Machine Learning strategies have been hailed as an exceptional approach for capture the local systems dynamics. Among other things, it can be used for regression, out of it output for near future time can be forecast. On the other hand, Nonlinear Time Series Analysis comprises a set of robust strategies that can be used to characterize dynamical behavior and even forecast dynamical changes. The combination of both approaches may provide an outstanding and robust tool to properly deal with complex systems, not only characterizing them, but also to forecast dynamical changes, like tipping points. This minisymposium aims to bring together experts in this area to provide an state of art view about this new approach.
Speakers:
Several dynamical systems are multistable: they exhibit a coexistence of stable solutions, formally called attractors, with examples including power grids, climate, the brain, and metabolic systems, to name a few. Perturbations, such as noise or external shocks, can induce such transitions between these attractors - which, depending on the application, may be either desirable or catastrophic. It becomes crucial therefore to study the stability in such multistable systems. Typically stability is studied using local bifurcation analysis and continuation, but this approach is limited and typically unsuitable for real-world applications where perturbations are finite-sized instead of infinitesimal. This calls for a global (or, non-local) view of stability. Recent progress has enriched the literature with various quantities that can be used as global quantifiers of stability: basin stability, basin entropy, or other indicators of so-called resilience like the width of the basin of attraction or the minimal fatal shock. In this minisymposium we want to highlight and promote recent research that focuses on global stability analysis of multistable systems, multistability in high-dimensional systems, and multistability in chaotic systems.
Nonlinear Dynamical Systems have been the vehicle for not only addressing important questions in Mathematics, Physics, and Engineering, but also for understanding real-world phenomena therein. Those include gravity (governed by the General Theory of Relativity), Nuclear Physics and Quantum Mechanics (Quantum Field Theory), Atomic Physics (atomic Bose-Einstein Condensation and related phase transitions), and Fluid Mechanics (governed by the Euler/Navier Stokes equations). Mathematicians and Theoretical Physicists have been working both on the theory and analysis of the respective governing equations whose studies have been complemented by numerical computations. The latter usually go hand-in-hand not only with theory but also with experimental realizations of the pertinent phenomena modeled by nonlinear dynamical systems.
This session will focus on the analysis and numerical computations of nonlinear dynamical systems through a variety of techniques from a broad range of different mathematical/physical areas. On one hand, the topics that will be discussed are relevant to forefront research in General Relativity, Topological/Quantum Field Theories, Nonlinear Waves, Pattern Formation, Solitons and Topological Defects. On the other hand, techniques that will be presented include, among others, integrability, local and global well-posedness, asymptotics, stability, and state-of-the-art numerical computing. This session will bring together high caliber experts in the above areas and highlight some of the newest findings and emerging trends.
Vegetation pattern formation is a relatively new research field where concepts and tools of pattern formation theory are applied to problems at the forefront of ecological research. The interest in vegetation patterns has surged since their widespread discovery in many parts of the world and their description in terms of spatially explicit mathematical models about two decades ago. Vegetation patterns comprise large populations of plants - the primary producers that form the basal level of the whole food web. Studying vegetation patterns is, therefore, essential for understanding ecosystem response to climate change and how to secure the many services ecosystems provide to humans.
Vegetation patterning is a population-level response to environmental stress that involves partial mortality of the plant population. A common type of stress is water deficit in drylands, which occupy more than 40% of the terrestrial earth area, and in non-drylands subjected to severe droughts. Indeed, most observations of vegetation patterns have been reported in drylands. Other kinds of stress include low nutrient levels in wetlands, stress related to soil biota such as pathogens or toxic materials, grazing by herbivores, and others.
The aim of the proposed mini-symposium is to present recent advances and challenges in understanding the various mechanisms that drive vegetation patterning, the resulting pattern dynamics, and the implications for ecosystem functioning in stressed environments. The presentations will cover various dynamical behaviors, including stationary and traveling-wave patterns, multistability of uniform and patterned states, front propagation and instabilities, multi-scale patterns, etc. Methods of model analysis to be presented will include bifurcation theory, singular perturbation theory, numerical continuation, direct numerical simulations, and others.
The proposed minisymposium aims at providing a bird’s-eye view of the current methodologies used to extract the salient features of high-dimensional, chaotic phase spaces, by means of an effective dimensional reduction. Understanding the dynamics of chaotic systems with many degrees of freedom is nowadays central in several disciplines spanning physics, engineering and biology, such as fluid dynamics in the study of turbulence, weather science, but also networks and wave dynamics, among others. Yet, direct numerical simulations often prove impractical due the high dimensionality and complexity of the state space, neither do they provide enough insight on the dynamics, let alone allowing efficient and accurate estimates of the relevant observables. Dimensional reduction techniques come to our rescue in this framework. The objective is that to determine an effective manifold that captures the dynamics in its full, by factoring out the symmetries of the flow, leveraging the compression produced by dissipation, partitioning the state space by means of unstable periodic orbits or other invariants, or transporting densities via a linear transfer operator such as Koopman’s or Perron-Frobenius’. These problems will be addressed by four of the proposed talks both from a theoretical and an operational standpoint. Describing the dynamics with a transfer matrix and reducing its effective dimensionality to facilitate its spectral analysis is also the topic of an additional lecture, in the context of network theory. The development of tools for the dimensional reduction of chaotic dynamics cannot ignore the state of the art of machine learning and related technologies, which have already been assisting practitioners in the recognition and computation of optimal Poincaré sections, or highly non-trivial smooth-conjugacy transformations for flattening convoluted neighborhoods, similarly to conformal mapping. This new frontier in the investigation of high-dimensional chaos will be the object of the remaining proposed reports.
The emergence of collective dynamical behaviours is a crucial topic for the understanding of the global evolution of systems, a task necessary to building predictable, controllable and robust ones, and finding efficient healthcare strategies. From power grids and laser dynamics to Parkinson and epidemic spreading, we are confronted in our everyday life to the difficulty of this task. In many cases, the specific global behaviour of a given system is due to its underlying graph structure, but it is well-known now that the structure is not the only factor responsible of the global dynamics: other factors can play a major role in its emergence, such as a particular coupling term, a noise, a delay term, etc., giving rise for instance to noise-induced stability or collective orders, and delay-induced clustering, among other phenomena. The purpose of this minisymposium is to address several aspects of such collective (or akin to collective) behaviours of dynamical systems which are not, or not only, structurally induced.
Analysing the long term and ergodic behaviour of stochastic weather and climate models (including their fluctuations and their statistical response to parameter changes) has attracted great interest from mathematicians as well as physicists, also due to the relevance of these topics for the current climate crisis. It turns out that ideas from data assimilation play an important role in this analysis.
At the same time, effectively combining observations with dynamical models for the purpose of prediction and modelling (validation) has lead to exciting recent works at the interface of machine learning, analysis of PDEs, and data assimilation. Clarifying the long term properties of the developed approaches however remains a challenge and relies on an improved understanding of the long term behaviour of the underlying systems. This mini-symposium will bring together researchers in the fields of SPDE´s, Data Assimilation, Machine Learning and Climate Science interested in working at the intersection of these areas, to join forces in order to further our understanding of the aforementioned problems.
A significant progress in understanding of typical processes in multispecies ecological systems, as well as neural and cognitive networks, has been achieved using cyclic dynamical systems. The goal of the minisymposium is to present and discuss recent advances in that field. The talks include the description and analysis of heteroclinic networks, propagation of fronts, as well as the stability of spatially extended cyclic systems.
Nonautonomous dynamical systems describe the qualitative behavior of evolutionary differential and difference equations whose right-hand side depends explicitly on time. The importance of nonautonomous dynamical systems is illustrated by the fact that a significant number of real-world applications are governed by time-dependent inputs, and the traditional mathematical theory of dynamical systems fails to address this. Weather phenomena associated to climate change such as El Niño, the carbon cycle, and even the stock markets, are examples of dynamical processes with a deep economic impact that require sophisticated models to take nonautonomous influences in form of changes of parameters into account.
The theory of nonautonomous dynamical systems has experienced a renewed and steadily growing interest in the last twenty years. It is the goal of this minisymposium to combine and explore recent developments in both theory and applications that address the understanding of phenomena in nonautonomous dynamical systems with focus on bifurcation-theoretical questions.
The Earth system is a highly complicated and complex system with nonlinear interactions on multiple spatial and temporal scales among multiple variables. Understanding the involved nonlinear and conceptually stochastic processes and the fundamental complexity in the Earth system is of great importance for better modeling, predictions, and projections. Recent scientific progress has led to the emergence of advanced approaches and theoretical developments in, for example, nonlinear time series analysis, dynamical and stochastic systems theory, causal inference, complex and neural networks, or the role of tipping points for complex systems. These approaches have demonstrated great potential to address general and specific problems in environmental science.
In this mini-symposium we aim to bring together mathematicians and environmental scientists to present recent research on the application of mathematical methods that facilitate modern dynamical system theory and stochastic approaches. We are looking forward to insighIul and extensive discussions on the relevance and necessary modifications of such approaches for Earth System science.
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Complex systems in applications ranging from ecology to game theory display dynamic transitions between different (quasi-)invariant states. A well-known example are fluctuations of species abundancies in ecosystems. When more than two such states are present, not all transitions are necessarily possible and intriguing patterns may emerge in the connection structure. These transitions can be modeled e.g. as heteroclinic or excitable connections between (quasi-)invariant sets. Recent years have seen remarkable developments when it comes to realizing specific transition structures—given by a directed graph—as dynamical structures in the phase space of an ODE system. Oftentimes, additional restrictions are to be met: for instance, additional dynamical properties such as robustness or stability might be desirable for the connection structure. On the other hand, the class of vector fields in which the design takes place can be restricted for example by underlying symmetries or coupling structure.
This minisymposium brings together leading experts in the area of restricted design to discuss recent advances and to uncover potential future paths.
The minisymposium highlights recent advancements on delay differential equations, functional differential equations and renewal equations, which play an important role in various application areas, including balancing and control, machining, nonlinear optics, neuroscience, population dynamics, climate modelling, among others.
Reservoir computers (RCs) are dynamical systems that are capable of computation and are becoming increasingly used in the areas of machine learning (ML) and neuromorphic computing. While RCs are known for outperforming other ML approaches with significantly less computational expense, there are various other factors that fuel their rising popularity. For instance, a RC’s dynamical nature facilitates us to ‘open the black-box’ and explain the underlying dynamics behind what permits learning, prevents learning, and produces unintended functionalities. This dynamics-driven mindset also helps us to build a bridge from the brain to new computing paradigms and to harness properties of physical and dynamical systems in order to invent new ways in which a machine can learn. This minisymposium showcases how concepts from dynamical systems, neuroscience, quantum mechanics and optics, such as generalised synchronisation, multifunctionality, many-body quantum systems and time-delay, can be used to describe the learning process of RCs, allows RCs to display a broader level of intelligence by performing multiple tasks without altering its structure, and obtain ultrafast information processing speeds in a highly efficient and minimalistic manner. The aim of this minisymposium is to highlight that the area of reservoir computing enables us to expand the capabilities of artificially intelligent systems while also confronting the ‘explainability’ elephant in the room.
Complex, dynamical systems are often comparatively easy to observe, but exceedingly difficult to capture and understand theoretically. To bridge the gap between theorized and observed dynamics, several data analysis tools and data-driven models developed in Machine Learning are now being applied to dynamical systems. For example, physics-informed neural networks and universal differential equations are two approaches in which (algebraic) knowledge of system dynamics is combined with an unknown, black box function whose parameters can be fit to data using gradient-based methods and learn to represent the unknown dynamical relationships in the data. The Sparse-Identification-of-Nonlinear-Dynamics regression framework enables users to learn simple, algebraic dependencies in differential equations from data on a system’s change over time. These tools promise to support classical modeling when precise knowledge of the system is lacking, but measurement data of the phenomenon of interest is available.
However, can we always be certain that the learned relationships reflect the full dynamics of the underlying system? How accurate are predicted future dynamics of the systems? And what are the consequence of such hybrid approaches for classical modeling of dynamical systems, e.g. in fluid dynamics, population dynamics, ocean, and climate modeling? Do we find or lose interactions and dynamics beyond what classical modeling provides?
This minisymposium invites contributions, from all areas of science, that apply machine learning to understand relationships between dynamic state variables and forecast future dynamic behavior of a system from data. We welcome both theoretical and applied work from a wide range of scientific disciplines. With this minisymposium, we aim to foster an interdisciplinary exchange about advantages and disadvantages, limits, and opportunities in applying Machine Learning to dynamical systems modeling and real-world modeling applications.
It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. This minisymposium provides a platform for dissemination of new results on the theory of adaptive dynamical networks and their applications.
Recent advances in networked physiology and brain dynamics in music perception as well as noval approaches in music production like nonlinear physical modeling of musical instruments, human-machine interaction or ensemble playing arrive at complex networks representing music as a self-organizing adaptive system. The minisymposium is to present brain models and measurements, noninear dynamical models of musical instruments or larger systems including musicans interactions to also extend into social and cultural systems. Findings include brain representation of musical tension and large-scale form, rhythm perception and groove interactions, adaptation of tempo among ensemble play, temporal-lobe epilepsy estimation correlating musical semantics with brain dynamics or synchronization of brain regions corresponding to musical content. Also musical instrument tone production is discussed in terms of turbluence in wind instruments, bow-string interactions or the complex interplay of musical instruments and their environments. Enlarging dynamical models into societies or cultural phenomena are discussed. The minisymposium is about to give insight into fundamental concepts, recent developments and findings as well as future questions and implementations.
Modern power systems are tasked with incorporating increasing shares of renewable energy sources. This ongoing drastic transformation is crucial to mitigating the climate crisis but increases the complexity of the energy systems and hence poses novel challenges to our understanding of the collective dynamics of these complex systems, including of power grids. Although the underlying equations have been known for a long time, the treatment of such grids on the systems level remains difficult, mainly due to their nonlinearities, distinct heterogeneities in the dynamics of the nodes, and their interaction topology, as well as the various kinds of perturbations and fluctuations present. Additionally, power grids are embedded in and interconnected to several other complex systems, such as electricity markets, fluctuating consumers, heating and transport sectors. We focus here on central challenges of decentralized and heterogeneous power grids and ways to analyze and improve their stability.
Nonequilibrium phase transitions and critical collective behavior in dynamical networks have recently attracted much attention. They are often associated with synchronization transitions of various kinds, giving birth to a plethora of synchronization patterns, and with critical states between different stable regimes. Examples are tipping transitions, noise-induced transitions, early warning signals, explosive synchronization, bistability, hysteresis, phase coexistence, nucleation, critical slowing down, critical fluctuations and correlations, and critical exponents. Applications can be found in many natural, socioeconomic, and technological systems, for instance, in neuroscience, power grids, social networks, etc.
Many real-world dynamical systems display network structure consisting of interacting agents. Due to the complex interaction topologies with corresponding distributed control laws and the resulting high dimensionality, the investigation of such coupled systems is challenging. In particular, non-static interaction structure (e.g. time-dependent or state-dependent coupling) as well as symmetries induced by the underlying network structure cause issues. Examples of systems with varying topologies range from neuroscience (active and inactive communication channels) over epidemiology (time-dependent social interactions allow for transmissions) to distributed computing (individual mobile agents communicate only when they are in each others vicinity). Difficulties arise on multiple sides: On the one hand, the dynamics of each interacting agent may be complicated (e.g. non-smooth). But on the other hand, structural properties such as symmetries may vary and have to be taken into account. In the analysis of distributed dynamical systems in terms of synchronization, pattern formation, and symmetries, these problems come together and require combined methods from network dynamics, geometric optimization, distributed computing, and graph theory.
In the field of mathematical biology, disease modeling utilizes mathematical frameworks to simulate and understand the dynamics of infectious diseases within populations. The integration of stability analysis and bifurcation theory is pivotal in unraveling the complexities of these models, offering crucial insights into the stability of equilibria and qualitative transformations that underpin disease dynamics, thereby advancing our knowledge of epidemiological phenomena. This mini-symposium aims to delve into the dynamical systems of disease models by combining modeling, bifurcation theory, and stability analysis. Researchers and practitioners are invited to contribute expertise on disease dynamics via various mathematical models, including both ordinary and partial differential equations. To emphasize the exploration of bifurcation phenomena, investigations into the qualitative changes induced by varying parameters are encouraged. Moreover, contributions that focus on stability analysis techniques to unravel the stability properties of disease equilibria and periodic solutions are welcomed. Through presentations and discussions, the mini-symposium aims to foster collaboration among interdisciplinary participants, facilitating the exchange of ideas and advancing our understanding of disease dynamics.
Turbulence is one of the major challenge of nonlinear science. It does not only play a crucial role for fundamental research but also for many applications. A very timely topic is the use of wind energy, as the energy resource, i.e. strong wind, is by its own nature highly nonlinear and turbulent. In the application one is typically confronted with many aspect of non ideal turbulent conditions. Thus besides the deeply elaborated homogeneous isotropic turbulence questions like nonequilibrium and non stationary turbulence attract more and more interest, as well as the transition to developed turbulence. In particular in the interaction of turbulence and wind energy systems the concepts of nonlinear dynamics become important to handle this complex system.
This minisymposium will discuss novel theoretical and data-driven methods to characterize non-autonomous dynamics. We present several examples of non- autonomous dynamics from different research areas in biology, particularly neuroscience and ecosystem dynamics as well as climate science. Neuronal systems, ecosystems as well as the climate system can be considered as interaction networks which are influenced by external time-dependent drivers like external stimuli in neuroscience. Changes in environmental conditions for ecosystems and the climate system are impacted by the climate crisis.
This minisymposium is devoted to such problems as mixing, ergodicity, and existence of invariant measures, frequently occurring while working with stochastic flows, generated e.g. by stochastic Navier-Stokes or Euler equations.
Reaction networks play a pivotal role in fundamental biochemical processes such as metabolism, cell-signaling, and cell-regulation. However, tackling the complexities of these networks poses tremendous challenges, encompassing network-reconstruction, parameter-fitting, and dynamical analysis. Consequently, it is not surprising that reaction networks are addressed with a diverse array of tools from various disciplines and many fields of mathematics ranging from dynamical systems to algebraic geometry. Unfortunately, there is still a limited cross-disciplinary exchange among researchers in different fields. This limitation impedes significant progress in comprehending these central structures. With this motivation, the minisymposium aims to bring together experts from different fields of mathematics and the natural sciences, all sharing a common interest in reaction networks. The overarching goal is to facilitate an interdisciplinary exchange of ideas, methods, perspectives, and questions, fostering collaboration that can propel advancements in understanding the dynamics of these intricate systems.
The concept of an energy budget for the Earth serves as a powerful sanity check for consistency with fundamental physical principles, and indeed climate models still exhibit energetic inconsistencies whereby energy either vanishes or is created from nowhere. Due to the immense size and complexity of the Earth system, approximations for the energy transfer throughout the scales of the atmosphere and ocean must be made, and it is here that these inconsistencies arise. The TRR 181 is an interdisciplinary research group of meteorologists, oceanographers and mathematicians working to develop physically consistent approximations for the phenomena which transfer energy through these scales, including turbulence, gravity waves, ocean/atmosphere coupling, and more. By implementing these improved approximations in existing Earth system models, we enhance climate analysis and forecast accuracy.
We tackle this problem on many fronts: with theoretical mathematicians investigating fundamental issues such as well-posedness of fluid flows and chaotic dynamics, with numerical analysts investigating robustness and spurious mixing, and with oceanographers and meteorologists providing physical insight and experimental observations.
Networks of coupled nonlinear units are relevant in many fields of physics, chemistry and biology, and constitute a fascinating area of research in nonlinear science. They are paradigmatic models of complex systems, being intensively studied nowadays. One of the most famous phenomenon described by these models is synchronization, which manifests itself in the form of various spatio-temporal patterns in different natural systems. Understanding the mechanisms underlying synchronization and the resulting collective behaviors is essential for predicting and controlling the dynamics of these systems. The aim of this minisymposium is to discuss the most recent developments and tendencies in the field. The topics covered will include (but will not be limited to) synchronization in neuronal ensembles, pattern formation in networks of coupled oscillators and networks of excitable units, and the mathematical methods used to analyze these phenomena.
The goal of medium-to-large scale quantum computation, a stage when the number of entangled qubits reaches quantum advantage where quantum processors are more efficient than classical computers, opens a broad spectrum of new challenges. These range from technological issues related to the preparation, protection and manipulation of highly entangled many-body states, to the need to improve computational algorithms that solve actual practical problems.
Here one important question refers to the interplay between entanglement growth vs. Hilbert space localization. Growing quantum correlations enable to exploit the advantages that entanglement brings into computation, with the closely related exploration of the set of computational states, relevant for dynamical control. These dynamical aspects clash with mechanisms that spread or delocalize quantum states outside of the subspace of computational states. Interestingly, the competition between fast scrambling, related to classical chaos, and localization has been the subject of quantum chaos where this dichotomy appeared in the form of the transition from integrability to chaos. In low-dimensional systems early seminal developments addressed this problem within the framework of asymptotic semiclassical analysis linking quantum mechanical amplitudes with classical phase space and the notion of chaos and integrability.
With the advent of experimentally realizable quantum simulators and corresponding advances in the control of many-body states, quantum chaos has again moved into the focus now addressing complex quantum systems with large number of degrees of freedom. This subject of many-body quantum chaos has provided the theoretical framework for studying the interesting competition between scrambling and localization.
In this Minisymposium we attempt to take a fresh look on this timely subject. The idea is to discuss aspects of the coexistence between fast scrambling and integrability as essential ingredients of quantum simulators, and to identify new physical concepts in the largely unexplored field of many body quantum chaos in quantum simulators.
This mini-symposium deals with current aspects of the non-linear dynamics of the heart. Top- ics range from signal analysis for the diagnosis of cardiac arrhythmias (e.g., atrial fibrillation), simulation and control of (chaotic) excitation dynamics in the myocardium and methods to terminate life-threatening ventricular fibrillation (low-energy defibrillation) to new methods of machine learning for the analysis and data-driven modeling of complex spatio-temporal data (e.g., from optical mapping or ultrasound).
The speakers come from theoretical and experimental basic research as well as clinical applications. We, therefore, expect that this mini-symposium will not only be a platform for the exchange of current research results but will also lead to new collaborations between the par- ticipating researchers and research areas that will address current and future medical needs using novel concepts from data analysis and dynamical systems theory.
The dynamics of the atmosphere, the ocean and the climate system as a whole are complex, nonlinear, and encompass a wide range of spatial and temporal scales. Reduced-order descriptions of coarse-grained variables are naturally stochastic due to the uncertainties relating to inevitable parametric and structural errors in weather and climate models as well as unresolved scales and processes. This minisymposium is centred around nonlinear and stochastic analysis and modelling of atmospheric, oceanic and climate phenomena; it aims at bringing together experts reporting on latest developments in these areas. This includes, for example, prediction and predictability, subgrid-scale parameterization, reduced-order models, extreme events and critical transitions; strategies may range from physics-based modelling to data-driven approaches using advanced statistics and machine learning techniques to hybrid modelling. While the applications are motivated in and geared towards weather and climate science, the discussed methodologies are relevant more generally for complex dynamical systems.
Hamiltonian systems constitute a significant class within dynamical systems, boasting numerous applications across various domains. In this mini-symposium, our goal is to incorporate some of the latest theoretical techniques and applications in this field, mainly regarding with the stability of solutions (equilibria, periodic and quasi-periodic solutions, and other related manifolds). Normally the stability character of these solutions depend on various parameters the Hamiltonian function depends upon, and varying these parameters lead to bifurcations of the solutions. The stability character (linear and non-linear) and the corresponding bifurcations are known for Hamiltonian systems up to two degrees of freedom but the analysis to higher degrees is far from trivial. We will explore topics such as the Kolmogorov–Arnold–Moser (KAM) theorem, non-linear stability theory for exponentially long times, geometric approaches based on symplectic reduction of the symmetries the equations enjoy, and different types of normal forms either of local or global nature. We plan to deal with recent approaches concerning the theoretical aspect of stability and bifurcation theory, but also with an eye to applications in celestial mechanics or atomic physics.
This minisymposium focuses on rigorous analysis of the Euler- and Navier-Stokes equations, with an emphasis on applications in geophysical fluid dynamics.
The theory of discrete holomorphic dynamical systems is the mathematical study of the iteration of holomorphic mappings on complex manifolds. This now-classical subject traces its origins back to the pioneering work of Fatou and Julia in the late 1910s on the iteration of rational maps on the Riemann sphere. The subject gained increased attention in the 1980s, partly due to the emergence of computer-generated images, and has remained very active since then. It has grown into a large area with broad connections to other branches of mathematics, such as algebra, analysis, geometry, measure theory, and number theory. The aim of this minisymposium is to present recent developments on diverse aspects of the subject and to foster collaborative discussions among participants.