The work on extension of symplectic techniques to non-toric settings (in collaboration with Prof.s Carlos Florentino, José Mourão and João Pimentel Nunes in Lisbon) concerns in particular the “wonderful compactifications” of DeConcini-Procesi. These are smooth projective varieties canonically associated to any reductive group over the complex field, which contain both the toric variety of the Weyl chamber decomposition of the Lie algebra, and a certain generalized flag variety. The toric and the flag variety are transverse subvarieties of complementary dimension with tightly related geometry: their Picard groups are isomorphic and ampleness conditions related. The role flag varieties play representation theorythrough the Borel–Weil–Bott theorem, and the lattice points in the toric moment polytope combine to give a decomposition of ample linear systems on the wonderful variety. Both the flag varieties and the wonderful compactifications are Fano varieties. It follows from known results that this is not the case for the toric part; existence and description of special metrics (Kähler–Einstein, or Kähler–Ricci-soliton, &c) on the different parts are conditioned by this fact, and extension of the explicit methods available for the description of special metrics on these subvarieties (such as Abreu’s equation for scalar curvature for the former, and invariant Kähler–Einstein metrics for the latter) have to be adapted to this circumstance. In the work on quivers (in collaboration with Luca Scala in Rio de Janeiro and Prof. Peter Gothen), we consider the relation between quiver representation in different categories. The rather well understood properties of representations on vector spaces serves as a guide for representations in the category of vector bundles or sheaves on algebraic curves, where usually furthermore we have to take stability parameters into account. As a concrete special case, we consider the moduli spaces of holomorphic triples on an algebraic curve, which also admits an interpretation in terms of equivariant vector bundles over a product with the projective line. The duality between dimension vectors (for representations in vector spaces) that yields spaces of semi-invariants of equal dimesion suggests that for this special kind of quiver sheaves, a statement analogous to Le Potier’s strange duality conjecture could be established this way. Pointwise over the moduli space of genus two curves, the anti-canonical bundle of the moduli space of genus two vector bundles with trivial determinant has been abelianized in work of Oxbury and Pauly. The two projective bundles (over the moduli of curves) identified in this way come equipped with natural connections: while Hitchin’s connection (on the non-abelian side) is known, in general and for monodromy obstructions, not to coincide with the natural connection on abelian theta functions, this is one of the exceptional cases where the isomorphism is potentially projectively flat. The work (in collaboration with Johan Martens in Edinburgh and Michele Bolognesi in Rennes) on abelianization in genus two and rank two considers this question, in the context of classical results relating the question to the geometry of moduli spaces of abelian surfaces. Progress in this project includes a clearer understanding of the role of Lagrangian level structures on linear systems on abelian varieties for the Mumford–Welters connection; in the case concerning us, these amount to the specification of a so-called Göpel tetrahedron in the projective space of the Kummer surface. This establishes a link between a classical topic of algebraic geometry and the deformation theory entering the abelian and non-abelian projective connections.