Abstract: We will talk about our contribution to a large project with the goal of a self-consistent numerical simulation of the evolution of the universe beginning soon after the Big Bang and ending with the formation of realistic stellar systems like the Milky Way. This is a multi-scale problem of vast proportions. It requires the development of new numerical methods that excel in accuracy, parallel scalability to the processes relevant in galaxy formation. These numerical methods themselves require the development of mathematical theory in order to guarantee the above mentioned requirements. in this talk we shall focus on our contribution to this effort. This is joint work among others with Volker Springel.

Abstract: The aim of the Minimal Model Program is to generalize the classification of complex projective surfaces, known in the early 20th century, to higher dimensional varieties. Besides providing a historical introduction, we will discuss some recent results and new aspects of this Program.

Abstract: The existence of transcendental numbers has been known only since 1844 by the work of Liouville. In our talk we shall give an overview on the development of transcendence theory since then. There are some milestone conjectures which belong to the most difficult problems in mathematics which we also shall discuss. The most recent is a conjecture of Kontsevich which comes out of Hodge Theory and the theory of mixed motives. It relates transcendence theory with very modern aspects of mathematics.

Abstract: Mathematical models and statistical arguments play a central role in the assessment of the changes that are observed in Earth's climate system. While much of the discussion of climate change is focused on large-scale computational models, the theory of dynamical systems provides the language to distinguish natural variability from change. In this talk I will discuss some problems of current interest in climate science and indicate how, as mathematicians, we can find inspiration for new applications.

Abstract: In this talk I shall discuss the relationship between the Painlevé equations, orthogonal polynomials and random matrices.

The six Painlevé equations were first discovered around the beginning of the twentieth century by Painlevé, Gambier and their colleagues in an investigation of nonlinear second-order ordinary differential equations. During the past 40 years there has been considerable interest in the Painlevé equations primarily due to the fact that they integrable equations and arise as reductions of soliton equations solvable by inverse scattering. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.

It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. In this talk I will show that for some weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are Wronskians, that also arise in the description of special function solutions of a Painlevé equation. These determinants also arise as partition functions in random matrix models and more generally, I will discuss the role of Painlevé equations in random matrix theory.