Colloquium abstracts

Nivedita Bhaskhar
UCLA, USA
December 20, 2018

Reduced Whitehead groups of algebras:  Any central simple algebra A over a field K is a form of a matrix algebra. Further A/K comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*]. One can define Whitehead groups more generally for any isotropic group and it turns out that the Tannaka-Artin question is a special case of the well-known Kneser-Tits conjecture. The Whitehead group detects the non-rationality of the underlying variety of the algebraic group and therefore is an interesting albeit difficult invariant to study. In this talk, we discuss these connections to rationality questions and trace the progress towards answering the Tannaka-Artin question, which become