#### Milestone Year

### 2013

#### Counting Representations

A representation of a group is a way of thinking of the group
in terms of linear algebra. Many interesting groups have only
finitely many different representations of given dimension.
The dependence of this upper bound as a function of the dimension
is a central topic of mathematical research in the 21st century.

In 2013 scientists from the Weizmann Institute together with
collaborators from US found an upper bound on the number of
representations for a large class of groups.
It turns out that the number of representations is controlled by
certain algebraic varieties (i.e. Geometric objects that are described
by polynomials) related to the group. Of particular importance are the
singularities of those varieties (i.e. the points where the varieties
are not smooth). In order to prove the upper bound, these scientists
showed that those singularities behaves almost as smooth points from
a certain analytic perspective. The proof is based on a surprising
relation between analytic and algebraic properties of the singularities.