Moduli spaces of framed instanton bundles on $\mathbb{CP}^3$

I will present the ADHM construction of framed instanton sheaves on $\mathbb{CP}^3$, which yields a parametrization of the (fine) moduli space of framed instanton bundles in terms of matrices satisfying some quadratic equations. Using twistor theory, one can show that such moduli space coincides with the space of twistor section of the moduli space of framed bundles on $\mathbb{CP}^2$. We also introduce a the notion of trisymplectic structures on a complex manifolds, and the notion of trisymplectic reduction. We show that the moduli space of framed instanton bundles can also be described in terms of a trisymplectic reduction, and conclude that it is a smooth trisymplectic manifold of the expected dimension. Joint work with Misha Verbitsky and with Amar Henni and Renato Vidal Martins.