Mathematical Forum

Nonparametric statistics aims to design statistical methods without specifying assumptions on the data-generating process. For one-dimensional data, such methods are based on the ranks, quantiles, and data ordering. For multivariate or more complex data, however, ranks or quantiles cannot be defined directly, as the data cannot be naturally ordered - how do you define a median for bivariate observations?

Depth functions introduce ordering in multivariate spaces in a data-dependent way. The points are ranked according to their "centrality" inside the dataset. We will discuss seminal data depths, and show that they are equivalent to well-established methods developed in geometry, some 150 years earlier. These unknown pairings allow us to resolve open questions in depth research, but also shed new light on old geometric problems.

Abstract: A complex number is called algebraic if it is the root of some polynomial with integer coefficients. Starting with efforts to solve Diophantine equations, extensions of the rational numbers by several algebraic numbers (ie, smallest fields that contain all of them) have been widely studied throughout algebra and number theory. I will focus on the more mysterious case of extensions by infinitely many numbers. Specifically, I will discuss

• how to measure the size of an algebraic number,
• how many "small" algebraic numbers does a given extension have, and
• the connections to logic and number theory (including a recent joint result with Nicolas Daans and Siu Hang Man).

Abstract: In this talks we study mappings that could serve as defomations in models on Nonlinear Elasticity. We have a body Omega in Rⁿ and we have a mapping f from Omega to Rⁿ that describes the deformation of this body.

Natural questions considered are when the mapping is continuous (the material does not break during the deformation), when is it invertible (no „interpenetration of matter“ happens) or when does it map sets of zero volume to sets of zero volume (no material is created or lost during our deformation). In particular we will study when the deformation does not change orientation (you cannot map your right hand to your left hand) or when can we approximate injective mappings by piecewise linear (or smooth) injective mappings.