Karsten Grove is a Danish-American mathematician working in metric and

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

and global analysis, mainly in topics related to global Riemannian geometry, Alexandrov geometry,

isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...

group actions and manifolds with positive or nonnegative sectional curvature.

Biography

Grove studied mathematics at

Aarhus University Aarhus University ( da, Aarhus Universitet, abbreviated AU) is a public research university with its main campus located in Aarhus, Denmark. It is the second largest and second oldest university in Denmark. The university is part of the Coimbra Gr ...

, where he obtained a Cand. Scient. (equivalent to a M.A.) in 1971 and Lic. Scient. (equivalent to a Ph.D.) in 1974. Between 1971 and 1972 he also acted as an instructor at Aarhus University. From 1972 to 1974 he had a postdoctoral position at the University of Bonn under the supervision of Wilhelm Klingenberg, despite not having yet formally concluded his doctoral degree. In 1974, Grove became an Assistant Professor at the University of Copenhagen and was promoted to Associate Professor in 1976, a position he held until 1987. He became a Professor at the University of Maryland in 1984, retiring from this position in 2009. Since 2007 he has held the endowed chair of "Rev. Howard J. Kenna, C.S.C. Professor" at the University of Notre Dame. Throughout his career, Grove has had 20 doctoral students, and 51 academic descendants. Grove was an invited speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

in 1990 in Kyoto (''Metric and Topological Measurements on manifolds''). He is a fellow of the American Mathematical Society.

Mathematical work

One of Grove's most recognized mathematical contributions to Riemannian Geometry is the Diameter Sphere Theorem, proved jointly with Katsuhiro Shiohama in 1977, which states that a smooth closed Riemannian manifold

(M,g)

with

\sec \ge 1

and

\operatorname > \pi/2

is homeomorphic to a sphere. Subsequently, the critical point theory for distance functions developed as part of the proof of this result led to several important advances in the area. Another result obtained by Grove, in collaboration with Peter Petersen, is the finiteness of homotopy types of manifolds of a fixed dimension with lower sectional curvature bounds, upper diameter bound, and lower volume bound.

References

{{DEFAULTSORT:Grove, Karsten Year of birth missing (living people) Living people Differential geometers Danish mathematicians Aarhus University alumni Fellows of the American Mathematical Society Academic staff of the University of Copenhagen University of Notre Dame faculty