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.

Please subscribe here to the seminars mailing list.

**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 2019/20

## October 17, 2019

*14:30 (A&G), seminar room 3517*

**Nikolay Martynchuk **(Friedrich-Alexander-Universität Erlangen-Nürnberg)** **

**Scattering monodromy in integrable Hamiltonian systems**

*Abstract:*The notion of Hamiltonian monodromy was introduced by Duistermaat, who defined this invariant as an obstruction to the existence of global action-angle coordinates in integrable Hamiltonian systems. Since then, non-trivial monodromy was observed in various specific examples of integrable systems, such as the spherical pendulum, the Lagrange top, and the hydrogen atom in crossed fields.

## October 24, 2019

*16:00 (Mathematical Colloquium Jena), *

** Ulf Hashagen **(Forschungsinstitut für Technik- und Wissenschaftsgeschichte, Deutsches Museum München)

**Rechnen, Denken und Erfinden im Zeitalter des Barock: Die Rechenmaschinen von Schickard, Pascal und Leibniz**

**Rechnen, Denken und Erfinden im Zeitalter des Barock: Die Rechenmaschinen von Schickard, Pascal und Leibniz**

## November 7, 2019

*14:30 (A&G), seminar room 3517*

** Uri Grupel **(Universität Innsbruck)

**Intersections with random geodesics in high dimensions**

*Abstract:* Abstract: Given a large subset of the sphere *A* ⊆ *S*^{n}^{-}* ^{1}* does the ratio of lengths between a random geodesic Γ and the intersection Γ ∩

*A*represent the size of

*A*or does it tend to a zero-one law (as the dimension grows)? We will show that for any large set

*A*we have a distribution that is not concentrated around neither zero nor one.

In contrast to the case of the sphere, for any convex body in high dimensions, we can find a subset, of half the volume, such that the ratio of lengths between the intersection of the random geodesic with the convex body and the intersection of the subset with the random geodesic will be close to zero-one law.

The analysis of the two cases has different flavors. For the sphere we analyze the singular values of the Radon transform, in order to bound the variance of the length of the random intersection. For convex bodies, we use concentration of measure phenomena.

The results on the sphere can be generalized to the discrete torus or to intersection on the sphere with higher dimensional subspaces.

(Contact: T. Wannerer)

## November 14, 2019

*14:30 (A&G), seminar room 3517*

**Jonas Knörr **(Goethe-Universität Frankfurt)** **

**Smooth valuations on convex functions**

*Abstract:* In recent years, valuations on functions arose as a natural generalization of valuations on convex bodies, and various types of valuations on different spaces of functions have been studied and classifed.

I will present some results from an ongoing project examining the space of* dually epi-translation* invariant valuations on convex functions.We will see how these functionals are related to translation invariant valuations on convex bodies and how one can exploit this relation to establish a notion of smoothness. It turns out that the dense subspace of smooth valuations can be described using integration of differential forms over the graph of the differential of a convex function (or more generally, the differential cycle) and I will present a sketch of proof for this result.

As an application, we will see that the subspace of smooth and rotation invariant valuations admits a very simple description.** **

(Contact: T. Wannerer)

*16:00 (Mathematical Colloquium Jena), *

**Prof. Nikola Sandrić **(University of Zagreb, z.Z. Humboldt-Stipendiat Jena)** **

**Stochastics stability of Markow processes **

*Abstract:* One of the classical directions in the analysis of Markov processes is studying their long-time behavior, the so-called stochastic stability. Stochastically stable Markov process naturally appear as mathematical models of many phenomena arising in nature and engineering, such as problems related to population dynamics, turbulent fuid flows, and homogenization of heterogeneous structures. The goal of this exposition is to motivate and gradually introduce the notion of stochastic stability, first by discussing the most simple probabilistic models (random walks and Lévy process), then classical diffusion processes, and finally diffusion processes with jumps.** **

## November 28, 2019

*14:30 (A&G), seminar room 3517*

** Igor Khavkine** (

**Czech Academy of Sciences, Prague)**

**IDEAL Characterization of Cosmological and Black Hole Spacetimes**

*Abstract*: On a (pseudo-)Riemannian manifold (*M*,*g*), an IDEAL characterization of a reference geometry (*M*_{0}*,g*_{0}) consists of a list of tensors {*T _{i}*[

*g*]} locally and covariantly constructed from the metric

*g*, such that

*T*[

_{i}*g*] = 0 iff (

*M*,

*g*) is locally isometric to (

*M*

_{0},

*g*

_{0}). Unfortunately, to date only a few IDEAL characterizations are known for interesting geometries. But if known, they have interesting applications to analysis and geometry on the reference background (

*M*

_{0},

*g*

_{0}). I will discuss how such characterizations were recently obtained for a class of cosmological and black hole spacetimes.

(Contact: V. Matveev)

## December 5, 2019

*14:30 (A&G), seminar room 3517*

**Siegfried Beckus ** (Universität Potsdam)** **

**Hunting the spectra via approximations**

*Abstract:* Is there a general method to approximate spectral properties of a given operator? If so, can we

control the approximations and which spectral properties are preserved? These questions are

addressed during this talk. We will put a special focus on operators defined via an underlying

dynamics and geometry. In this case, we study how deformations of our dynamical system

respectively its geometry lead to suitable approximations. A particular focus is put on the

Kohmoto butterfly.

(Contact: M. Schmidt)

## December 12, 2019

*14:30 (A&G), seminar room 3517*

**Jonas Franke** (iDiv Leipzig/FSU Jena)** **

**Food webs exposed to press perturbations - the accuracy of predictions made by the net efficiency matrix**

The interactions of different species in natural communities are commonly represented by food webs. Mathematically, a food web is a directed graphs describing the who-eats-who of the community. By feeding and feeding on each other, species' biomasses change over time. This time-dependent behavior can be studied as a dynamical system, called trophic network.

Of particular interest is a trophic network's behavior under external perturbations. A classical method to approximate the effect of a press perturbation on a trophic network's steady state uses the net effects matrix (NEM). The NEM is given by the inverse of the jacobian at the respective steady state. Even though this method has been widely used for years, its reliability has scarcely been studied and the existence of a new steady state under press perturbation never been proven. In my master's project, I used the implicit function theorem to deliver such a proof. The related considerations yield a maximum strength of the press perturbation for which the existence of a new steady state can be assured. Furthermore, the proof shows that the NEM-approximation is but the first of many possible iterates and how the method may easily be improved by using these.

Based on this, I investigate the new steady state and its approximation for allometric trophic networks. On the one hand, a theoretical boundary for the maximum strength of the press perturbation and the approximation error for allometric trophic networks is derived analytically. On the other hand, I use numerical simulations to study the NEM-method's qualitative behavior. As it turns out, the respective approximation is very reliable at predicting if a species experiences an advantage or diasadvantage from a press perturbation. Moreover, almost always an upper bound for this effect is provided. Finally I present how the complexity of the trophic network and the strength of the press perturbation influence these results.** **

(Contact: T. Jäger)

## January 16, 2020

*16:00 (Mathematical Colloquium Jena), *

**Daniel Rosen ** (Ruhr-Universität Bochum)** **

**TBA**

*Abstract:* t.b.a.** **

(Contact: T. Wannerer)