Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study, Princeton from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. He died in Princeton. (He used to answer the question of whether he was related to Emile Borel alternately by saying he was a nephew, and no relation.)

He studied at the ETH Zürich. He came under the influence of the topologist Heinz Hopf, and the Lie group theorist Stiefel . He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Leray and Henri Cartan.

He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group G a Borel subgroup H is one such that the homogeneous space G/H is a projective variety, and as small as possible. For example if G is GL_n then we can take H to be the subgroup of upper triangular matrices. In this case it turns out that H is a maximal solvable subgroup, and that the parabolic subgroups P between H and G have a combinatorial structure (in this case the G/P are the various flag manifolds). Both those aspects generalize, and play a central role in the theory.

The Borel-Moore homology theory applies to general locally compact spaces, and is closely related to sheaf theory.

He published a number of books, including work on the history of Lie groups.