Multisymplectic Field Theory
Multisymplectic geometry gives an intrinsic formulation of the Hamiltonian formalism of field theories. It is a natural generalization of symplectic geometry, which is the geometric framework for classical mechanics. This approach allows for a covariant description of the dynamics of fields, and it provides a natural setting for the study of symmetries and conservation laws.
Beyond Integrability
This line of research focuses on systems that go beyond traditional integrability. We focus on systems which are not integrable in the classical Arnold-Liouville sense, but which share some of their properties, as having conserved quantities or being non-chaotic. Nevertheless, they might have a more complex structure, as having folliations by submanifolds whith different topology than tori (higher genus, non-compact), or situations in field theories instead of mechanics. We use Lie-groupoid theory in order to describe their symmetries, generalizing the role of abelian Lie groups.
Quantum systems and topological phases of matter
We study quantum systems whose physical behaviour is governed by topology, algebraic structures and generalized notions of symmetry. A central focus is the mathematical description of anyonic excitations: quasiparticles in two-dimensional systems whose exchange properties are governed by braid-group representations rather than ordinary bosonic or fermionic statistics. We also aim to place anyonic statistics within a common operational framework, with applications to the foundations of quantum theory and to the information-theoretic structure of topological quantum computation.