Research problems on relations between Nijenhuis geometry and integrable systems

TL;DR

This paper reviews open problems and questions at the intersection of Nijenhuis geometry and integrable systems, aiming to guide future research in understanding their complex relationship.

Contribution

It provides a comprehensive survey of unresolved issues and challenges in the interplay between Nijenhuis geometry and integrable systems, based on recent research symposia.

Findings

Identifies key open problems in Nijenhuis geometry and integrable systems

Highlights the importance of these problems for advancing the field

Provides a resource for researchers to explore future directions

Abstract

The paper surveys open problems and questions related to interplay between the theory of integrable systems with infinitely and finitely many degrees of freedom and Nijenhuis geometry. This text has grown out from preparatory materials for the series of research symposia and workshops on Nijenhuis geometry and integrable systems held at SMRI (Sydney) and MATRIX (Creswick) in February 2022 and at La Trobe University (Melbourne) and MATRIX in February 2024, and from the open problem sessions at these events. It includes both relatively simple questions to get familiar with the topic, as well as challenging problems that are of great importance for the field.