He is known for (with Michael Weiss) proving the Mumford conjecture on the cohomology of the stable mapping class group, and for developing topological cyclic homology theory.
[1][2][3] In 1971 he took a faculty position at Aarhus University, and he remained there until 2008, when he moved to Copenhagen.
[1][2] His doctoral students have included Søren Galatius and Lars Hesselholt.
[3] Madsen was elected as a member of the Royal Danish Academy of Science and Letters in 1978,[4] as a foreign member of the Royal Swedish Academy of Sciences in 1998,[1][5] and as a foreign member of the Royal Norwegian Society of Sciences and Letters in 2000.
[1] In 2011, he won the Ostrowski Prize for outstanding achievement in pure mathematics, shared with Kannan Soundararajan and David Preiss.