Numerical simulations produce data that contains hidden information. This hidden information can be exploited to create even more accurate representations of the data. This presentation will address how these more accurate representations of data can be constructed using information from higher order methods for numerical partial differential equations. Although the discussion will be framed in the context of discontinuous Galerkin finite element methods, the techniques demonstrated naturally apply to other methods as well. This presentation will focus on identifying where this hidden accuracy comes from, why the hidden accuracy is important, and what are the different manifestations of the power of this hidden information.

We present the first Q-superquadratically convergent version of Newton's method for solving operator equations in Banach spaces that requires only one operator value and one inverse of the Fréchet derivative per iteration. The R-order of convergence is at least 2.4142. A semi-local analysis provides Sufficient conditions for existence of a solution and convergence. The local analysis assumes that a solution exists and shows that the method converges from any starting point belonging to an explicitly defined neighbourhood of the solution called the ball of attraction.

In this talk we review some results on random Schrödinger operators, a standard framework for the study of disordered quantum systems and the absence of wave propagation in random media, a phenomenon known as Anderson localization. Our goal is to apply these results to the study of spectral and dynamical properties of Delone operators, which serve to model aperiodic structures (quasi-crystals). As a consequence, we will show that aperiodic media, although usually associated to singular continuous spectrum and anomalous transport, does exhibit often Anderson localization and, in particular, pure point spectrum.

An element x of a group G is called a test element if for any endomorphism ϕ of G, ϕ(x) = x implies that ϕ is an automorphism. The first non-trivial example of a test element was given by Nielsen in 1918, when he proved that every endomorphism of a free group of rank 2 that fixes the commutator [x, y] of a pair of generators must be an automorphism. In this talk I will discuss the distribution of test elements in free groups and surface groups.

About 30 years ago Nigel Hitchin introduced a new class of so called completely integrable systems. These geometric objects had their origin in a set of differential equations but surprisingly turn out to have various alternative descriptions that allow for arithmetic and topological applications. Thus a wide range of methods have been applied to get a better understanding of the underlying spaces.

In this talk I will try to explain a little bit about the history of these spaces and of some of the methods used to study them. After giving some more recent insights into their geometry I would like to explain what the P=W conjecture of de Cataldo, Hausel and Migliorini is about, which gives an unexpected link between two distinct algebraic structures of the manifolds appearing in Hitchin's construction.

Typical properties of groups formulated in terms of actions on metric spaces are not axiomatizable. The most important examples of such prop- erties are amenability, property (T) of Kazhdan or property FR that any continuous isometric action of G on a real tree has a fixed point.

One of substitutes of axiomatizability is the following notion of Ph.Hall. A class of topological groups K is called bountiful if for any infinite group G from K and any subset C of G there is H in K which is a subgroup of G, contains C and the density character of C coincides with the density character of H.

We introduce a stronger property and verify it in many situations. In order to realize this we show how isometric actions of metric groups on Hilbert spaces and real trees can be presented in continuous logic.

Locally compact (l.c.) groups are fundamental to many areas of mathematics and their group structure is important in many of the contexts in which they appear. Each l.c. group divides into a connected factor and a totally disconnected factor. Understanding connected l.c. groups reduces, via the solution of Hilberts 5th Problem achieved in the 1950s, to the study of real Lie groups, but a comparable understanding of t.d.l.c. groups is only now being developed.

Part of this new understanding, analogous the role of eigenvalues and eigenvectors in Lie theory, is the concept of the scale on the t.d.l.c. group and associated actions of sub-quotients of the group on regular trees.

In biological cells a tight regulation of processes in the cell and on the cell membrane is for many functions of the cell essential.

We introduce different mathematical models in the form of coupled bulk-surface reaction-drift-diffusion systems and investigate in particular pattern forming properties.

Certain polynomials f(x_1,..,x_n) in n variables and with integer coefficients have the remarkable property that, for every number ring A, the equation f(a_1,...,a_n)=0 has only finitely many solutions with a_1,...,a_n elements of A. The Lang-Vojta conjecture in fact predicts that polynomials with this property are precisely those whose zero set defines a "hyperbolic" variety. In particular, Lang-Vojta's conjecture relates certain "arithmetic" properties of a polynomial with certain "complex analytic" properties. In this talk I will explain this deep conjecture through many simple examples, and present several results predicted by this conjecture.