In pure math, he works in homotopy theory, having resolved the Segal Conjecture, as well as in manifold topology, with cases of Borel and Novikov conjectures, and also in algebraic K-theory.
In applied math, he is one of the founders of the field of topological data analysis and among the first (if not the first) to develop persistent homology. With Gurjeet Singh he created the Mapper algorithm based on the Reeb graph. That has been used in many, many settings, for example finding a new genetic marker for breast cancer. He co-founded the company Ayasdi, which continues to develop topological (and topology-inspired) tools along often in conjunction with machine learning, with clientele which ranges from academics to government to industry in an impressive array of areas.