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 Jäger 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)

# Summer term 2019

## 11. April 2019

*16:00 (DS&MP), seminar room 3517*

**Francisco Lopez Hernandez** (Universidad Autonoma de San Luis Potosi)

**Some aspects of dynamics on solenoids**

We will define solenoids from different points of view in order to describe some dynamical properties of solenoidal homeomorphisms.

(Contact: T. Jäger)

## 25. April 2019

*14:30 (DS&MP), seminar room 3517*

**Tobias Weihrauch **(Universität Leipzig)

**Resistance Forms of Graphs**

We review basic properties of Resistance Forms as introduced by Kigami and investigate the special case of energy forms of infinite graphs. In particular, we discuss the connection of their associated resistance metric to the graphs random walk.

(Contact: M. Schmidt)

## 09. May 2019

*14:30 (DS&MP), seminar room 3517*

**Anna Muranova **(Universität Bielefeld)

**Two approaches to the notion of effective impedance**

Abstract: It is known that electrical networks with resistors are related to the Laplace operator and Dirichlet problem on weighted graphs. In this talk we consider more general electrical networks with coils, capacitors, and resistors. Such electrical networks give rise to complex-weighted graphs. The corresponding Dirichlet problem with complex-valued coefficients does not necessary have a solution, and if it has, it may be not unique. This creates difficulties in definition of the effective impedance. We present two approaches to overcoming this difficulty. In the first approach we show that, in the case of multiple solutions, all they have the same energy and, therefore, the effective impedance is well-defined. In the second approach we consider weights of the edges as rational functions of λ=iω, here ω is a frequency of the current and use the fact that rational functions form an ordered field. Then we develop a theory of weighted graphs with weights from an ordered field and prove that the effective impedance is always well defined in this case.

(Contact: D. Lenz)

## 23. May 2019

*14:30 (DS&MP), seminar room 3517*

**Martin Schneider **(TU Dresden)

** Concentration of measure and its applications in topological dynamics**

Abstract: The phenomenon of measure concentration, in its modern formulation, was isolated in the late 1960s and early 1970s by Vitali Milman, extending an idea going back to Paul Levy's work on the geometry of (high-dimensional) Euclidean spheres, and has since led to numerous interesting applications in geometry and combinatorics. In their groundbreaking 1983 joint work, Gromov and Milman linked this phenomenon with topological dynamics: they proved that any Levy group, i.e., any topological group G admitting a dense exhaustion by an increasing sequence of compact subgroups with Levy-Milman concentration of the corresponding normalized Haar measures, is extremely amenable, which means that every continuous action of G on a non-void compact Hausdorff space must have a fixed point. Examples of such Levy (hence extremely amenable) groups include the unitary group of the infinite-dimensional separable Hilbert space equipped with the strong operator topology, the isometry group of the Urysohn space with the topology of point-wise convergence, and the automorphism group of the non-atomic standard Lebesgue space with the weak topology.

In my talk, I will survey some very recent developments concerning applications of measure concentration in topological dynamics. This will include a generalization of the above-mentioned result by Gromov and Milman, a correspondence principle connecting Gromov's metric measure geometry with dynamics and ergodic theory of large topological groups, as well as some new manifestations of concentration of measure.

(Contact: T. Hauser, T. Jäger)

*16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307*

**Prof. Dr. Edriss S. Titi** (University of Texas, Weizmann Intitute of Science, University of Cambridge, zurzeit Einstein-Visiting-Fellow FU Berlin)

### Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background flows

Abstract: In this talk I will present a unifed approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit "Landua-damping" mechanism due to large spatial average in the initial data.

## 6. June 2019

*14:30 (DS&MP), seminar room 3517*

**Nikolai Edeko **(Universität Tübingen)

**Equicontinuous factors of flows on locally path-connected compact spaces **

Given a quotient map f of topological spaces X and Y, it is generally difficult to relate geometric properties of X to those of Y without assumptions on the quotient map f. A useful assumption is monotonicity of f, i.e., connectedness of its fibers, since it ensures that a quotient map between appropriate spaces induces a surjective homomorphism on the level of fundamental groups. Motivated by a result of T. Hauser and T. Jäger proving the monotonicity of the maximal equicontinuous factor of a flow on a locally connected compact space, we provide a general functional analytic criterion for monotonicity of factors of topological dynamical systems which allows to understand the monotonicity of the maximal equicontinuous factor under a different perspective. We then show that this monotonicity result and the above-mentioned geometric property of monotone quotient maps allow, in the absence of non-trivial fixed functions, to represent equicontinuous factors of (abelian) flows on compact manifolds as rotations on compact abelian Lie groups. We discuss how this can be used to link the spectral theory of such flows with the first Betti number of the underlying manifold.

(Contact: T. Hauser, T. Jäger)

*16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307*

**Peter Bürgisser **(TU Berlin)

**On the number of real zeros of structured random polynomials**

We plan to report on two recent results stating that structured (systems of) random polynomials typically only have few real zeros.The first result is on random fewnomials: it says that a system of polynomials in $n$ variables with a prescribed set of $t$ terms and independent centered Gaussian coefficients has an expected number of positive real zeros bounded by $2 {t \choose n}$.

The second result is on Koiran's Real Tau Conjecture, which claims that the number of real zeros of a sum of $m$ products of $k$ real sparse univariate polynomials, each with a fixed set of at most $t$ terms, is bounded by a polynomial in $m,k,t$. The Real Tau Conjecture implies Valiant's Conjecture $VP \ne VNP$. We have confirmed the conjecture on average: if the coefficients in these structured polynomials are independent standard Gaussians, then theexpected number of real zeros is bounded by O(mkt).

The proofs are based on the Rice formula and methods fromintegral geometry.

This is joint work with Alperen Erguer, Josue Tonelli-Cueto and Irenee Briquel.

(Contact: T. Wannerer)

## 13. June 2019

*16:30 Mathematical Colloquium Jena, Carl-Zeiß-Straße 3, SR 307*

**Aleksandr Koldobskiy **(University of Missouri-Columbia / Max Planck Institute for Mathematics Bonn)

**Slicing and distance inequalities for convex bodies**

Slicing inequalities provide estimates for the volume of a solid in terms of areas of its plane sections.One of the problems is the Busemann-Petty problem asking whether convex bodies with uniformly

smaller areas of their central hyperplane sections necessarily have smaller volume. Another is the

problem of Bourgain asking whether every symmetric convex body of volume one has a hyperplane

section whose area is greater than an absolute constant. We show new estimates of this kind depending

on the (outer volume ratio) distance from the body to the class of intersection bodies or to the class of

unit balls of subspaces of

*L*. Many of the results hold for arbitrary measures in place of volume.

_{p}## 27. June 2019

*14:30 (DS&MP), seminar room 3517*

**Nicolae Strungaru **(MacEwan University, Edmonton)

**On the diffraction of Meyer sets**

In this talk we will review some recent progress in the diffraction of measures with Meyer set support. We will prove the existence of a generalized Eberlein decomposition, and the norm almost periodicity of the diffraction, and of each of the three spectral components.

(Contact: D. Lenz)