I'm interested in applications of Riemann-Hilbert problems in Complex Geometry. In particular, certain complex integrable systems arising from the study of moduli spaces of Higgs bundles have a very rich geometry. These are hyperkahler manifolds and in certain cases, it is possible to obtain the hyperkahler metric through solutions of an infinite-dimensional Riemann-Hilbert problem.
I also study the analytic aspects of the Riemann-Hilbert problems and their relation to canonical factorization, state space realizations and isomonodromic deformations from a geometrical perspective.
2017: A construction of hyperkähler metrics through Riemann-Hilbert problems I (submitted)
2017: A construction of hyperkähler metrics through Riemann-Hilbert problems II (submitted)
2016, 2015: IUPUI Athletics Favorite Professor