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Department of Mathematics
Broadly, my research lies in nonlinear differential equations, integrability and solitons, mathematical physics and analysis.
My interests in nonlinear differential equations center on applications of symmetry analysis and conservation laws to the study of PDEs, particularly nonlinear wave equations and soliton equations, as well as ODEs connected with exact solutions through group invariance and other methods. Some applications are also devoted to extending standard methods and developing new approaches both for symmetries and conservation laws. I am co-authoring two books in this area with George Bluman (Department of Mathematics, UBC).
In mathematical physics my interests include nonlinear aspects of the Yang-Mills equations, the Einstein gravitational field equations and massless fields on curved spacetime, wave maps (nonlinear sigma/chiral models) and Schrodinger maps (Heisenberg models). One recent focus is on their symmetry and conservation law structure as well as integrable reductions and local/global analysis of solutions. Another direction for many years has been the study of deformations (novel nonlinear generalizations) of gauge theories like Yang-Mills and gravity theories.
Recently I have begun working on computational and geometric aspects of integrable PDEs and solitons. One direction has been classifying integrable hyperbolic systems and group-invariant soliton equations. Related work has studied the geometric origin of such PDEs and their Hamiltonian structure arising from curve flows in various curved generalizations of Euclidean space.