Nicolau C. Saldanha has experience in Topology, Combinatorics and Analysis. One recent research topic is the study of spaces of curves. He determined the homotopy type of the space of closed locally convex curves in the sphere S2, and also of the related space of curves with prescribed initial and final jets (DOI: 10.2140/gt.2015.19.1155). This work uses ideas related to the h principle together with algebraic and geometric tools (including Bruhat cells). Four PhD students and two postdocs worked in related problems, including the study of the analogous problems in higher dimension and the study of spaces of curves with curvature in a prescribed interval in the sphere S2, the euclidean plane and other surfaces. Such problems were initially motivated by the study of certain sets in spaces of functions defined by differential equations or relations: more precisely, we try to extend Sturm theory to higher orders. Another area of recent work is the study of 3D domino tilings. Using topological methods, together with a graduate student we defined the twist of a 3D tiling: the twist is an integer and is invariant under flips (the simplest local moves). Recent work indicates that for connected and simply connected regions, the twist is essentially the only invariant; if the region is not simply connected, the flux is another invariant, in a sense more fundamental. On the one hand, two tilings with the same twist can be joined by a sequence of flips provided some extra space is allowed.

On the other hand, for some regions it can be shown that for almost any two tilings with the same twist no extra space is required.