## Mathematisches Institutskolloquium

Das Mathematische Institutskolloquium richtet sich an ein breites mathematisches Publikum (mit Bachelor Abschluss in Mathematik). Es soll die Diskussionen über die mathematischen Spezialisierungen der verschiedenen Arbeitsgruppen am Institut fördern. Außerdem sollen auch Studierende (Master-Studierende und fortgeschrittene Bachelor-Studierende) durch das Kolloquium die Gelegenheit erhalten sich über aktuelle Themen der Mathematik zu informieren.

### Sommersemester 2023

**Herr Prof. Gary Froyland**(School of Mathematics and Statistics, University of New South Wales)**"Transfer operator analysis of the ocean and climate”**

"I will illustrate how operator-theoretic methods can help us learn about our oceans and climate by discussing two examples: estimating basins of attraction on the global surface ocean and constructing improved characterisations of the El Nino Southern Oscillation. I will draw on data sourced from fleets of ocean drifters, and satellite-derived sea-surface temperature observations."**05.04.2023, 15:15 Uhr, Ulmenstraße 69, Haus 3, SR 326**

### Wintersemester 2022/2023

**Herr Dr. Stefan Tappe**(Institut für Mathematik, Universität Rostock)**"****Stochastic partial differential equations and invariant manifolds in embedded Hilbert spaces”**

In this presentation we investigate stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. More precisely, we are interested in the question when a finite dimensional submanifold is locally invariant for such an equation. We start with reviewing some classical results for ordinary differential equations (ODEs) and stochastic differential equations (SDEs). Afterwards, we move on to general SPDEs and establish an invariance result where the coefficients of the SPDE are merely assumed to be continuous. As a concrete example we consider SPDEs with values in the space of tempered distributions; in this situation the continuously embedded Hilbert spaces are given by Hermite-Sobolev spaces and the coefficients are differential operators of first and second order. At this juncture there is an interplay between finite dimensional SDEs and particular types of SPDEs. This presentation is based on joint work with Rajeev Bhaskaran (Indian Statistical Institute, Bangalore Centre). Das Kolloquium findet diesmal*ausschließlich als Zoom-Meeting online*statt.**11.01.2023, 15:15 Uhr**

**Herr Dr. Matthias Schymura**(Institut für Mathematik, Universität Rostock)**"****A geometric view on the Lonely Runner Problem: Coverings by Zonotopes”**

The Lonely Runner Conjecture concerns k runners on a circular track of length 1 who start running from a common starting position with pairwise distinct constant velocities. The claim is that for each runner there is a certain point in time at which she is at distance at least 1/k from all the other runners on the track. This is a picturesque formulation of a conjecture by Jörg Wills (1967) in the theory of Diophantine Approximation. The problem is strongly linked to Dirichlet's classical approximation theorem and has found various reinterpretations, for instance, in terms of view-obstructions and billiard ball motions. Many efforts on the conjecture have been taken over the last five decades, yielding important partial results by attacking the problem from many different angles. However, a complete solution to the Lonely Runner Conjecture is only known for up to 7 runners to date.The talk aims to give an overview over the history of the problem, its various interpretations, and the state of the art. In particular, we introduce yet another interpretation in terms of the existence of integer points in certain zonotopes (projections of the cube). We report how this geometric approach leads to new insights on the Lonely Runner Problem, and how it motivates the investigation of covering properties of zonotopes in general.**14.12.2022, 15:15 Uhr, Ulmenstr. 69, Haus 3, HS 326/327**

**Herr Prof. Dr. Viktor Avrutin**(Institut für Systemtheorie und Regelungstechnik, Universität Stuttgart)**"A hidden bridge between continuous and discontinuous worlds (and how period two implies chaos)"**

Many problems in engineering and applied science lead us to consider piecewise smooth maps. Examples of such systems include applications in electronics (switching circuits), mechanics (systems with dry friction or impacts), economics and social sciences (systems involving decision making processes), as well as other systems with thresholds and constraints. From the mathematical point of view, piecewise smooth maps are extremely challenging and can be subdivided in two classes, namely, continuous and discontinuous maps. Possibly dynamics and bifurcation phenomena in both classes differ significantly, and at present, they are understood much better for the former class than for the latter one. Recently, a novel approach for investigation of discontinuous maps has been suggested which surprisingly combines several aspects of the dynamics commonly observed in both classes of maps. The idea of this approach is to extend the definition of a discontinuous map in such a way that at the points of discontinuities, the function is considered to be set-valued. This unifies the bifurcation analysis for continuous and discontinuous maps; makes several theorems proven for continuous maps applicable to discontinuous ones; and helps us to understand the dynamics of maps with steep branches which are hard to deal with otherwise.**16.11.2022, 15:15 Uhr, Ulmenstr.****69, Haus 3, HS****326/327**

**Dr. Dirk Hartmann**(Siemens Digital Industry Software)**"Scalable Digital Twins - Combining Machine Learning and Physics-based Simulations"**

Abstract: Digital Twins, tightly connecting the real and the digital world, are a key enablers to support industrial decision making for complex systems. They allow informing operational as well as strategic decisions upfront through accepted virtual predictions and optimizations of their real-world counter parts. Sufficiently accurate and fast digital models are required to do so, which today involves significant manual expert efforts limiting industrial scalability.

Integrating machine learning and physics-based methods offers opportunities to overcome these limitations. In this context, we will review selected use cases covering 3D and lumped system simulation and demonstrate limitations and potential of hybrid methods along these. Finally, we will indicate fields requiring more research from an industrial point of view in the field of computational science & engineering and machine learning.**19.10.2022, 14:00 Uhr, Ulmenstr. 69, Haus 3, SR 322**(Hybrid-Veranstaltung)

### Sommersemester 2022

**Prof. Dr. Thomas Mikosch**(University of Copenhagen, Denmark)**"Richard von Mises and the development of modern extreme value theory"**

Abstract: This talk has the following goals:

(1) To present Richard von Mises’s achievements in extreme value theory in their historical context.

(2) To provide a short history of extreme value theory and statistics since the 1920s.

(3) To discuss some of the challenges in modern extreme value theory.**22.06.2022, 15:15 Uhr, Ulmenstr. 69, Haus 3, HS 326/327**(Hybrid-Veranstaltung)

**Dr. rer. nat. Isabelle Schneider**(University of Rostock)**"Pattern-Selective Feedback Stabilization of Ginzburg-Landau Spiral Waves"**

Abstract: The complex Ginzburg-Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg-Landau m-armed spiral waves have been investigated extensively. However, most spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this talk I selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. The tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

This is joint work with Jia-Yuan Dai (National Chung Hsing University, Taiwan) and Babette de Wolff (Vrije Universiteit Amsterdam, Netherlands).**18.05.2022, 15:15 Uhr, Ulmenstr. 69, Haus 3, HS 326/327**(Hybrid-Veranstaltung)

### Wintersemester 2021/2022

**Dipl.-Ing. Alexander Steinicke, PhD**(Montanuniversitaet Leoben)**"From Stochastic Differential Equations to Piecewise Lipschitz Functions"**

Abstract**16.02.2022, 15:15 Uhr**, Online-Veranstaltung

**Prof. Dr. Jan Sieber**(University of Exeter, UK)**"Delay differential equations - numerical treatment and the case of large delay"**

Abstract: Delay differential equations (DDEs), in which some of the arguments enter with a time delay, but which otherwise look like ordinary differential equations, occupy a place somewhere between ordinary differential equations and partial differential equations (PDEs) with one space dimension. This is noticeable when implementing numerical methods, especially for finding equilibria, periodic solutions and the spectra of the linear DDEs. When the delay becomes large the limiting behaviour of some periodic oscillations approaches that of patterns in PDEs. A particular example are "temporal dissipative solitions" periodic pulses caused by delayed feedback into an excitable system.

(Joint work with Serhiy Yanchuk, Stefan Ruschel, Matthias Wolfrum)**12.01.2022, 15:15 Uhr**, Online-Veranstaltung

**Prof. Dr. Ivan Veselić**(Technische Universität Dortmund)**"Uncertainty relations and applications"**

Abstract: Uncertainty relations or unique continuation estimates for various classes of functions are investigated in several fields of mathematical analysis. They have also a number of interesting applications, among them those in mathematical physics and the theory of partial differential equations. While in other areas uncertainty implies less knowledge or weaker mathematical results, in these cases uncertainty principles in fact improve our knowledge about certain mathematical objects.

The talk aims at shedding a bit of light on these aspects of uncertainty principles.**15.12.2021, 16:00 Uhr**(Uhrzeit aktualisiert am 09.12.), Online-Veranstaltung

**Prof. Dr. Wilhelm Stannat**(TU Berlin)**"Mean-field approach to Bayesian estimation of Markovian signals"**

Abstract: Estimating Markovian signals X from noisy observations is an important problem in the natural and engineering sciences. Within the Bayesian approach the underlying mathematical problem essentially consists in the (stochastic) analysis of the conditional law of X with a view towards its efficient numerical approximation.

In this talk I will discuss mean-field type descriptions of the conditional law of X, when X is the solution of a stochastic differential equation, and present recent results on corresponding ensemble-based numerical approximations.

The talk is based on joint work with T. Lange, S. Pathiraja and S. Reich.

References:

[1] S. Pathiraja, S. Reich, W. Stannat: McKean-Vlasov SDEs in nonlinear filtering,

SIAM J. Control Optim. 59 (2021), no. 6, 4188–4215.

[2] S. Pathiraja, W. Stannat: Analysis of the feedback particle filter with diffusion

map based approximation of the gain, Foundations of Data Science 3 (2021): 615-645.

[3] T. Lange, W. Stannat: Mean field limit of Ensemble Square Root Filters - discrete

and continuous time, Foundations of Data Science 3 (2021): 563-588.**17.11.2021, 15:15 Uhr, Ulmenstr. 69, Haus 3, HS 326/327**(Hybridveranstaltung)

**Prof. Constantinos Siettos**(University of Naples Federico II, Italy)**"Numerical Solution of Nonlinear PDEs and Stiff Problems of ODEs with Random Projection Networks and Extreme Learning Machines"**

Abstract: We use a class of machine learning the so-called Random Projections Networks and Extreme Learning Machines to numerically solve nonlinear partial differential equations (PDEs) and stiff problems of ODEs. For our demonstrations, we study several benchmark problems, namely (a) the Rober and Van-der Pol ODEs and, (b) the one-dimensional viscous Burgers and, the one- and two-dimensional Bratu PDEs. We also show how one can expolit the proposed methodology to construct bifurcation diagrams past limit points. The numerical efficiency of the proposed numerical machine learning scheme is compared against well established numerical analysis methods such as the adaptive Runge-Kutta ode45 and the ode15s a variable-step, variable-order solver based on the numerical differentiation formulas for solving the ODEs and central finite differences (FD) and Finite-element (FEM) methods for solving PDEs. We show that the proposed machine learning framework, regarding the solution of ODEs yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the ode45 and ode15s integrators, while regarding the solution of PDEs outperforms FD and importantly FEM for medium to large sized grids in terms of computational times and numerical accuracy.**13.10.2021, 15:15 Uhr**, Online-Veranstaltung

### Sommersemester 2021

**Dr. Paolo Di Tella**(University of Rostock)**"On Martingale Representation Theorems"**

Abstract: A central result in Stochastic Analysis is the martingale representation theorem of the Brownian motion. In this talk we are going to present classical result and extensions of the Brownian martingale representation theorem in more general contexts. We shall then present some applications to mathematical finance.**14.07.2021**, 1**5:15 Uhr**, Online-Veranstaltung

**Prof. Kim Knudsen**(Technical University of Denmark)**"Electromagnetic imaging – mathematical analysis and computations"**

Abstract:

In this talk we will look at inverse problems related to electromagnetic imaging. One example is Electrical Impedance Tomography (EIT), mathematically known as the Calderón problem, where the goal is to identify a body’s interior electrical conductivity distribution from measurements of voltages and currents on the surface of the body. This problem is severely ill-posed and requires heavy regularization techniques to be implemented before allowing for image reconstruction even with low resolution and contrast.

Recently, novel hybrid imaging methods such as Acousto-Electric Tomography and Magnetic Resonance EIT have appeared. These approaches exploit different coupled physical phenomena and therefore hold promise for much more accurate and stable methods than EIT.

From a mathematical analysis and computational perspective we consider the three different examples; we will formulate the relevant models, pose fundamental questions and give (partial) answers.**16.06.2021, 15:15 Uhr**, Online-Veranstaltung