## Periodic Reporting for period 1 - IPaDEGAN (Integrable Partial Differential Equations: Geometry, Asymptotics, and Numerics.)

Reporting period: 2018-01-01 to 2019-12-31

Partial Differential Equations (PDEs) are among the main tools for an efficient modelling of physical phenomena. Integrable PDEs, exhibiting infinite-dimensional analogues of regular (integrable) behaviour displayed by finite dimensional systems, began to be studied in the middle of the XX century in fluid dynamics, field theory and plasma physics.

It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with the discovery of the recurrence behaviour of soliton solutions, the Lax pair formulation, and the infinite number of conservation laws. Over the last 40 years refined analytical geometrical as well as numerical tools have been developed to study integrable or nearly integrable systems.

The idea that an integrable behaviour persists in real (non-integrable) systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs is the leitmotiv of this research project. The predictive power of numerics and scientific computing is used both as a testing tool for theoretical models and as a generator of new conjectures.

Random matrices, introduced in the study of the statistical properties of quantum systems with a great number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications.

Just like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour, conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix grows indefinitely.

One of the principal scientific aim of the Network is to create a fertile research environment for younger scientists (especially Ph.D. students and Post-docs) working in the fields of Mathematical Physics by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.

The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The power of efficient numerical tools is instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bi-Hamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals are being used in the pursue of our goals.

It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with the discovery of the recurrence behaviour of soliton solutions, the Lax pair formulation, and the infinite number of conservation laws. Over the last 40 years refined analytical geometrical as well as numerical tools have been developed to study integrable or nearly integrable systems.

The idea that an integrable behaviour persists in real (non-integrable) systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs is the leitmotiv of this research project. The predictive power of numerics and scientific computing is used both as a testing tool for theoretical models and as a generator of new conjectures.

Random matrices, introduced in the study of the statistical properties of quantum systems with a great number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications.

Just like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour, conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix grows indefinitely.

One of the principal scientific aim of the Network is to create a fertile research environment for younger scientists (especially Ph.D. students and Post-docs) working in the fields of Mathematical Physics by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.

The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The power of efficient numerical tools is instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bi-Hamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals are being used in the pursue of our goals.

The project has resulted in 23 published papers (and 12 more preprints) by the end of Period 1, covering our progress in the three scientific themes of IPaDEGAN:

Work Package II – (Near-to) Integrable PDEs and Applications

Work Package II – Isomonodromic Deformations, Painlevé equations, and random objects

Work Package IV – Numerical Approaches.

Significant achievements were obtained in the theory of Random Matrices, in the theory of stratified fluid dynamics, and in the numerical study of integrable equations in 2D, such as the Dawey-Stewartson equation.

These publications were mainly the result of secondments in 2018 and 2019. More secondments are scheduled in the nearest future, which will enable the teams to make further progress towards the project’s goals.

We have held so far three Network conferences. The first one in Milano in June 2018, the second one at C.I.R.M. Marseille in June 2019, and the third one in Dijon in September 2019. A fourth major conference is scheduled to happen in Trieste, July 2020, and we are considering the possibility of organizing a final Conference in June 2021, with an eye towards possible follow-ups of the IPaDEGAN research activities.

All Conferences saw the participation of an enlarged scientific community, thus enhancing the dissemination strategy of the Network. Special care was (and will be) given to younger members of the Network in terms of the possibility of presenting their results and discuss with more experienced specialists in their field of research. Care was also given to outreach events, resulting in the participation (among other public events) in two European Researchers’ Nights in France and three in Italy.

Work Package II – (Near-to) Integrable PDEs and Applications

Work Package II – Isomonodromic Deformations, Painlevé equations, and random objects

Work Package IV – Numerical Approaches.

Significant achievements were obtained in the theory of Random Matrices, in the theory of stratified fluid dynamics, and in the numerical study of integrable equations in 2D, such as the Dawey-Stewartson equation.

These publications were mainly the result of secondments in 2018 and 2019. More secondments are scheduled in the nearest future, which will enable the teams to make further progress towards the project’s goals.

We have held so far three Network conferences. The first one in Milano in June 2018, the second one at C.I.R.M. Marseille in June 2019, and the third one in Dijon in September 2019. A fourth major conference is scheduled to happen in Trieste, July 2020, and we are considering the possibility of organizing a final Conference in June 2021, with an eye towards possible follow-ups of the IPaDEGAN research activities.

All Conferences saw the participation of an enlarged scientific community, thus enhancing the dissemination strategy of the Network. Special care was (and will be) given to younger members of the Network in terms of the possibility of presenting their results and discuss with more experienced specialists in their field of research. Care was also given to outreach events, resulting in the participation (among other public events) in two European Researchers’ Nights in France and three in Italy.

In its first half of lifespan, the Network’s activities have developed novel methodologies combining techniques coming from apparently unrelated areas of Mathematics such as Geometry, Analysis and Numerical Analysis, applied to problems originating from Physics and Mathematics. For instance, asymptotic methods were applied in models of integrable and near-to integrable equations in one space dimension, and problems in the theory of random matrices have been solved by using techniques coming from Conformal Field Theory. Also, new numerical techniques were used to study semi-classical limits of integrable models in two space dimensions.

An immediate impact was given in terms of dissemination to Workshops (around 100 talks) of the project’s major results, as well as of the publications of papers in renowned Mathematical and Physical Journals. We deem important to remark that that IPaDEGAN younger members are obtaining considerable success in their careers both within Academia and outside.

Form a broader perspective, later impact will certainly come out of the realizations of the goals concerning the application of our results to a variety of unsolved problems in Mathematics and mathematical Physics. Potential social and socio-economic impact will come from the expected development of ∂-bar techniques to inverse problems of medical imaging, and, possibly, by the development of our research about stratified fluid flows, to be potentially applied in oceanography and marine sciences

An immediate impact was given in terms of dissemination to Workshops (around 100 talks) of the project’s major results, as well as of the publications of papers in renowned Mathematical and Physical Journals. We deem important to remark that that IPaDEGAN younger members are obtaining considerable success in their careers both within Academia and outside.

Form a broader perspective, later impact will certainly come out of the realizations of the goals concerning the application of our results to a variety of unsolved problems in Mathematics and mathematical Physics. Potential social and socio-economic impact will come from the expected development of ∂-bar techniques to inverse problems of medical imaging, and, possibly, by the development of our research about stratified fluid flows, to be potentially applied in oceanography and marine sciences