Research
My main research interests are in the field of symplectic topology and geometry, which is a branch of mathematics concerned with the study of symplectic manifolds i.e. geometric structures that arise naturally in the study of classical and quantum mechanics. My work particularly focuses on several areas within this field:
Symplectic and Hamiltonian Group Actions: A fundamental problem is to better understand actions of Lie groups on symplectic manifolds, that is, symmetries of the manifold that preserve its symplectic structure. This can be broken into two broad questions: how the knowledge of Lie group actions can be used to understand the manifold’s geometry and topology and, conversely, how the geometry of the manifold restricts the groups that can possibly act on it. This includes extending various classification results of Hamiltonian Lie group actions.
Topology of Symplectomorphism Groups: More generaly, I am interested in structural and topological properties of infinite dimensional symplectomorphism groups, such as their homology and homotopy groups, or the geometries induced on them by various invariant metrics. This involves understanding how the symplectic structure influences and constrains the possible topological configurations of auxiliary infinite dimensional spaces such as spaces of almost complex structures.
Embedding Problems: Thanks to the foundational work of Gromov in the 1980’s, we known that key properties of a symplectic space are captured by looking at how standard symplectic balls $B(c)\subset\mathbb{R}^{2n}$ (which, by Darboux theorem, are the building blocks of symplectic manifolds) embbed into it. Another aspect of my research involves studying how symplectomorphism groups act on spaces of symplectic embeddings, and what geometric and topological information can be extracted from this action.
Quantization and Mathematical Physics: I am also exploring the connections between symplectic geometry and mathematical physics, particularly in the context of geometric quantization (à la Souriau). This involves understanding how classical mechanical systems described by symplectic manifolds transition to quantum mechanical systems.
Interplay with Algebraic Geometry: My work often intersects with algebraic geometry, particularly in the study of Kähler manifolds and deformation of complex structures. This includes investigating how techniques from algebraic geometry can be applied to solve problems in symplectic topology and vice versa.
Finally, an interesting feature of the above areas is that they usually involve a variety of mathematical tools and techniques borrowed from other fields, including real and complex differential geometry, algebraic topology, and geometric analysis.