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Dr. Sören von der Gracht, (geb. Schwenker)

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Dr. Sören von der Gracht, (geb. Schwenker)

Lehrstuhl für Angewandte Mathematik

Mitglied - Postdoc - DFG-Projekt "Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme"

Büro:
TP21.1.17
Besucher:
Technologiepark 21
33100 Paderborn
Forschungsinteressen
  • Dynamische Systeme mit Netzwerkstruktur
    • Generizität
    • Bifurkationstheorie
    • Anwendung von Darstellungstheorie
    • Verbindung zu äquivarianter Dynamik
    • Heterokline Dynamik
    • Higher-Order Interactions
    • Anwendungen
  • Äquivariante Dynamik
    • Generizität
    • Bifurkationstheorie
  • Darstellungstheorie
    • Zerlegung von Darstellungen
    • Darstellungen von Monoiden
    • Darstellungen von Köchern
    • Verbindung zu Netzwerken
Forschungsprofile

ORCiD: 0000-0002-8054-2058

WOS ResearcherID: AAU-9040-2020

Google Scholar: Link

Research Gate: Link

 

Dr. Sören von der Gracht, (geb. Schwenker)
05.09.2022 - heute

Postdoc - Universität Paderborn

Projekt: "Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme"

01.10.2019 - 04.09.2022

Postdoc - Universität Hamburg

01.01.2020 - 31.12.2020

Postdoc - HafenCity Universität Hamburg

Projekt: "Linear Algebra Driven by Data Science"

01.10.2015 - 30.09.2019

Promotion - Universität Hamburg - Mathematik

Dissertation: "Genericity in Network Dynamics"

Betreuer: Prof. Reiner Lauterbach

01.10.2013 - 01.10.2015

Master of Science - Universität Hamburg - Mathematics

Abschlussarbeit: "Equivariant Bifurcations in R8 and the Ize Conjecture"

Betreuer: Prof. Reiner Lauterbach

01.10.2010 - 30.09.2013

Bachelor of Science - Universität Hamburg - Mathematik

Abschlussarbeit: "Instabilität der 'logrolling' Lösung in Flüssigkristallen unter Scherströmung"

Betreuer: Prof. Reiner Lauterbach


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2022

Amplified steady state bifurcations in feedforward networks

S. von der Gracht, E. Nijholt, B. Rink, Nonlinearity (2022), 35(4), pp. 2073-2120

We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.


2020

Linear Algebra driven by Data Science

T. Schramm, I. Gasser, S. Schwenker, R. Seiler, A. Lohse, K. Zobel, Hamburg Open Online University, 2020

Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik, der neben der Optimierung und der Stochastik die Grundlage für praktisch alle Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt werden können. Behandelte Themen sind: Der Vektorraum der reellen Zahlentupel, reelle Vektorräume allgemein Lineare Abbildungen Matrizen Koordinaten und darstellende Matrizen Lineare Gleichungssysteme, Gaußalgorithmus Determinante Ein Ausblick auf nichtlineare Techniken, die für neuronale Netzwerke relevant sind.



A new algorithm for computing idempotents of ℛ-trivial monoids

E. Nijholt, B. Rink, S. Schwenker, Journal of Algebra and Its Applications (2020), 20(12)

The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ZM, after which we prove that it also works for RM where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for RM is obtained from the one our algorithm yields for ZM in a straightforward manner. In other words, for any finite R-trivial monoid M our algorithm only has to be performed for ZM, after which a system of idempotents follows for any ring with a given system of idempotents.


Quiver Representations and Dimension Reduction in Dynamical Systems

E. Nijholt, B.W. Rink, S. Schwenker, SIAM Journal on Applied Dynamical Systems (2020), 19(4), pp. 2428-2468

Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.


2019

Genericity in Network Dynamics

S. Schwenker, Universität Hamburg, 2019

This thesis deals with the investigation of dynamical properties – in particular generic synchrony breaking bifurcations – that are inherent to the structure of a semigroup network as well the numerous algebraic structures that are related to these types of networks. Most notably we investigate the interplay between network dynamics and monoid representation theory as induced by the fundamental network construction in terms of hidden symmetry as introduced by RINK and SANDERS. After providing a brief survey of the field of network dynamics in Part I, we thoroughly introduce the formalism of semigroup networks, the customized dynamical systems theory, and the necessary background from monoid representation theory in Chapters 3 and 4. The remainder of Part II investigates generic synchrony breaking bifurcations and contains three major results. The first is Theorem 5.11, which shows that generic symmetry breaking steady state bifurcations in monoid equivariant dynamics occur along absolutely indecomposable subrepresentations – a natural generalization of the corresponding statement for group equivariant dynamics. Then Theorem 7.12 relates the decomposition of a representation given by a network with high-dimensional internal phase spaces to that induced by the same network with one-dimensional internal phase spaces. This result is used to show that there is a smallest dimension of internal dynamics in which all generic l-parameter bifurcations of a fundamental network can be observed (Theorem 7.24). In Part III, we employ the machinery that was summarized and further developed in Part II to feedforward networks. We propose a general definition of this structural feature of a network and show that it can equivalently be characterized in different algebraic notions in Theorem 8.35. These are then exploited to fully classify the corresponding monoid representation for any feedforward network and to classify generic synchrony breaking steady state bifurcations with one- or highdimensional internal dynamics.


2018

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

S. Schwenker, SIAM Journal on Mathematical Analysis (2018), 50(3), pp. 2466-2485

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in [B. Rink and J. Sanders, SIAM J. Math. Anal., 46 (2014), pp. 1577--1609]. It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group---finite or compact Lie. Our generalization also includes noncompact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.


2016

Equivariant bifurcations in four-dimensional fixed point spaces

R. Lauterbach, S. Schwenker, Dynamical Systems (2016), 32(1), pp. 117-147

In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in previous papers we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behaviour is different from what we have seen in the known examples.


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