Tag - Geometric measure theory

Paul Minter: An Overview of Geometric Measure Theory, Area Minimising Currents, and Recent Progress

8th April, 2024

Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide a way to study the interplay between geometry and topology, as has historically been the case for variational methods in geometry.

Almgren developed a theory which established that these area minimising structures are manifolds away from a codimension 2 ‘singular set’. The singular set itself, however, remained rather mysterious, including whether it necessarily has locally finite measure, unique tangent cones, or geometric structure (rectifiability).

In this talk I will attempt to give an overview of these ideas, as well as of recent work (joint with Camillo De Lellis and Anna Skorobogatova) answering some of the questions above related to singularities of area minimizers.

Izabella Łaba: Cyclotomic divisibility – from tiling to harmonic analysis and geometric measure theory

30th March, 2022

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ZM of a finite cyclic group. Coven and Meyerowitz (1998) proposed a characterization of all finite tiles in terms of the cyclotomic divisors of associated mask polynomials, and proved it when the tiling period M has at most two distinct prime factors. In joint work with Itay Londner, we extended it to the case when M=(pqr)2, where p,q,r are distinct primes. The methods we developed can be applied to other questions that hinge on cyclotomic divisibility, ranging from number theory to harmonic analysis and geometric measure theory. In particular, Caleb Marshall and I were able to use cyclotomic divisibility methods to prove new Favard length estimates for product Cantor sets. The talk will provide an introduction to this group of problems.