Kronheimer's early work was on
gravitational instantons, in particular the classification of
hyperkähler 4-manifolds with asymptotical locally
Euclidean geometry (ALE spaces), leading to the papers "The construction of ALE spaces as hyper-Kähler quotients" and "A Torelli-type theorem for gravitational instantons." He and
Hiraku Nakajima gave a construction of
instantons on ALE spaces generalizing the
Atiyah–
Hitchin–
Drinfeld–
Manin construction. This constructions identified these
moduli spaces as moduli spaces for certain
quivers (see "Yang-Mills instantons on ALE gravitational instantons.") He was the initial recipient of the
Oberwolfach prize in 1998 on the basis of some of this work. Kronheimer has frequently collaborated with
Tomasz Mrowka from the
Massachusetts Institute of Technology. Their collaboration began at the
Mathematical Research Institute of Oberwolfach, and their first work developed analogues of
Simon Donaldson's invariants for 4-manifolds with a distinguished surface. They used the tools developed to prove a conjecture of
John Milnor, that the four-ball genus of a (p,q)-torus knot is (p-1)(q-1)/2. They then went on to develop these tools further and established a structure theorem for
Donaldson's polynomial invariants using
Kronheimer–Mrowka basic classes. After the arrival of
Seiberg–Witten theory their work on embedded surfaces culminated in a proof of the
Thom conjecture—which had been outstanding for several decades. Another of Kronheimer and Mrowka's results was a proof of the
Property P conjecture for knots. They developed an instanton Floer invariant for knots which was used in their proof that Khovanov homology detects the unknot. Besides his research articles, his writings include a book, with
Simon Donaldson, on 4-manifolds, and a book with Mrowka on
Seiberg–Witten–Floer homology, entitled "Monopoles and Three-Manifolds". This book won the 2011
Doob Prize of the AMS. In 1990 he was an
invited speaker at the
International Congress of Mathematicians (ICM) in
Kyoto. In 2018 he gave a plenary lecture at the ICM in
Rio de Janeiro, together with Tomasz Mrowka. In 2023 he was awarded the
Leroy P. Steele Prize for Seminal Contribution to Research. Kronheimer's PhD students have included Ian Dowker, Jacob Rasmussen,
Ciprian Manolescu, Olga Plamenevskaya and Aliakbar Daemi. ==References==