Local properties of many spaces are well understood because there are simple local models: the surface of the earth cannot be distinguished from the plane at small scales.
Although the sphere has a well-understood local model, it has interesting and surprising global features. For example there are always antipodal points on the sphere where both the temperature and the barometric pressure are equal. Differential topology is the study of global features of spaces with simple local models. Our expertise in differential topology lies in the differential topology of infinite dimensional spaces. Our research specializes in research areas such as geometric partial differential equations, symplectic topology, and geometric dynamics.
For example in symplectic topology we are interested in the definition and computation of symplectic invariants of contact and symplectic manifolds, with an eye towards the classification of symplectic and contact structures, but also towards the application of these invariants to the study of dynamics on these manifolds, for instance in connection with the question of existence of periodic Hamiltonian and Reeb orbits both finite and infinite dimensional.
In the area of dynamical systems we study knot and braid invariants using techniques from finite dimensional algebraic topology such as Conley index theory and the theory of partial differential equations via pseudo-holomorphic curves. The algebraic invariants that are constructed this way are used to study for example parabolic and Hamiltonian dynamics.
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