Abstract: Abstract: Recent advances in laser technology have made it possible to sample the boundary of a real world object to produce a large amount points. Many applications require to reconstruct the object from this point sample. We will describe algorithms for this reconstruction which have topological and geometric guarantees. In the process I shall introduce the various complexes that can be built from point cloud data which help in deciphering geometry and topology of the hidden sampled object. In a follow up lecture I shall describe an algorithm to decipher the topology of a hidden manifold from point cloud data in high dimensions.