My current research uses techniques from pure mathematics (notably symplectic geometry, the natural mathematical framework for classical mechanics) to prove results obtained by theoretical physicists using the methods of quantum field theory. In my doctoral thesis (under the supervision of Michael Atiyah) I provided a mathematically rigorous proof of results on the asymptotics of the three-manifold invariants of Witten and Reshetikhin-Turaev which Witten had conjectured based on his approach to these invariants using quantum field theory.
In recent joint work with Frances Kirwan I have proved formulas of Witten which encode the structure of the cohomology ring of the moduli space of holomorphic vector bundles on a Riemann surface: the main technique used is a method from symplectic geometry and equivariant cohomology known as nonabelian localization, which Kirwan and I developed in our initial paper. Later developments are joint work with Young-Hoon Kiem, Frances Kirwan and Jonathan Woolf.
In joint work with Jonathan Weitsman I have studied these moduli spaces using techniques from symplectic geometry (the theory of Hamiltonian group actions): these methods endow the moduli spaces with Hamiltonian flows, in some cases leading to a structure of integrable system on them, and yielding a very transparent description of the formulas for their symplectic volumes.
In joint work with Megumi Harada, Tara Holm and Augustin-Liviu Mare, we have shown that the level sets of the moment map for the natural torus action on the based loop group are connected.
In joint work with Jacques Hurtubise and Reyer Sjamaar (following an earlier paper joint with Victor Guillemin and Reyer Sjamaar) we study imploded cross-sections. This is a refinement of the symplectic cross section.
Symplectic geometry, geometric applications of quantum field theory
Awards and Grants:
- Fellow of the American Mathematical Society, 2013
- Fellow of the Royal Society of Canada, 2007
- Steacie Fellowship, 2004
- Premier's Research Excellence Award, 2004
- Coxeter-James Prize, 2002
- Alfred P. Sloan Research Fellowship, 1997
A.B.(Princeton), M.A.(Cambridge), Ph.D.(Oxford), F.R.S.C.