Office 209, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, CANADA

rayan (at) math.usask.ca | 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 hyperkähler 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.

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

- 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