I’m currently supervising two graduate students: PhD candidates Nasser Heydari, Ibrahem Al-Jabea.

Former Master’s students:
Peter Deal, Toward a non-Abelian Atiyah-Bredon sequence, completed Spring 2013.

Former NSERC USRA students:
Brad Dart, Topological gauge theory and representation varieties, Summer 2011.
Adam Gardner, Symplectic geometry and integrable systems, Summer 2011.

Projects

Ph.D. Students

Nasser Heydari, Non-abelian Kirwan surjectivity for real loci, Fall 2013-present.

Consider the Hamiltonian action of a  compact group G on a symplectic manifold (M,w). The Kirwan Surjectivity theorem relates the G-equivariant cohomology of M with the cohomology of the symplectic quotient M//G.  In the presence of a compatible anti-symplectic involution t of (M,w) and when G is a torus, Goldin-Harada have proven a similar result relating the equivariant cohomology of the fixed point Lagrangian M^t with a fixed point Lagrangian in the quotient (M//G)^t.    Nasser aims to generalise Goldin and Harada’s result to non-abelian compact groups G.

Ibrahem Al Jabea, Higher cohomology of GKM-sheaves, Winter 2013 – present.

In a paper to be published in the Journal of Symplectic Geometry, I introduced a new class of sheaves, called GKM-sheaves, that establish a general framework able to incorporate many constructions in GKM-theory (a branch of equivariant topology named after Goresky, Kottwicz, and MacPherson). These sheaves $F$ have the property that their zeroth cohomology $H^0(F)$ (i.e. the ring of global sections), is isomorphic to the equivariant cohomology of some $G$-space. Ibrahim’s problem is to calculate and find a geometric interpretation for the positive degree cohomology of these sheaves. We have made good progress on this problem and I am confident based on our preliminary results that this project will lead to a published research paper.

Master’s Students

Peter Deal, Toward a non-Abelian Atiyah-Bredon sequence, Fall 2010- Fall 2012.

Peter’s thesis problem was to generalize a cornerstone result about the cohomology theory of connected abelian lie groups actions (the Atiyah-Bredon sequence) to non-abelian groups actions. Although the problem was not successfully resolved, Peter learned a great deal of advanced material about fibre bundles and equivariant cohomology. His thesis paper presented an introduction to these topics, leading up to a description of my conjecture and an illustration of the conjecture in some simple examples.

USRAs

Adam Gardner, Symplectic Geometry and Integrable Systems, Summer 2011.

Adam made substantial progress on his summer research project to study the “Moduli space of Delzant polytopes” which classify compact toric manifolds, including successfully proving that the moduli space is connected.

Brad Dart, Topological Gauge Theory, Summer 2011.

Brad’s project was to study the topology of a class of gauge theoretic moduli spaces associated to the Klein bottle.

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