Journal

The article Maslov $S^1$ Bundles and Maslov Data with Bohuan Lin and Holger Waalkens has been accepted for publication in the Journal of Mathematical Physics. In the article we introduce the notion of Maslov $S^1$ bundles over general symplectic manifolds, generalizing similar constructions on cotangent bundles. We analyze the properties of Maslov $S^1$ bundles and define a generalization of the Maslov index when the Maslov bundle is trivial. In the case where the Maslov bundle over the manifold $M$ is not trivial, but there is a symplectic $S^1$ action on $M$, we introduce the notion of Maslov data which serves as a non-integrable version of the Maslov index. Finally, we use Maslov bundles to prove results about homogeneous spaces and monotone symplectic manifolds, and we consider applications to integrable Hamiltonian systems.

Our recent preprint Hamiltonian Monodromy in a Tavis-Cummings System with an $A_2$ Singularity, with Gabriela Gutierrez-Guillen, Pavao Mardešić, and Dominique Sugny, analyzes the topology of the singular Lagrangian fibration of the classical two-spin Tavis–Cummings system. We show that this three-degree-of-freedom integrable Hamiltonian system has a singularity of $A_2$ type which has not been previously observed in “well-behaved” singular Lagrangian fibrations and goes beyond the familiar focus–focus ($A_1$) singularity.

The Zu Chongzhi Center conference Advances in Applied Mathematics and its Applications takes place October 26-29, 2023 at Duke Kunshan University. The conference is co-organized by Konstantinos Efstathiou, Jian-Guo Liu, Shixin Xu, and Xiaoqian Xu.

I am in Antwerp for the 7th International Conference on Finite Dimensional Integrable Systems in Geometry and Mathematical Physics (FDIS 2023) organized by Sonja Hohloch. I gave a presentation on Rotation 1-Forms and Non-Compact Monodromy based largely on earlier joint work with Andrea Giacobbe and Pavao Mardešić.