# Summer term 2016

## 07. April 2016

*14:30 (A&G)***Christoph Richard** (Friedrich-Alexander-Universität Erlangen-Nürnberg)

**Poisson summation and pure point diffraction**

The diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. We will explain this result, which is obtained by Fourier analysis of unbounded measures on locally compact second countable abelian groups, as developed by Argabright and de Lamadrid. This is joint work with Nicolae Strungaru (Peterborough).

*16:00 (DS&MP)***Dominik Kwietniak** (Uniwersytet Jagiellonski, Krakow, Poland & UFRJ, Rio de Janeiro, Brazil)

**Dbar metric and invariant measures of hereditary shift spaces**

I will describe some properties of the simplex of invariant measures of a hereditary shift space. We say that a shift space over {0,1} is hereditary if it is closed with respect to coordinatewise multiplication by an arbitrary 0-1 sequence. Using dbar pseudometric as a tool we prove that the set of ergodic measures is arcwise connected for every hereditary shift space and it is even entropy-dense for a particular class of hereditary shifts, which include many B-free shifts.

B-free shifts are defined as follows: Given a set of integers A we identify its characteristic function with an infinite 0-1 sequence x(A). The closure of the orbit of x(A) with respect to the shift generates a symbolic dynamical system X_A. Recall that an integer is B-free if it has no factor in a set B contained in N. For example square-free integers are Bsq-free where Bsq is the set of squares of primes. Abdalauoi, Lemańczyk and De La Rue, extending an idea of Sarnak, studied B-free integers F_B through dynamics of the shift space generated by the characteristic sequence of F_B.

I am going to present how using dbar metric one can obtain results about invariant measures of B-free shifts and their entropy. (This is a joint work with Jakub Konieczny and Michal Kupsa.)

## 14. April 2016

*14:30 (A&G)***David Damanik** (Rice University, Houston, USA)

**Almost periodicity in time of solutions of the KdV equation**

We describe joint work with Ilia Binder, Michael Goldstein and Milivoje Lukic, which is motivated by the following conjecture of Percy Deift: Solutions of the KdV equation with almost periodic initial data are almost periodic in time. Our work confirms this conjecture in the so-called Sodin-Yuditskii regime, that is, assuming that the Schrödinger operator whose potential is given by the initial datum has purely absolutely continuous spectrum (along with some assumptions on the topological structure of the spectrum). The talk will explain the overall structure of our approach and some of the key ideas.** **

## 21. April 2016

### Mathematisches Kolloquium

*16:30 Carl-Zeiß-Straße 3, SR 309***Israel Michael Sigal **(University of Toronto)

**Magnetic Vortices, vortex lattices and automorphic functions**

## 12. May 2016

*14:30 (A&G)*

**Johannes Kellendonk**(Université Claude Bernard Lyon I, France)

**Cyclic cohomology for graded Real C*-algebras and with an application to topological insulators**

The classification of topological insulators is based on K-theory. In the non-commutative approach (which allows for the inclusion of disorder) one needs to work with Real C*-algebras, their K-theory and an appropriate dual theory. Response coefficients are then obtained as pairings. There are two options for the dual theory and the associated pairings: K-homology with index pairings or cyclic cohomology with Connes pairings. We explore the latter possibility.** **

## 19. May 2016

### Mathematisches Kolloquium

*16:30 Carl-Zeiß-Straße 3, SR 309***Eldar Straume** (University of Trondheim)

**On the collaboration of Felix Klein and Sophus Lie in the early 1870's,**** seen for the first time in the light of Klein's letter to Lie**

**seen for the first time in the light of Klein's letter to Lie**

## 26. May 2016

*16:00 (DS&MP)***Carlos Sierra** (MPI Biogeochemie, Jena)

**The global carbon cycle as a dynamical system**

In this talk I will introduce a research program that aims at conceptualizing models of the global carbon cycle as dynamical systems and use existing mathematical theory to improve our understanding of the interaction between climate and the carbon cycle. This program has four main themes: 1) identification of isomorphic models, 2) characterization of system invariants, 3) assessment of stability, and 4) determination of the statistical behavior of classes of dynamical systems used to model the global carbon cycle. I will provide examples of how existing mathematical results can greatly help in doing synthesis of computer models, and will pose some questions for mathematical results that are needed to solve some outstanding scientific questions.** **

## 2. June 2016

*16:00 (DS&MP)*

**Katrin Gelfert**(Universidade Federal do Rio de Janeiro, Brazil)

**Dimensions and critical regularity of hyperbolic graphs**

Wir diskutieren die fraktale Struktur invarianter Mengen bestimmter Abbildungen. Im Allgemeinen - insbesondere in Phasenraum mit Dimension > 2 - kann die Struktur nur schwer analysiert werden und ein natürlicher Zugang ist deshalb die Untersuchung schrittweise komplizierterer (z.B. höherdimensionaler) Systeme. Diesem Zugang folgend, betrachten wir hier zunächst Mengen, die als Graphen über bestimmten Mengen in einem 2-dimensionalem Phasenraum aufgefasst werden können; und wir beschränken uns auf Diffeomorphismen mit gleichzeitig partiell-hyperbolischer und hyperbolischer Struktur. Wir beschreiben die (kritische: Lipschitz oder auf allen Skalen Hölder-stetige) Regularität solcher Graphen und ziehen Rückschlüsse über deren Box-counting-Dimension. Die Ergebnisse resultieren teilweise aus gemeinsamer Arbeit mit L. Diaz (PUC-Rio), M. Gröger und T. Oertel-Jäger.

## 9. June 2016

*14:30 (A&G)*

**Gabriel Fuhrmann**(Friedrich-Schiller-Universität Jena)

**Nonsmooth saddle-node bifurcations in quasiperiodically forced systems**

### Mathematisches Kolloquium

*16:30 Carl-Zeiß-Straße 3, SR 309*

**Manfred Einsiedler **(ETH Zürich)

**Integer points on spheres and their orthogonal complement**

## 16. June 2016

*14:30 (A&G)*

**Nobuaki Obata**(Tohoku University, Sendai, Japan)

**Quantum probability and spectral analysis of graphs**

Quantum probability is an algebraic extension of classical (Kolmogorovian) probability, tracing back to von Neumann who gave the mathematical formalism of statistical problems in quantum mechanics. Thus, quantum probability is based on the pair $(\mathcal{A},\varphi)$ of a *-algebra and a state on it. The algebraization is also useful for the study of spectra of (growing/random) graphs. I like to overview the method of quantum decomposition, which enables to study symmetric matrix (adjacency matrix) in terms of annihilation and creation operators in Fock space and is closely related to the theory of orthogonal polynomials.If time permits, I like to refer to some concepts of independence and application to graph products.

## 23. June 2016

*14:30 (A&G)*

**Dmitry Faifman**(University of Toronto, Canada)

**Some integral-geometric formulas for O(p,q)**

A valuation is a finitely additive measure on convex bodies. Valuation theory traces its origins to Hilbert's 3rd problem, and has since become an integral part of convex geometry.

Among the central pieces of the theory are Hadwiger's theorem, classifying all continuous valuations that are invariant under Euclidean motions; and the Poincare-Blaschke-Chern kinematic formulas, which evaluate the averages of such valuations over the different relative positions of two convex bodies. In this talk, we will put those results in the framework of Alesker's theory of valuations. I will then describe what happens when the Euclidean group of motions is replaced with the indefinite orthogonal group, e.g. the Lorentz group.

*16:00 (DS&MP)*

**Jing Wang**(Friedrich-Schiller-Universität Jena & Nanjing University, China)

**Mode-locking in quasiperiodically forced circle maps**

## 30. June 2016

### Mathematisches Kolloquium

*16:30 Carl-Zeiß-Straße 3, HS 6***Klaus Böhmer **(Philipps-Universität Marburg)

**Math Goes Public: Interessante Phänome bei Nichtlinearen Problemen, Tau-Tropfen auf Spinnen-Netzen**

## 07. July 2016

*16:00 (DS&MP)*

**Tony Samuel**(California Polytechnic State University, San Luis Obispo, USA)

**Aperiodic order and Jarnik sets**

Sturmian words (subshifts) are combinatorial objects that are quite remarkable just by the fact that they can be formulated in terms of a variety of mathematical framework, for instance:

- Billiards - Movement of a ball on a square billiard table.
- Combinations - Aperiodic words that are balanced.
- Geometry - Digitised straight lines or circle rotations.
- Dynamical Systems - Minimal factors.
- Number Theory - Continued fractions and semi-group morphisms.

_{1}, a

_{2}, ...] with unbounded continued fraction entries, we will discuss new characterising relations of Sturmian subshifts with slope θ with respect to the regularity properties of spectral metrics as introduced by Kellendonk and Savinien, level sets defined in terms of the Diophantine properties of θ and complexity notions which are generalisations and extensions of the combinatorial concepts of linearly repetitive, repulsive and power free.