Integrable Hamiltonian Systems

My research studies mathematical models from classical mechanics known as integrable Hamiltonian systems. These systems are special because they have enough integrals, that is, quantities that stay constant over time, to make their motion highly structured and, in principle, exactly solvable. Locally, the behavior of such systems is very simple: motion takes place on invariant tori — smooth, torus-shaped surfaces in an abstract space that records position and momentum — and this motion can be described using special coordinates called action–angle variables.

The main questions I work on arise when one looks at these systems globally rather than locally. Even though integrable systems look simple in small regions, they become more complicated when looking across the entire space of possible motions. Understanding why this happens, and what it reveals about the system, is a central theme of my work.

One focus of my research is the global geometry and topology of invariant tori. I study how these torus-shaped surfaces are arranged, how they change near special or singular motions, and how global features emerge that cannot be detected by local calculations alone. This leads to mathematical invariants — quantities or structures that remain meaningful regardless of the coordinates used.

A key concept here is monodromy, which describes an obstruction to defining action–angle coordinates globally. I have worked on several variants of this idea (standard monodromy, fractional monodromy, quantum monodromy, scattering monodromy), showing how different types of singular behavior give rise to different forms of monodromy. These results illustrate that even in systems that are considered “solvable,” global structure can still be rich, subtle, and informative. A brief exposition of my work on monodromy can be found in the linked article “Global geometry of physical systems and its quantum manifestations”.

Another aspect of my work concerns Hamiltonian bifurcations, where changes in system parameters lead to qualitative changes in the organization of invariant tori in phase space. I study how these transitions occur and how they affect the global structure of the system.

Many of these questions are motivated by concrete physical examples, such as models from molecular physics, where integrable systems help explain observed patterns in the quantum spectrum of atoms and molecules. The mathematics provides here a clean framework for understanding which features are universal and which depend on specific modeling assumptions.

To prepare for research in these areas, it is important to study Multivariable Calculus and Linear Algebra, which introduce the ideas of change, conservation, and geometry in higher dimensions. The course ODE and Dynamical Systems provides the basic tools for studying evolving systems and emphasizes structure and qualitative behavior. Throughout, developing geometric intuition and abstract reasoning is more important than mastering long computations; nevertheless, symbolic and numerical tools such as Mathematica are often used to explore and illustrate representative systems.