Garding studied at Lund University, where he was a PhD with Marcel Riesz, whose collected works he published in 1988. After that, he was also in Lund Professor of Mathematics. He was several times at the Institute for Advanced Study in Princeton (eg 1959/60, 1958, 1958, 1960, 1966 /7), where the Swedes Arne Beurling and Lars Hörmander ( who studied with him) were as professors and where he also worked with Michael Atiyah and Raoul Bott.
From his study of the Dirichlet problem comes his " Garding 's inequality " for solutions of linear elliptic partial differential equations. Garding also dealt with linear hyperbolic partial differential equations ( eg linear hyperbolic differential equations with constant coefficients Acta Mathematica Bd.85, 1950, p.1 -62, with Bott, Atiyah. Lacunas for hyperbolic differential operators with constant coefficients. Acta Mathematica Bd.124, 1970, p.109 -189 ) and partial differential equations of mixed type. With Arthur Strong Wightman, he worked on axiomatic quantum field theory ( " Garding - Wightman axioms ").
Garding also interested in the history of mathematics and wrote a book on the history of mathematics in Sweden.
In 1958 he gave a plenary lecture at the International Congress of Mathematicians in Edinburgh ( Some Trends and Problems in linear partial differential equations ) and in 1970 he was invited speaker on the ICM in Nice ( Sharp fronts and lacunas ).
- Some Points of Analysis and Their History. AMS, Providence 1997.
- Encounter with Mathematics. Springer -Verlag 1977, 2nd edition 1983, ISBN 3-540-90229-5.
- Cauchy 's Problem for hyperbolic equations. Lecture Notes Chicago in 1957.
- . Mathematics and Mathematicians - Mathematics in Sweden before 1950 AMS 1998 (. Swedish: Matematik och Matematiker Matematik i Sverige före 1950 Lund University Press, 1994. ).
- Singularities in Linear Wave Propagation. Lecture Notes in Mathematics, Springer-Verlag 1987.
- With Torbjörn Tambour: Algebra for Computer Science. Springer -Verlag, 1988, ISBN 3-540-96780- X.
- The Dirichlet problem. Mathematical Intelligencer Vol.2, 1979, No.1.