Local analytic geometry, singularities, local algebra
My main research interests are in (real and complex) local analytic geometry. More specifically, I study algebraic aspects of the local geometry of analytic mappings, as well as algebro-geometric and differential properties of subanalytic sets, and analytic invariants.
I am also interested in discrete mathematics. Particularly, extremal graph theory, long cycles and paths, and the combinatorics of desingularization.
Symplectic geometry, Gromov-Witten invariants, Hamiltonian groups actions, low dimensional topology
My main interest is in symplectic topology and geometric topology. Currently, my research focuses on the homotopy theoretic, geometric, and algebraic properties of symplectomorphism groups. Symplectic geometry was initially developped as the mathematical framework of classical physics. It evolved into an independent field of research, very topological in nature, at the crossroads of differential topology, algebraic geometry, dynamics, and gauge theory. Nowadays most of the new developments involve Gromov's theory of $J$-holomorphic curves, Hofer geometry, Donaldson's theory of topological Lefschetz fibrations, and Floer homology theory.
Associative algebra, Lie algebra, and group theory; the relationship between identities, gradings, and actions.
The specific branch of Noncommutative Algebra that I study is sometimes called Combinatorial Algebra. The term derives from its ancestor, Combinatorial Group Theory, which traditionally focuses on such topics as groups presented by generators and relators, growth in groups, the Burnside Problem, as well as computational and algorithmic problems. Similar problems are now studied for infinite-dimensional associative algebras, Lie algebras, and Jordan algebras.
I served as Department Chair during the period 2006-2011 and was appointed Faculty Scholar in 2010.
Before joining Western University in 1999, I held various positions at the University of Alabama, the University of Alberta, the University of Oxford, and the Università degli Studi di Palermo. I have also made extended research visits to the University of Auckland, the Vrije Universiteit Brussel, Lunds Universitet, the University of New Brunswick, and the Università di Milano-Bicocca.