At Harvard Mumford became a student of Oscar Zariski, and his work in geometry always combined the traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book *Geometric Invariant Theory*, on the equations defining an abelian variety, and on algebraic surfaces. His books *Abelian Varieties* (with C.P. Ramanujam) and *Curves on an Algebraic Surface* combined the old and new theories (to the disadvantage of the former, it has been claimed by Sheeram Abhyankar). His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were the only accessible introduction.

Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and a study of Shimura's many papers from the 1960s.

Mumford’s research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, and enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group. He published some further books of lectures on the theory.

He also was one of the founders of the toroidal embedding theory; and sought to apply the theory to Grobner basis techniques, through students who worked in algebraic computation

He was awarded a Fields Medal in 1974. During the 1980s he left algebraic geometry, in order to study brain structure. He was a MacArthur Fellow from 1987 to 1992.

He has recently published a joint book on the visual geometry of limit sets.