Assistant Professor, Department of Mathematics & Statistics, University of Saskatchewan
Office 209, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, CANADA

rayan (at) | Dept of Math & Stats | U of S


In my research, I am attempting to pursue a deeper understanding of problems at the interface of complex algebraic geometry, differential geometry, and mathematical physics.

Moduli spaces, coming either from gauge theory or from geometric invariant theory, are central to my work. I am particularly interested in computing topological invariants of moduli spaces, such as Betti numbers, using a combination of representation theory and analysis. To this end, I have proven that the genus zero limit of the ADHM formula of Chuang, Diaconescu, and Pan gives correct Betti numbers for moduli spaces of twisted Higgs bundles. Some things I have written on these topics: 1010.2526, 1309.7014, 1406.1693, 1609.08226. Recently, Jonathan Fisher and I computed the Betti numbers for the rational cohomology of hyperpolygon space — the natural hyperk√§hler analgoue of polygon space — for all ranks: 1410.6467. Fascinating links between the moduli space of Higgs bundles and mirror symmetry are a driving factor in my work.

I am also interested in the geometry of manifolds admitting Ricci-flat metrics, including but not limited to Calabi-Yau manifolds and, in particular, hyperkähler manifolds. See 1706.05819, for instance. Mirror symmetry is again the impetus here, along with integrability.

Research interests broadly

  • complex algebraic geometry
  • symplectic geometry
  • differential geometry
  • mathematical / theoretical physics
  • gauge theory
  • integrable systems

Research interests less broadly

  • moduli spaces of vector bundles and sheaves on complex varieties
  • moduli of Higgs bundles and geometry / topology of the Hitchin fibration
  • representation spaces of quivers in various categories, including (hyper)polygon space
  • hyperkähler geometry
  • deformations and stability of vector bundles on Calabi-Yau manifolds
  • mirror symmetry, especially with regards to Higgs bundles
  • Morse theory, especially for noncompact and singular spaces
  • inverse problems and transforms in geometry
  • applications of geometry outside mathematics and physics

You can find my works on the arXiv, Google Scholar, and ORCiD.