Scaling limits of random structures, in two dimensions in particular, often posses some conformal invariance properties, giving ways to methods of geometric analysis. Among the fascinating questions here are the configurations of random tilings under scaling limits, and the boundaries between their ordered and disordered, or liquid, limit regions. The liquid domains carry a natural complex structure, which turns out can be described by a quasilinear Beltrami equation with very specific properties. In this talk, based on joint work with E. Duse, I. Prause and X. Zhong, I show how this approach leads to understanding and classifying the geometry of the frozen boundaries for different random tilings and other dimer models.

The aim of the talk is to characterize quasiconformal mappings in terms of the distortion of the vertices of equilateral triangles. This is joint work with Colleen Ackermann and Aimo Hinkkanen.

We consider the maximal size of sets in which homeomorphisms with integrable distortion can attain some predefined stretching and rotation. Establishing the sharp bounds for the size of these sets amounts to finding the multifractal spectra of these mappings. This gives a lot of information about the geomeric properties of these mappings, for example, we will see that the maximal pointwise stretching and rotation can only be obtained in small sets. Furthermore, we use the multifractal spectra to study compression of the Hausdorff measure under these mappings. Using this approach we improved the previously known bounds for the compression, proven by Clop and Herron in 2014, to the bounds that can be seen to be sharp from the examples they constructed.

We study the boundary rigidity problem for compact manifolds with boundary when the boundary is not strictly convex. We show boundary and lens rigidity in dim 2 if the manifolds has no conjugate points and is non-trapping, and some results about the linearised problem in higher dimensions. Joint work with Mazzucchelli and Tzou.

The anisotropic Calderón problem is the inverse problem consisting in determining a metric on a compact Riemannian manifold with boundary from the Dirichlet-to-Neuman map. The resolution of the problem in a conformal class follows from a similar inverse problem on the Schrödinger equation and remains an open question in dimensions higher than 3. In previous works, we could solve this inverse problem under structural assumptions on the known metric (namely that it takes the form of a warped product with an Euclidean factor) and additional geometric assumptions on the transversal manifold. The proof of uniqueness relies on the high frequency limit in a Green identity involving pairs of complex geometrical optics solutions to the Schrödinger equation. This talk will be concerned with our attempts to remove the additional transversal assumptions on the geometry by refraining from passing to the limit. Unfortunately, this path only leads to partial results on the linearized problem for the time being. This a joint work with Yaroslav Kurylev, Matti Lassas, Tony Liimatainen and Mikko Salo.

In this talk we will discuss some near optimal results about the singular set for mean curvature flow.