While classical lattices are central to geometry, approximate lattices offer a generalization that relaxes the subgroup requirement while preserving discreteness and "finite covolume" properties. Introduced by Yves Meyer in the 1970s for the abelian case (linked to Pisot numbers and quasi-crystals), the theory has recently been successfully extended to all locally compact groups.
This talk will provide an overview of this non-abelian structure theory, which relies on a synthesis of additive combinatorics, model theory, dynamics and bounded cohomology. After establishing the fundamentals, I will discuss open problems and highlight a striking connection to uniform Ulam stability. We will explore how the geometry and dynamics of these sets may offer a new perspective on stability questions.
