamic PET acquisitions are often limited by long scan durations and high computational demands. Long axial field-of-view (LAFOV) dynamic PET imaging offers enhanced sensitivity and extended coverage, making it a promising tool for improved quantification. However, its complex tissue environment, large-scale data size, and organ-specific physiological variations pose significant challenges. The direct application of classical kinetic models without proper validation in LAFOV PET may lead to inaccuracies due to unique tracer kinetics and regional heterogeneity across different organs. This talk explores novel kinetic modeling approaches and optimized scan strategies to address these challenges.

Kirill Kovalenko  (Scuola Superiore Meridionale, Italy) 

Title: Higher-order interactions and hypergraphs: dynamics and structure

Abstract:  In this talk, I am going to discuss the differences between higher-order networks and classical pairwise networks from dynamical and structural perspectives. I will illustrate this with two recent studies: one on the D-dimensional Kuramoto model and how the introduction of higher-order interactions enables a partially synchronized state of pure contrarians, and another on the generalization of centrality measures to hypergraphs and how it allows one to differentiate individuals who are central at different levels of interaction.

Peter J. Olver (University of Minnesota, US) 

Title: Fractalization, Quantization, and Revival in Dispersive Systems

 Abstract: Dispersive quantization, also known as the Talbot effect, describes the remarkable evolution, through spatially periodic linear dispersion, of rough initial data, producing fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures and revivals at rational times.  Such phenomena have been observed in dispersive waves, optics, and quantum mechanics, and have intriguing connections with number theoretic exponential sums.   I will survey results on the analysis and numerics for linear and nonlinear dispersive wave models, both integrable and non-integrable, integro-differential equations modeling interface dynamics, and, time permitting, Fermi-Pasta-Ulam-Tsingou systems of coupled nonlinear oscillators.

Andreas Amann (UCC, Ireland) 

Title: Symmetries in adaptively coupled networks

 Abstract: Traditional complex network dynamics deals with nodes which are connected via static links. In contrast, in adaptive networks the strength of individual links becomes itself a dynamical quantity.  This enables the emergence of complex dynamical patterns including the formation of synchronized clusters.  I show how symmetries in the network can be used in this case to simplify the analysis.

Sarah Selkirk (University of Warwick, UK) 

Title: The distribution of the maximum protection number in random trees

 Abstract: The protection number of a vertex $v$ is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this joint work with Clemens Heuberger and Stephan Wagner, we study the maximum protection number in simply generated trees using techniques from analytic combinatorics. We develop new methods with which to study generating functions dependent on multiple parameters, and obtain the distribution for the maximum protection number by applying them.

Anca Mustata (UCC, Ireland) 

Title: Compactifications of the Space of Rational Pointed Curves and their Symmetries

 Abstract: In how many ways can n points be arranged on a complex projective line, up to Möbius transformations? The set of all such configurations forms a smooth (n-3)-dimensional variety known as the moduli space of pointed rational curves. It admits many different compactifications, the most notable being the moduli space of stable pointed rational curves (Deligne and Mumford 1969), whose points parametrise trees of projective lines with n marked points and no point-fixing automorphisms. Thus, interesting properties of these spaces can be expressed in terms of combinatorics of stable trees. Natural symmetries of these spaces are given by the symmetric group of permutations on n elements, and it turns out that this forms the entire automorphism group of the space of pointed rational stable curves (A.Bruno and M.Mello, 2013). In this talk we will discuss related compactifications of the moduli space of pointed rational curves which exhibit more ample symmetry, and discuss some of their remarkable geometric properties

Leo Creedon (Atlantic Technological University, Ireland) 

Title: Group algebras, derivations and new partial algebraic structures

 Abstract: Group algebras are rings and vector spaces with an intuitive construction from a group and a field. Their study combines many aspects of group theory, ring theory and linear algebra. They have been used in representation theory and have emerging applications in error correcting codes and cellular automata.

The derivations of an associative algebra form a Lie algebra, but it is rarely the case that this set of derivations forms an associative algebra. This leads to the question: "When does the set of derivations of a ring itself form a ring?" This question is answered here for finitely generated group algebras.

The composition of derivations is rarely a derivation, but in positive characteristic $p$, derivations of an algebra form a restricted Lie algebra. So the set derivations of an algebra (with positive characteristic $p$) form a vector space where (repeated) p-th powers are allowed, but the set is not closed under the obvious multiplication (composition of derivations).

This leads to the definition of new partial algebraic systems. These new structures are defined, and their theory is developed. Derivations of associative algebras motivate and give examples of such structures. 

This is joint work with Kieran Hughes of ATU.

Timothy Linehan (Central Statistics Office (CSO), Ireland) 

Title: Official Demography, Fertility and Mortality Statistics in Ireland- Recent Developments and their Implications

 Abstract: This seminar discusses recent developments and projects in official Irish Demography, fertility and mortality statistics, as well as their potential applications, and implications, for researchers.