Every locally compact group divides into a connected and a totally discon- nected factor. It has been known since the solution of Hilbert’s fifth problem in the 1950s that connected locally compact groups can be approximated by Lie groups but there is no corresponding theorem accounting for the structure of totally disconnected, locally compact (t.d.l.c.) groups.
Our understanding of t.d.l.c. groups has advanced significantly in recent years, however, and this talk will survey that progress. Current investi- gations of the structure of t.d.l.c. groups will be reviewed, and the aims and ideas compared with methods used to understand connected and Lie groups.
George Willis is known internationally as an innovator and problem solver in diverse fields of mathematics. In topological algebra, his insights into locally compact groups and fundamental concepts that he has introduced, such as the scale function and flatness, have initiated new research directions and made new applications possible. He has made advances in harmonic analysis by combining algebraic and probabilistic methods in a novel way, and has made decisive contributions in functional analysis through the construction of key examples.