Geometric measure theory

Geometric measure theory is a part of analysis, and is not just an independent theory but has close connections to other areas of analysis and mathematics including PDEs, harmonic analysis, ergodic theory, number theory, combinatorics and much more. The importance of geometric measure theory and its subfield, fractal geometry is not restricted to pure mathematics, as they have applications in physics, chemistry and biology.The proposed research focuses on geometric measure theory and on its interplay with various other fields of mathematics. A famous example, where such important interplay occurs, is the Kakeya conjecture originating from the needle problem: what is the minimum area of a region in the plane inside which a needle (a unit line segment) can be rotated through 180 degrees? One of the aims of this research programme is to study certain phenomena related to the Kakeya conjecture and to solve a particular problem about a generalization of this needle problem.Another example, where such an interplay occurs, is the extremely challenging problem of Paul Erdos related to the local properties of the Lebesgue measure on the real line. He raised the following question about 40 years ago: Does there exist an infinite set S such that every set of positive measure contains an affine copy of S? The solution of this problem would have a large impact not only on measure theory, but also on combinatorics and on many other fields of mathematics.