The Research of Thomas P. Branson

Abstract

Abstract. The Midwest Geometry Conference 2007 was devoted to the substantial mathematical legacy of Thomas P. Branson who passed away unexpectedly the previous year. This contribution to the Proceedings briefly introduces this legacy. We also take the opportunity of recording his bibliography. Thomas Branson was on the Editorial Board of SIGMA and we are pleased that SIGMA is able to publish the Proceedings.Key words: conformal differential geometry; Q-curvature; spectrum geometry; intertwining operators; heat kernel asymptotics; global geometric invariants 2000 Mathematics Subject Classification: 01A70; 22E46; 22E70; 53A30; 53A55; 53C21; 58J52; 58J60; 58J70In memory of Thomas P. Branson (1953 Tom's research interests ranged broadly; this is perhaps best indicated by the diverse group of collaborators listed implicitly in his bibliography, which follows this section. Tom was also known for his general knowledge in many areas of mathematics, and many of his colleagues will remember well his prompt and in-depth replies to email questions.In the 1980s Tom's research on conformal invariance was significantly ahead of its time. An enduring theme of his research was the natural interplay between invariance and the underlying symmetry groups. Tom's work continues to motivate and inspire a thriving and impressive international research effort. Many of the conference speakers have contributed to the various research trends that Tom Branson started.It is impossible in a written summary to do justice to Tom's research career. Here we shall outline just a few directions that we feel to be especially significant. Q-curvature and extremal problemsTom Branson is perhaps most well known for his definition, development, and application of a new curvature quantity in Riemannian geometry. For dimension 4 this first entered the public arena in his joint work [22] with Ørsted, but it was extended to all even dimensions and developed significantly in [29] and [36]; these days it is usually termed "Branson's Q-curvature". A key feature is its conformal transformation law in dimension n, e −nω Q = Q + P ω,