Harmonic analysis and discrete differential geometry
Discrete differential geometry is a fairly new field that has sparked intense research over the past three decades. Its appeal stems largely from its wide ranging applications in various non-mathematical fields such as architecture and computer graphics. The subject aims to find analogues of notions from smooth differential geometry for discrete objects like polygons and polyhedra. There are several different approaches towards this end, depending on the notions one wishes to discretise.
We are also interested in some problems of harmonic analysis on infinite graphs, which can be thought as the discrete counterpart of corresponding problems on manifolds. In particular, we study singular integrals associated with Laplacians and some geometric inequalities on graphs.