Research Interest
My research area is the geometry of differential equations. My basic tools include the theory of exterior differential systems and the method of equivalence, both initiated by Élie Cartan. The former is a geometric, coordinate-free theory for differential equations; the latter is a way to tell when two geometric structures are equivalent. A combination of the two allows me to study differential equations intrinsically via their geometric invariants. A main topic of my past and current research is Bäcklund transformations of second-order PDEs. Originally discovered in classical differential geometry, Bäcklund transformations have been playing an important role in mathematical physics since the mid 20th century. By using the geometric tools mentioned above, my papers [2], [4] and [6] provided new generality results and new constructions of Bäcklund transformations. Another topic that I'm interested in is the absolute equivalence of control systems. It is well-known that, if a control system admits a so-called dynamic feedback linearization, then one can express all its solutions by differentiation alone, a property much desired in applications. It has been a long-standing open problem to determine exactly when such a linearization is possible. In [3] and [5], my collaborators and I were able to make some progress in this direction.