The Geometry & Topology research group at the University of Waterloo is a young and very active group. We have frequent seminars, specialized graduate courses, visitors, and postdoctoral fellows. If you are interested in graduate studies or postdoctoral fellowship opportunities in geometry and topology at Waterloo, please contact one of the group's faculty members.
Some of the research interests of our group include
- low dimensional topology
- gauge theory
- symplectic and complex geometry
- arithmetic geometry
- special holonomy and calibrations
- applications of geometry to physics
- toric geometry
Ákos Nagy, BEIT Canada
Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform for Fuchsian groups which I will call the hyperbolic Bloch transform (HBT). The HBT transforms wave functions on the hyperbolic plane to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. I will prove that the HBT is injective and “asymptotically unitary”. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan (arXiv:2208.02749).
Guillermo Gallego, Universidad Complutense de Madrid
Multiplicative Higgs bundles are a natural analogue of Higgs bundles on Riemann surfaces, where the Higgs field now takes values on the adjoint group bundle, instead of the adjoint Lie algebra bundle. In the work of Charbonneau and Hurtubise, they have been related to singular monopoles over the product of a circle with the Riemann surface.
In this talk we study the natural action of an involution of the group on the moduli space of multiplicative Higgs bundles, also from the point of view of monopoles. This provides a "multiplicative analogue" of the theory of Higgs bundles for real groups.
Andy Royston, Penn State University
The extended Bogomolny equations are a system of PDE's for a connection and a triplet of Higgs fields on a three-dimensional space. They are a hybrid of the Bogomolny equations and the Nahm equations. After reviewing how these latter systems arise in the study of magnetic monopoles, I will present an energy functional for which solutions of the extended Bogomolny equations are minimizers in a fixed topological class. In this construction, the connection and Higgs triplet are defined on all of R^3 and couple to additional dynamical fields localized on a two-plane that are analogous to jumping data in the Nahm equations. Solutions can therefore be interpreted as finite-energy BPS solitons in a three-dimensional theory with a planar defect. This talk is based on work done in collaboration with Sophia Domokos.