The research seminars Analysis & Geometry (A&G) and Dynamical Systems & Mathematical Physics (DS&MP) are organised by the research groups of Prof. David Hasler, Prof. Daniel Lenz, Prof. Vladimir Matveev, Prof. Tobias Oertel-Jäger, Prof. Anke Pohl and Prof. Thomas Wannerer.

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**When**: Thursdays @ 14:30-16:00 (A&G) and 16:00-17:30 (DS&MP)

**Where**: Ernst-Abbe-Platz 2, seminar room 3517 (OpenStreetMap)

# Winter term 2016/17

## 18. August 2016

*14:15, seminar room 3517***Michael Baake** (Universität Bielefeld)

**Renormalisation theory for primitive substitutions**

## 20. October 2016

*14:30 (A&G)***Jakub Konieczny** (University of Oxford, United Kingdom)

**Automatic sequences, generalised polynomials and nilmanifolds**

*16:00 (DS&MP)*

**Tanja Eisner** (Universität Leipzig)

**Weighted Ergodic Theorems**

## 27. October 2016

*14:00 (A&G)***Aapo Kauranen** (Jyväskylä/Prague)

**Sobolev spaces and Lusin's condition (N) on hyperplanes**

## 03. November 2016

*14:30 (A&G)***Alexander Teplyaev** (University of Connecticut, USA)

**Existence, uniqueness and vector analysis on fractals**

The talk will describe how the heat kernel estimates, which are mainly due to Grigor'yan and Telcs, and related functional spaces and potential theory, imply the existence and uniqueness of self-similar Dirichlet forms (and hence Laplacians and diffusion processes) on generalized Sierpinski carpets. This is a joint result with Barlow, Bass and Kumagai. The second part of my talk will review recent results on vector analysis on such spaces, such as quasilinear PDEs, Dirac and magnetic Schrödinger operators, spectral triples, Hodge theory, Navier-Stokes equations, and some unusual properties of the classical curl operator. This includes joint results with J.P. Chen, M. Hinz, D. Kelleher, M. Röckner. The motivation for this vector analysis come from physics, such as studying magnetic properties of fractal structures.

## 10. November 2016

*14:30 (A&G)***Franz Schuster** (Technische Universität Wien, Austria)

**Affine vs. Euclidean isoperimetric inequalities**

In this talk we explain how every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. This is joint work with Christoph Haberl.

## 24. November 2016

*14:30 (A&G)***Dan Rust** (Universität Bielefeld)

**Ordered cohomology and co-dimension one cut and project sets**

We'll discuss how to study aperiodic tilings in Euclidean space from a topological point of view and how tools from algebraic topology, namely Cech cohomology, can be used to distinguish them. For some classes of tilings, cohomology groups are an extremely strong invariant, but for others, one needs to enrich this invariant in order to extract finer information to distinguish within a class. Tilings coming from cut-and-project methods happen to be such a class where the cohomology on its own isn't very useful. We'll talk about how the top degree Cech cohomology of a tiling comes equipped with a natural order structure. Our main result is that ordered cohomology completely classifies codimension-one cut-and-project tilings up to homeomorphism. Moreover, isomorphism of two ordered cohomology groups is equivalent to a very concrete condition involving the existence of a unimodular integer matrix.## 01. December 2016

*14:30 (A&G)***Matthias Reitzner** (Universität Osnabrück)

**Random polytopes: limit theorems**

Let η be the set of random points of a Poisson point process in R^{d}, and let K be a convex set of volume 1. Denote by s the mean number of random points in K, and by K_{s} the convex hull of these points. We are interested in properties of K_{s} as s tends to infinity: expectations, variances, limit theorems and large deviations for functionals of K_{s}.

*16:00 (DS&MP)*

**Alexey Bolsinov** (Loughborough University, United Kingdom)

**Stability analysis, singularities and topology of integrable systems**

In the theory of integrable systems, there are two popular topics:

1) Topology of integrable systems, which studies stability of equilibria and periodic trajectories, bifurcations of Liouville tori, singularities and their invariants, topological obstructions to the integrability and so on.

2) Theory of compatible Poisson brackets, which studies one of the most interesting mechanisms for integrability based on the existence of a bi-Hamiltonian representation.

The aim of the talk is to construct a bridge between these two areas and to explain how singularities of bi-Hamiltonian systems are related to algebraic properties of compatible Poisson brackets. This bridge provides new stability analysis methods for a wide class of integrable systems.

## 08. December 2016

*14:30 (A&G)***Nina Lebedeva** (Steklov Institute of Mathematics at St.Petersburg, Russia)

**Total curvature of geodesics on convex surfaces**

We prove that the total curvature of a minimizing geodesic segment on a convex surface in the 3-dimensional Euclidean space can not be arbitrarily large (joint with Anton Petrunin).

*16:00 (DS&MP)***Marc Rauch** (Universität Jena)

**The inverse variational principle**

## 15. December 2016

*14:30 (A&G)***Jun Luo**(Universität Jena/College of Mathematics and Statistics, Chongqing University, China)

**Self-similar sets, simple augmented trees, and Lipschitz equivalence**

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graphs on the IFS has been established recently. In this talk, we introduce a notion of simple augmented tree on the IFS which is a Gromov hyperbolic graph. By using a combinatoric device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this result to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition, which is an important topic in fractal geometry and geometric measure theory.

## 05. January 2017

*14:30 (A&G)***Felix Dorrek** (Technische Universität Wien, Austria)

**Minkowski Endomorphisms**

A Minkowski endomorphism is a continuous and SO(n)-equivariant map, from the space of convex bodies to itself, such that Φ(K + L) = Φ(K) + Φ(L), for all K, L ∈ K. These endomorphisms were first considered by Schneider in 1974. In this talk a few open questions about Minkowski endomorphisms are going to be discussed. Among other things, it will be shown that Minkowski endomorphisms are uniformly continuous. This, in turn, implies a stronger form of a representation result for Minkowski endomorphisms due to Kiderlen.

## 19. January 2017

*14:30 (A&G)***Andreas Bernig** (Goethe-Universität Frankfurt)

**Integral geometry of the complex projective space**

Two complex submanifolds of the complex projective space of complementary dimension and in general position will intersect in a constant number of points which is given by Bezout's theorem. If we take two real submanifolds of complementary dimension, the number of intersection points will no longer be constant and one may ask about the average number of intersection points. More generally, given two geometric objects A,B in complex projective space (compact submanifolds with boundaries or corners; sets of positive reach or subanalytic sets) and some isometry invariant functional (for instance Euler characteristic or volume), one may ask about the expected value of this functional applied to A intersected with gB, where g is an element of the isometry group. The solution to this old problem was recently obtained, in collaboration with Joseph Fu (Univ. of Georgia) and Gil Solanes (UA Barcelona) in the form of a kinematic formula in complex projective space.

## 02. February 2017

*14:30 (A&G)***Andreas Knauf** (Friedrich-Alexander-Universität Erlangen-Nürnberg)

**Symplectic aspects of scattering**

*16:30 (Mathematical Colloquium)***Fabio Tal** (Universidade de Sao Paulo/Friedrich-Schiller-Universität Jena)

**Entropy zero dynamics in dimension two**