A representation of a group is a way of thinking of the group in terms of linear algebra. In this context, irreducible representations plays the role analogous to that of prime numbers in number theory. Specifically, any representation can be decomposed to irreducible parts, and a multiplicity-free representation is one in which any irreducible component appears only once. This makes the decomposition much more rigid and accessible. The space of functions on a geometric object with symmetries is sometimes a multiplicity-free representation. A classical example of such situation is the space of functions on the sphere. In general, such situations give rise to Gelfand pairs, and they are very useful in harmonic analysis: They provide nice ways to present a function on a symmetric geometric object (like, for example, the sphere) as a combination of functions that behave in a predictable way under symmetries.
In the years 2007-2010, Weizmann scientists found several series of Gelfand pairs. Unlike the majority of the earlier examples of Gelfand pairs, the news pairs were not compact (i.e., are not of bounded nature). Based on these developments, many scientists all over the world significantly extended the collection of known examples of non-compact Gelfand pairs, and by doing so got us closer to a full classification of such pairs. The Gelfand pairs discovered at the Weizmann Institute already found applications beyond representation theory and harmonic analysis, for example, in number theory.