This is to provide updates on my research as a backup communication channel for my project Multidimensional Integrable Systems
https://www.researchgate.net/project/Multidimensional-integrable-systems
should Researchgate go through with its plan to discontinue the project feature altogether on March 31, 2023.
For the previous update log please see this snapshot
https://archive.ph/kvObF
made on April 1, 2023.
You are welcome to follow this question, especially if you are interested in my research, and/or have already followed or intend to follow the above project.
For now, here are the links
0) to my lab
https://www.researchgate.net/lab/Integrability-Online-Lab-Artur-Sergyeyev
1) to the presentation explaining the most important research results in the project
Presentation From Contact Geometry to Multidimensional Integrable Systems
2) to the key paper of the project
Article New integrable (3+1)-dimensional systems and contact geometry
and to the other works in the project
Article Recursion Operators for Multidimensional Integrable PDEs
which is now a spotlight, see
https://www.researchgate.net/spotlight/64143f68cb436c2c2d0f3049
Article Integrable (3+1)-dimensional system with an algebraic Lax pair
Article Integrable (3+1)-dimensional systems with rational Lax pairs
Article A Simple Construction of Recursion Operators for Multidimens...
As an AI language model, I don't have access to up-to-date information on ongoing research projects. However, I can provide general information about the field of multidimensional integrable systems and recent developments in the field.
Multidimensional integrable systems refer to systems of partial differential equations (PDEs) that can be solved explicitly by quadratures or integrals. These systems arise in various areas of mathematics and physics, including algebraic geometry, mathematical physics, and differential geometry.
Recent developments in the field of multidimensional integrable systems have focused on the study of the geometric and algebraic structures underlying these systems. In particular, there has been significant progress in the study of cluster algebras, which provide a unifying framework for many different types of integrable systems. Other areas of active research include the study of integrable systems on curved spaces, the classification of integrable systems, and the development of numerical methods for solving multidimensional integrable systems.
Overall, the study of multidimensional integrable systems remains an active and rapidly developing area of research, with important implications for many different fields of mathematics and physics.
An important correction to M.e.I.'s answer of March 24, 2023: for many integrable systems, especially integrable partial differential systems, it is possible to write plethora of particular solutions but not necessarily the general solution, see e.g. https://mathoverflow.net/a/9873 and https://w.wiki/6VdL .
For advice on how to preserve the project data & share research updates after discontinuation of the projects feature on March 31, see here:
https://www.researchgate.net/post/The_end_of_Researchgate_projects_what_to_do
IMPORTANT: Fresh snapshot of the project log: https://archive.ph/fCsC1
I just finished giving an invited talk Recent Advances in Multidimensional Exactly Solvable Models at the International Webinar on Advances in Mathematical Sciences organized by the West Bengal State University in Kolkata, India based on the papers
Article New integrable (3+1)-dimensional systems and contact geometry
Article Integrable (3+1)-dimensional system with an algebraic Lax pair
See also the project update:
https://www.researchgate.net/project/Multidimensional-integrable-systems/update/6426a42b28b5df6cef0b33f4
Recent talk Infinitely many new (3+1)-dimensional integrable systems from contact geometry at the Nagoya Math-Phys Seminar mainly based on
Article New integrable (3+1)-dimensional systems and contact geometry
Seminar web site: https://www.math.nagoya-u.ac.jp/~hamanaka/NMPS.html
Slides: http://www.math.nagoya-u.ac.jp/~hamanaka/Artur20231109.pdf
Recording: https://www.youtube.com/watch?v=p6fhhebM_lY
The project's main paper
Article New integrable (3+1)-dimensional systems and contact geometry
has recently reached 40 citations (without self-citations) in Google Scholar
https://scholar.google.com/scholar?start=0&hl=en&cites=4464507104104868530,3265588241197881773