Research Interests: Local analytic geometry, singularities, local algebra

Research Profile: 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.

Research Interests: Analysis on manifolds, complex geometry, symplectic geometry, automorphic forms, quantization

Research Interests: Complex analysis, complex approximation theory

Research Interests: Classical analysis, summability theory

Research Interests: Algebraic topology, derived categories, modular representation theory, mathematical physics, algebraic and scientific computation

Research Interests: Algebraic combinatorics; arrangements of hyperplanes and subspaces; configuration spaces; combinatorial aspects of rational homotopy theory

Research Interests: Algebraic geometry, moduli of bundles, algebraic stacks

Research Interests: Transformation groups, toric topology, homological algebra, computer algebra

Research Interests: Homotopy theory and its applications, algebraic geometry, category theory

Research Profile: See Personal Webpage.

Research Interests: Noncommutative geometry, cyclic cohomology, operator algebras, quantum groups and Hopf algebras.

Research Interests: Birational invariant theory, multiplicative invariants, algebraic groups, Galois cohomology, Lie theory, representation theory, lattices, essential dimension, vector bundles over curves

Research Interests: Class field theory, Galois cohomology, absolute Galois groups, group cohomology, motivic cohomology, anabelian geometry, Langlands program

Research Interests: Symplectic geometry, Gromov-Witten invariants, Hamiltonian groups actions, low dimensional topology

Research Profile: 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.

Research Interests: Associative algebras, knot theory, graph theory

Research Interests: Algebraic groups and monoids, group actions in algebraic geometry, invariant theory, reflection groups

Research Interests: Associative rings and algebras, Lie algebra, group theory

Research Profile: 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 and Lie algebras. I am particularly interested in Burnside-type problems. First posed early in the twentieth century, the famous Burnside Problem for groups asked: Is every finitely generated periodic group finite? The Kurosh-Levitzki Problem is an associative algebra analogue: Is every finitely generated nil algebra finite-dimensional? Counterexamples to both problems were first constructed by Golod and Shafarevich in the 1960's. It was natural, therefore, to reformulate these problems with additional hypotheses in order to obtain a positive solution. Zelmanov won the prestigious Fields Medal in 1994 for his proof that every finitely generated residually finite group is finite, thereby solving the so-called Restricted Burnside Problem for groups. I have extended Zelmanov’s seminal work in group and Lie theory to a single unified theory that has direct applications to other branches of algebra. In particular, I have proved that a finitely generated nil algebra is finite-dimensional if it is infinitesimally-PI, generalizing Shirshov's result for PI-algebras. I have also had some success in applying this new theory to Kaplansky’s Problem which addresses the structure of group algebras whose augmentation ideal is Jacobson radical. Kaplansky’s Problem has been described as the next big hurdle in group theory after the Restricted Burnside Problem. The solution of the Restricted Burnside Problem is equivalent to saying every pro-finite group of finite exponent is locally finite. Zelmanov went on to prove even more: every periodic pro-finite group is locally finite. Together, Wilson and Zelmanov proved a related result every Engelian pro-finite is locally nilpotent. These results led to the obvious generalizations to all compact groups. I have been working on the Lie-theoretic analogues of these problems. In particular, McInnes and I have proved that every Engelian pro-finite Lie ring is locally nilpotent. It is unknown whether or not this result extends to all compact Lie rings.

Research Interests: Holomorphic mappings, CR geometry, analytic continuation, biholomorphic equivalence of domains and real submanifolds

Research Interests: Hardy inequalities, Fourier inequalities, function spaces, positive operators