Random sampling in convex polytopes with applications to additive combinatorics
There are many situations in additive combinatorics in which it is useful to know that sumsets -- that is sets of the form $A+B = \{ a+b : a \in A, b \in B\}$ in abelian groups -- are somehow quite structured. I shall describe how one can prove (rigorous versions of) such statements using random sampling to approximate points in convex polytopes by low-dimensional faces.