Gregory D. Landweber

Research

My research interests include Clifford algebras, graph theory, error correcting codes, supersymmetry, string theory, K-theory, equivariant symplectic geometry, and representation theory of Lie groups and loop groups. For a complete description of my research, see my research statement. I am currently working on two collaborative research projects:

Supersymmetry and Adinkras

Collaborators:

In physics, there are two types of elementary particles: bosons, such as photons, and fermions, such as electrons and quarks. Supersymmetry is the principle that this dichotomy is not accidental, but rather is due to a requirement that the universe contain both bosons and fermions, paired off as supersymmetric partners. Supersymmetry is a vital ingredient in string theory, and it elegantly explains the appearance of both types of particles in nature. My collaborators and I are attempting to classify all of the possible collections of particles that can appear in a supersymmetric physical theory using diagrams we call "Adinkras", graphs with vertices corresponding to the particles in the theory and edges corresponding to the supersymmetry pairings. This project also involves the representation theory of Clifford algebras, error correcting codes, and K-theory.

Quotients in symplectic and related geometries

Collaborators:

Megumi Harada (McMaster University)
Graeme Wilkin (Johns Hopkins University)
Lisa Jeffrey (University of Toronto)
Eckhard Meinrenken (University of Toronto)
Anton Alekseev (University of Geneva)
Paul Selick (University of Toronto)
Reyer Sjamaar (Cornell University)

Symplectic geometry is a branch of differential geometry that generalizes the mathematical formalism underlying classical and quantum mechanics. A symplectic structure on a curved space can be described in terms of complex numbers; in my research, I also study hyperkähler structures which can be described similarly in terms of quaternions. One way to obtain such spaces is to start with a simpler space that admits a symmetry group and a momentum map, and then construct a quotient by restricting to a fixed momentum and then dividing by the symmetry. My collaborators and I study the topology of such quotients in terms of the equivariant K-theory and other algebraic topology invariants of the original simpler space.