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.