The history of constrained optimization spans nearly three centuries. It goes back to a letter Johann Bernoulli sent in 1715 to Varignon, announcing a very simple rule with which the many hundreds of different problems in fluid and solid mechanics considered in detail by Varignon can be solved in the blink of an eye. Varignon then explains this rule at the end of his book, but unfortunately cites the letter of Johann Bernoulli with an incorrect date. Bernoulli's rule, based on virtual velocities, was later carefully explained by Lagrange, and led to the discovery of the famous multiplier method of Lagrange, with which many optimization problems can be easily treated. Using so called Lagrange multipliers is however a much more far reaching concept, and we will see that one can, armed only with Lagrange multipliers, discover the important primal and dual equations in optimal control and the famous maximum principle of Pontryagin. Pontryagin himself however did not discover his maximum principle using Lagrange multipliers, he used a more geometric argument. We will finally give the complete formulation of PDE constrained optimization based on adjoints introduced by Lions.

Die Approximation der Verteilung einer Summe von unabhängigen zufälligen Vektoren durch die Gauss-Verteilung ist fundamental für viele Anwendungen der Wahrscheinlichkeitstheorie. Insbesondere ist dabei die Verteilung der euklidischen Länge von solchen Summen von Interesse. Im Vortrag wird dargestellt wie die Frage der Approximationsgüte mit klassischen Gitterpunktproblemen von Landau und Hardy aus der Geometrie der Zahlen für grosse Kugeln und Ellipsoide zusammenhängt. Insbesondere werden jüngste Resultate zu Gitterpunktproblemen und der Gleichverteilung auf Bahnen von eindimensionalen unipotenten Untergrupppen eingesetzt, um diese verwandten Probleme für alle Dimensionen ab 5 zu lösen.

Let G be a finitely generated group. A rich interplay between algebra and geometry arises by viewing G as a metric space, or as a metric measured space. I will describe two invariants of finitely generated groups, namely growth and Poisson boundary, and explain by new examples that their relationship is deep, but still mysterious.
Its growth function Γ(n) counts the number of group elements that can be written as a product of at most n generators. This function depends on the choice of generators, but only mildly.
I will show that, for almost any function Γ that grows sufficiently fast, there exists a group with growth asymptotic to Γ. These give also the first examples of groups for which the growth function is known, and is neither polynomial nor exponential.
The Poisson boundary of a random walk (given by a one-step measure) describes the tail events of the walk. If the random walk takes place on a group G, then the Poisson boundary is very much connected to classical invariants of G: if the boundary is trivial, then the group generated by the measure's support is amenable. If the boundary is non-trivial for a finitely supported measure, then G has exponential growth.
The connection between growth and Poisson boundary, however, is still mysterious. Kaimanovich and Vershik asked in 1983 whether the converse holds; namely, whether there exist groups of exponential growth such that all measure with finite support have trivial boundary.
I will show that such examples exist. Curiously, the constructions of all examples are based on a common method, that of "permutational wreath products". I will outline a few other consequences of the construction to geometric group theory.
This is joint work with Anna Erschler.

Using Gromov's h-principle theorems for open partial differential relations, one can obtain with very little effort results in differential geometry which would be nearly impossible to prove by other methods. In particular, Riemannian and semi-Riemannian metrics whose curvatures satisfy certain strict inequalities can be shown to exist on noncompact manifolds. I will explain the technique and present many examples.