The main focus of the workshop as expressed in its title is extreme waves in various physical media. These include extreme waves in the open ocean, experimental water facilities, and optical fibers. Extreme events in nature by now are well known and happen frequently in all parts of the globe. These include bushfires, floods, tornadoes, earthquakes, and tsunamis. Their frequency is visibly and noticeably increasing in recent years. Extreme waves are part of these phenomena that can be studied with relatively high precision. Moreover, they can be modeled theoretically using mathematical tools and reproduced experimentally in the lab. The possibility of common description of extreme waves in such different scales as optics and ocean waves allows interdisciplinary communication between scientists that otherwise would work separately without using the knowledge achieved in other areas.

Leading experts on these subjects present at the workshop included:

Prof. Norbert Hoffmann from Hamburg University of Technology (Germany), Prof. Joachim Peinke from Carl von Ossietzky University of Oldenburg (Germany), Prof. Majid Taki from the University of Lille (France), Prof. Matthias Fink from ESPCI Paris (France), Prof. Orazio Descalzi from the University of Andes (Chile), Prof. Cristina Masoller from the Polytechnic University of Catalonia (Spain), Dr. Matteo Conforti the University of Lille (France), Prof. Goery Genty from Tampere University (Finland), Prof. Arnaud Mussot from the University of Lille (France), Prof. Fabio Baronio from the University of Brescia (Italy), Prof. Victor Shrira from Keele University (UK), Prof. Paolo Santini from the Sapienza University of Rome (Itali), Prof.Mustapha Tlidi from the Free University of Brussels (Belgium), Dr. Maura Brunetti from the University of Geneva (Switzerland), Dr. Margarida Facao from the University of Aveiro (Portugal), Prof. Thomas Adcock from the University of Oxford (UK), Prof. Uwe Bandelow from WIAS, Berlin (Germany), Prof. Wonkeun Chang from Nanyang Technological University (Singapore), Prof. Michelle Sander from Boston University (USA), Prof. Brian Haus from the University of Miami (USA), Prof. Stefan Wabnitz from the Sapienza University of Rome (Italy), Prof. Takuji Waseda from the University of Tokyo (Japan), Prof. Holger Kantz, MPIPCS, Dresden (Germany).

Among the newcomers to the field and the young researchers who made visible contributions to the state of the research area we mention:

Prof. Yan Li from the University of Bergen (Norway), Dr. Ioannis Karmpadakis from the Imperial College London (UK), Dr. Alberto Alberello from the University of East Anglia (UK), Dr. Raphael Stuhlmeier from the University of Plymouth (UK), Mr. Maxime Canard a Master student from the Ecole Centrale de Nantes (France), Dr. Ludovica Dieli from the Sapienza University of Rome (Itali), Ms. Tatiana Tarasova from the Passau University (Germany), Dr. Andrei Gelash from the University of Burgundy (France).

An important part of extreme wave research is its practical applications. First, let us mention the novel directions that appeared in recent years. The role of ice on the surface of water in generation of extreme waves, spectral downshift of waves and the wave dissipation has been considered in a few talks. As significant areas of the world ocean are covered by ice, this research reveals the specific dynamics of waves in such areas. Counter-propagating waves and their influence on extreme wave dynamics is another area considered at the workshop. This is relevant to the so-called crossing seas when the waves are propagating in all directions of the two-dimensional ocean surface. Extreme weather events have been discussed by Prof. Kantz. He used daily precipitation data and water levels of the river Elbe for the illustration of extremes. These can be further extended to illustrate weather changes due to climate change.

Modulation instability (MI) is at the root of spontaneous appearance of extreme waves, as has been confirmed in the majorities of the talks given. Despite many years of research efforts, MI remains a complex phenomenon that is at the center of attention of many researchers. The number of publications on this subject has increased exponentially over the last few years. Significant attention to this subject has also been paid at the workshop. Its tight relation to Akhmediev breathers is one of the major points that reveals the long-term evolution of MI. Recent discoveries include the wider range of frequencies that lead to MI.

A deeper look into the mathematical concepts of rogue waves has been given by Prof. Santini. Using the theory developed for the NLSE, he extended it to the case of 2+1 dimensions covering Davey-Stewartson II equation that is also integrable and admits exact solutions. Further extension of these results to the case of KP equation, have been presented in the talk by Prof. Maruno.

Statistical analysis is an important part of rogue wave studies. Multi-point statistical analysis and comparison of empirical data of waves in turbulent state has been presented by Prof. Peinke. An interesting point of his analysis is that the extreme waves are associated with negative entropy. This conclusion could be one of the fundamental results that may have far-reaching consequences. A statistical approach based on kinetic equations has been presented by Ms. Tarasova. The latter can be applied to a large variety of media where extreme waves appear in the form of multi-soliton gas in shallow water using the Korteweg-de Vries formalism. Experimental statistical data on extreme waves in a 150 m water tank have been presented by Mr. Canard. These results provide links between the extreme crest distributions, the modulations of the wave envelop and the nonlinearity of the system. Extreme wave statistics in the cases of intermediate and shallow water depths have been presented by Dr. Karmpadakis. These studies are based on both, sea surface measurements and the lab data. These experimental data can be used to create a predictive model for crest height statistics.

An important aspect of research on extreme waves is their prediction. Extreme waves that appear in ocean currents have been analyzed by Prof. Shrira. The ocean currents (like notorious Agulhas current in the Indian Ocean) allow for resonant triad interactions that do not exist without the current. One of the interacting waves becomes longer and may serve as a precursor of an extreme wave. This possibility has been theoretically investigated.

Optical extreme waves have been presented in a number of talks. Quadratic optical media is one of the possibilities. Prof. Schiek addressed this issue in his experimental studies of Akhmediev breathers and modulation instability that essentially lead to extreme waves. These studies are done in spatial geometry that is a counterpart of the wave evolution in time. Prof. Baronio made further steps in this idea and investigated resonant radiation emitted by the Akhmediev breathers and the Peregine soliton that is a particular case of the Akhmediev breather at zero frequency. The studies of extreme waves in multi-mode fibers have been presented by Prof. Wabnitz. These are also related to the second-harmonic generation. Beam breaking into filaments revealed extreme local intensity peaks.

Last, but certainly not least, novel machine-learning techniques for ocean wave prediction for the purpose of operational prediction of rogue waves in the ocean have been reported by Prof. Hoffmann. These advances have a strong potential to be used on ships or fixed marine installations and could predict the formation of large-amplitude waves within a few minutes. It is also worthy to mention that for popularizing the subject Prof. Hoffmann presented public lecture for the citizens of Dresden informing them about the extreme waves in nature and their most recent scientific studies. The lecture attracted significant number of people interested in extremes.

It is hard to list all impactful talks made at the workshop. Overall, significant progress has been achieved in the area of extreme waves that have been amplified by additional three years of intensive research on the subject during the pandemics time. There is no doubt that this workshop will serve as one of the significant roadmaps in further progress in the quickly developing area of extreme waves, which appears to become a discipline of its own.