# Summer term 2018

# Summer term 2018

## 17. May 2018

*14:30 (A&G), seminar room 3517***Gil Solanes **(Universitat Autònoma de Barcelona)

**Integral geometry of isotropic spaces**

**Abstract: **The kinematic formula of Blaschke and Santaló measures the set of euclidean motions that bring a convex body to intersect another one. Similar formulas exist for other kinds of objects (e.g. smooth submanifolds), and more general ambients, including rank one symmetric spaces. The explicit determination of the kinematic formulas in these spaces is an ongoing program, which has seen important progress based on fundamental results by S. Alesker in the theory of valuations. After reviewing the classical case of real space forms, we will see how the kinematic formulas have been obtained in complex space forms, and other isotropic spaces.

(Contact person: T. Wannerer)

## 24. May 2018

*14:30 (A&G), seminar room 3517***Nguyen-Bac Dang** (École Polytechnique)

**A positive cone in the space of continuous translation invariant valuations**

**Abstract:** I will discuss a joint work with Jian Xiao. In this talk, I will exploit some ideas coming from complex geometry to define a cone in the space of continuous translation invariant valuations. This cone allows us to define a subspace of valuations V' and a norm on it. I will then explain how the convolution of valuations on smooth valuations in V' extends continuously with respect to the topology induced by this norm so that V' has a structure of graded Banach algebra. Finally, I will give two applications of our construction.

(Contact person: T. Wannerer)

## 31. May 2018

*16:00 (DS&MP), seminar room 3517***Felipe Ramos-Garcia **(Universidad Autónoma de San Luis Potosí)

**Title: Weak forms of equicontinuity **

**Abstract: **We will show how to use different forms of equicontinuity to classify dynamical systems with discrete spectrum (or Kronecker).

(Contact person: T. Jäger)

## 7. June 2018

*16:00 (DS&MP), seminar room 3517***Michael Hartl** (Imperial College London)

**Asymptotically autonomous random dynamical systems**

**Abstract:** Random dynamical systems (RDS), i.e. dynamical systems on metric spaces driven by some noise, have been broadly studied in the past. They emerge naturally in many situations, such as solutions of SDE's or a model iterated function systems. Due to the influence of the noise, those systems are per se non-autonomous, but using a dynamical model for the noise, one can write them as autonomous skew product flows defined on the product of a probability space and the metric phase space.

We study random dynamical systems, where even the deterministic part is non-autonomous already. By this we mean noise driven systems whose skew product flow has an explicit dependence on time. Especially we are interested in systems that are asymptotically autonomous, which means that the skew product flow converges as time tends to infinity.

In this talk I will present a general framework for autonomous and non-autonomous RDS. Then I will focus on stochastic approximations, a particular class of asymptotically autonomous RDS, including examples and a bifuraction result. At the end I will address some open questions concerning generalizations of results for autonomous RDS to the non-autonomous setting.

(Contact person: T. Jäger)

## 14. June 2018

*16:30 Mathematical Colloquium Jena, CZ3, SR 308*

**Michael Dellnitz**(Universität Paderborn)

**Glimpse of the Infinite – on the Approximation of the Dynamical Behavior for Delay and Partial Differential Equations**

**Glimpse of the Infinite – on the Approximation of the Dynamical Behavior for Delay and Partial Differential Equations**

**Abstract**: Over the last years so-called *set oriented *numerical methods have been developed for the analysis of the long-term behavior of finite-dimensional dynamical systems. The underlying idea is to approximate the corresponding objects of interest – for instance *global attractors *or related *invariant measures *– by box coverings which are created via multilevel subdivision techniques. That is, these techniques rely on partitions of the (finite-dimensional) state space, and it is not obvious how to extend them to the situation where the underlying dynamical system is infinite-dimensional.

In this talk we will present a novel numerical framework for the computation of finitedimensional dynamical objects for infinite-dimensional dynamical systems. Within this framework we will extend the classical set oriented numerical schemes mentioned above to the infinite-dimensional context. The underlying idea is to utilize appropriate *embedding techniques *for the reconstruction of global attractors in a certain finitedimensional space. We will also illustrate our approach by the computation of global attractors both for delay and for partial differential equations; e. g. the Mackey-Glass equation or the Kuramoto-Sivashinsky equation.

(Contact person: T. Jäger)

## 21. June 2018

*16:00 (A&G), seminar room 3517***Florian Besau** (Goethe-Universität Frankfurt)

**Weighted Floating Bodies**

**Abstract:** Imagine for a moment a solid body floating in water. From a physics point of view, Archimedes’ principle states that the buoyant force acting on the body in an upward direction is equal to the weight of the water displaced by it. In other words, the body is floating if the weight of the body is equal to the weight of the water displaced.

If the body rolls around, then there is a part inside of it that will always be below the water surface. This kernel is the floating body. To be more precise, for a convex body, i.e., a compact convex subset of R^{d} with non-empty interior, we may define the floating body as the subset that is obtained by cutting away all caps of volume equal to a given positive constant δ. This classic construction can be traced back to C. Dupin in the 19th century.

Remarkably, the floating body behaves covariant with respect to affine transformations. Of particular interest has therefore been the volume difference between the body and its floating body as δ goes to zero. This gives rise to an (equi-)affine invariant, namely Blaschke’s affine surface area. This affine surface area was introduced by Blaschke in the 1920s for smooth convex bodies in dimension two and three and an extension to all convex bodies in all dimensions was established by C. Schütt and E. M. Werner in 1990 using the floating body. Independently E. Lutwak and also K. Leichtweiß gave extensions of the affine surface area to all convex bodies around the same time and all three extensions later turn out to be equivalent. C. Schütt also established that for a convex polytope the derivative of the volume difference between the polytope and its floating body gives rise to the total number of full flags of the polytope—an important combinatoric invariant.

In the construction of the floating body one measures the volume of caps, or in other words, one assumes a uniform density. A natural generalization is to introduce a positive continuous weight function which leads to the notion of weighted floating bodies. We started investigating the limit behavior of weighted floating bodies of convex bodies and polytopes. These weighted floating bodies can also be related to intrinsic floating body constructions in spaces of constant curvature, e.g. the Euclidean unit sphere or hyperbolic space.

In this talk I will present a short overview of our results for weighted floating bodies and our applications in spherical and hyperbolic spaces. (Based on joint works in part with Monika Ludwig, Carsten Schütt and Elisabeth M. Werner.)

(Contact person: T. Wannerer)

## 28. June 2018

*16:30 (MP&DS), seminar room 3517***Daniel Karrasch** (TU München)

**Lagrangian Coherent Structues**

**Abstract:** In this talk, I give an introduction to Lagrangian Coherent Structures (LCSs). LCSs have been introduced as distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a finite time interval. Of particular interest are Lagrangian coherent vortices in turbulent fluid flows, which can be viewed as finite-time analogues to regular islands in a surrounding chaotic sea.

Usually, LCSs are considered in a purely advective (i.e., deterministic dynamics) framework. I will indicate issues with this approach and instead provide an advection–diffusion-based framework. This opens new avenues to analytical, computational and visualization approaches to study, for instance, turbulent fluid flows.