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Dr. rer. nat. Siegfried Beckus

Lecturer and Researcher

Kontakt
Raum:
2.09.2.11
Telefon:
+49 331 977 2748
E-Mail:
beckus@uni-potsdam.de
...

Research interests & Preprints

Preprints

Research interests

  • spectral theory of Schrödinger operators on graphs with aperiodic ordered potentials
  • dynamical Systems
  • Delone sets
  • graph limits and graphings
  • aperiodic tilings
  • operator algebras
  • fields of C*-algebras (especially C*-algebras induced by groupoids)

PhD thesis

Spectral approximation of aperiodic Schrödinger operators
Friedrich-Schiller Universität Jena, October 2016

Diploma thesis

Generalized Bloch Theory for Quasicrystals
Friedrich-Schiller Universität Jena, February 2012

Scientific Output

@ googlescholar, mathscinet

Publications

2024 | Linear repetitivity beyond abelian groups | Siegfried Beckus, Tobias Hartnick, Felix PogorzelskiZeitschrift: to appear in Ergodic Theory Dynam. SystemsLink zum Preprint

Linear repetitivity beyond abelian groups

Autoren: Siegfried Beckus, Tobias Hartnick, Felix Pogorzelski (2024)

We show that linearly repetitive weighted Delone sets in groups of polynomial growth have a uniquely ergodic hull. This result applies in particular to the linearly repetitive weighted Delone sets in homogeneous Lie groups constructed in the companion paper arXiv:2109.15210 using symbolic substitution methods. More generally, using the quasi-tiling method of Ornstein-Weiss, we establish unique ergodicity of hulls of weighted Delone sets in amenable unimodular lcsc groups under a new repetitivity condition which we call tempered repetitivity. For this purpose, we establish a general sub-additive convergence theorem, which also has applications concerning the existence of Banach densities and uniform approximation of the spectral distribution function of finite hopping range operators on Cayley graphs.

Zeitschrift:
to appear in Ergodic Theory Dynam. Systems

2023 | MFO Report: The dry ten Martini problem for Sturmian dynamical systems | Ram Band, Siegfried Beckus, Raphael LoewyZeitschrift: Oberwolfach Reports - OWRLink zum Preprint

MFO Report: The dry ten Martini problem for Sturmian dynamical systems

Autoren: Ram Band, Siegfried Beckus, Raphael Loewy (2023)

This extended Oberwolfach report (to appear in the proceedings of the MFO Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling theorem) are open for all nonzero couplings and all irrational rotations. We present here the proof strategy.

Zeitschrift:
Oberwolfach Reports - OWR

2021 | Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems | Siegfried Beckus, Baptiste DevyverZeitschrift: Journal d'Analyse MathematiqueReihe: 146Seiten: 165-203Link zur Publikation , Link zum Preprint

Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems

Autoren: Siegfried Beckus, Baptiste Devyver (2021)

Let H be a generalized Schrödinger operator on a domain of a non-compact connected Riemannian manifold, and a generalized eigenfunction u for H: that is, u satisfies the equation Hu = λu in the weak sense but is not necessarily in L2. The problem is to find conditions on the growth of u, so that λ belongs to the spectrum of H. We unify and generalize known results on this problem. In addition, a variety of examples is provided, illustrating the different nature of the growth conditions.

Zeitschrift:
Journal d'Analyse Mathematique
Reihe:
146
Seiten:
165-203

2021 | Eigenfunctions growth and R-limits on Graphs | Siegfried Beckus, Latif EliazZeitschrift: Journal of Spectral TheoryReihe: 11Seiten: 1895-1933Link zur Publikation , Link zum Preprint

Eigenfunctions growth and R-limits on Graphs

Autoren: Siegfried Beckus, Latif Eliaz (2021)

A characterization of the essential spectrum of Schrödinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on the natural numbers and the d-dimensional integer lattice as “right-limits,” captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in the essential spectrum of H corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.

Zeitschrift:
Journal of Spectral Theory
Reihe:
11
Seiten:
1895-1933

2020 | Spectral continuity for aperiodic quantum systems: applications of a folklore theorem | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Mathematical PhysicsReihe: 61Link zur Publikation , Link zum Preprint ,

Oberwolfach Preprint

Spectral continuity for aperiodic quantum systems: applications of a folklore theorem

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2020)

This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson–Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay–Rudin–Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.

Zeitschrift:
Journal of Mathematical Physics
Reihe:
61

2019 | Hölder Continuity of the Spectra for Aperiodic Hamiltonians | Siegfried Beckus, Jean Bellissard, Horia CorneanZeitschrift: Annales Henri PoincaréReihe: 20Seiten: 3603 - 3631Link zur Publikation , Link zum Preprint ,

Oberwolfach Preprint

Hölder Continuity of the Spectra for Aperiodic Hamiltonians

Autoren: Siegfried Beckus, Jean Bellissard, Horia Cornean (2019)

We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.

Zeitschrift:
Annales Henri Poincaré
Reihe:
20
Seiten:
3603 - 3631

2019 | Corrigendum to “Spectral continuity for aperiodic quantum systems I. General theory | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Functional AnalysisReihe: 277Seiten: 3351-3353Link zur Publikation , Link zum Preprint

Corrigendum to “Spectral continuity for aperiodic quantum systems I. General theory

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2019)

A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.

Zeitschrift:
Journal of Functional Analysis
Reihe:
277
Seiten:
3351-3353

2018 | Delone dynamical systems and spectral convergence | Siegfried Beckus, Felix PogorzelskiZeitschrift: Ergodic Theory and Dynamical SystemsReihe: 40Seiten: 1510-1544Link zur Publikation , Link zum Preprint

Delone dynamical systems and spectral convergence

Autoren: Siegfried Beckus, Felix Pogorzelski (2018)

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

Zeitschrift:
Ergodic Theory and Dynamical Systems
Reihe:
40
Seiten:
1510-1544

2018 | Shnol-type theorem for the Agmon ground state | Siegfried Beckus, Yehuda PinchoverZeitschrift: Journal of Spectral TheoryReihe: 10Seiten: 355 - 377Link zum Preprint

Shnol-type theorem for the Agmon ground state

Autoren: Siegfried Beckus, Yehuda Pinchover (2018)

Let H be a Schrödinger operator defined on a noncompact Riemannian manifold Ω, and let W∈L∞(Ω;R). Suppose that the operator H+W is critical in Ω, and let φ be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction of H satisfying |u|≤φ in Ω, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K⋐Ω the operator H admits a positive solution in Ω'=Ω∖K, and |u|≤ψ in Ω', where ψ is a positive solution of minimal growth in a neighborhood of infinity in Ω.
Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

Zeitschrift:
Journal of Spectral Theory
Reihe:
10
Seiten:
355 - 377

2018 | Spectral continuity for aperiodic quantum systems I. General theory | Siegfried Beckus, Jean Bellissard, Giuseppe De NittisZeitschrift: Journal of Functional AnalysisReihe: 275Seiten: 2917 - 2977Link zur Publikation , Link zum Preprint

Spectral continuity for aperiodic quantum systems I. General theory

Autoren: Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis (2018)

How does the spectrum of a Schrödinger operator vary if the corresponding geometry and dynamics change? Is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures? In this work a positive answer is provided using the rather general setting of groupoid C*-algebras. A characterization of the convergence of the spectra by the convergence of the underlying structures is proved. In order to do so, the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles. With this at hand, magnetic Schrödinger operators on dynamical systems or Delone systems fall into this unified setting. Various approximations used in computational physics, like the periodic or the finite cluster approximations, are expressed through the tautological groupoid, which provides a universal model for fields of groupoids. The use of the Hausdorff topology turns out to be fundamental in understanding why and how these approximations work.

Zeitschrift:
Journal of Functional Analysis
Reihe:
275
Seiten:
2917 - 2977

2018 | Note on spectra of non-selfadjoint operators over dynamical system | Siegfried Beckus, Daniel Lenz,Marko Lindner,Christian SeifertZeitschrift: Proceedings of the Edinburgh Mathematical SocietyReihe: 61Seiten: 371 -386Link zur Publikation , Link zum Preprint

Note on spectra of non-selfadjoint operators over dynamical system

Autoren: Siegfried Beckus, Daniel Lenz,Marko Lindner,Christian Seifert (2018)

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Zeitschrift:
Proceedings of the Edinburgh Mathematical Society
Reihe:
61
Seiten:
371 -386

2017 | On the spectrum of operator families on discrete groups over minimal dynamical Systems | Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian SeifertZeitschrift: Mathematische ZeitschriftReihe: 287Seiten: 993 - 1007Link zur Publikation , Link zum Preprint

On the spectrum of operator families on discrete groups over minimal dynamical Systems

Autoren: Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian Seifert (2017)

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in ℓp(ℤ). Here, we generalize this to a large class of bounded linear operator families on Banach-space valued ℓp-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

Zeitschrift:
Mathematische Zeitschrift
Reihe:
287
Seiten:
993 - 1007

2016 | Continuity of the Spectrum of a Field of Self-Adjoint Operators | Siegfried Beckus, Jean BellissardZeitschrift: Annales Henri PoincaréReihe: 17Seiten: 3425 - 3442Link zur Publikation , Link zum Preprint

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Autoren: Siegfried Beckus, Jean Bellissard (2016)

Given a family of self-adjoint operators (At)t∈T indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum σ(At) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T,d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.

Zeitschrift:
Annales Henri Poincaré
Reihe:
17
Seiten:
3425 - 3442

2013 | Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals | Siegfried Beckus, Felix PogorzelskiZeitschrift: Mathematical Physics, Analysis and GeometryReihe: 16Seiten: 289 -308Link zur Publikation , Link zum Preprint

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

Autoren: Siegfried Beckus, Felix Pogorzelski (2013)

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.

Zeitschrift:
Mathematical Physics, Analysis and Geometry
Reihe:
16
Seiten:
289 -308

CV

CV in english

Academic career

University of Potsdam since 10/2022 Senior Lecturer and researcher

University of Potsdam since 10/2018 - 09/2022 Post-Doc with Prof. Dr. Matthias Keller

Israel Institute of Technology (Technion), Haifa 10/2016-09/2018 Postdoctoral Fellowship with Prof. Dr. Yehuda Pinchover and Prof. Dr. Ram Band

Georgia Institute of Technology, Atlanta, USA 02-04/2014 Visiting Student/Research Collaborator

Friedrich Schiller University Jena 03/2012-09/2016 PhD Student

Education

PhD 10/2016, Friedrich Schiller University Jena

Diploma 02/2012, Friedrich Schiller University Jena

Grants and Projects

Marie-Weber-Grant 2022 "Aperiodische Ordnung: Die Entschlüsselung versteckter Strukturen" starting 09/2022 (PhD position for 1 year) - announcement by the Hans Böckler Stiftung

DFG Project "Periodic approximations of Schrödinger operators associated with quasicrystals" since 12/2018 (PostDoc position for 2 year and travel money)

ZiF Summer Camp 2020 Research stay at the Center of Interdisciplinary Research at the University Bielefeld, Bielefeld, Germany

Scholarship of the DAAD Program to participate at congresses, International Workshop on Operator Theory and its Applications, Lisbon, Portugal

Scholarship for "Research in Pairs" at the Mathematisches Forschungsinstitut Oberwolfach, Germany, 01/2018

Postdoctoral Fellowship Israel Institute of Technology (Technion), Haifa, Israel, 10/2016 - 09/2018

Scholarship of the DAAD Program to participate at congresses, XVIII International Congress on Mathematical Physics, Santiago de Chile, Chile

Funding for the PhD Seminar "Förderung von interdisziplinären Arbeitsgruppen und Nachwuchsnetzwerken" funded by the Graduierten-Akademie in Jena

(Co-)Supervision

Yannik Thomas (PhD thesis)

Yannik Thomas (MSc thesis), Non-Euclidean Estimates for Sturmian Hamiltonians, 2023

Lior Tennenbaum (PhD thesis), joint with Prof. Ram Band (Israel Institute of Technology)

Alberto Takase (PhD thesis, joint with Prof. Anton Gorodetski (University of California Irvine)

Franziska Sieron (Master thesis), The density of periodic configurations in strongly irreducible subshifts of finite type (joint with Prof. Dr. Daniel Lenz), 2016 

Daniel Sell (Master thesis), Topological groupoids and Matuis spatial realization theorem (joint with Prof. Dr. Daniel Lenz), 2015

Franziska Sieron (Bachelor thesis), The balanced property of primitive substitutions (joint with Prof. Dr. Daniel Lenz), 2014

(Co-)Organized Scientific meetings

DOGS - a series of joint workshops between research groups in Haifa, Jena, Leipzig and Potsdam, Link

Summer school on "Spectral theory and geometry of ergodic Schrödinger operators", Universität Potsdam, 07/2023-08/2023, Link

Workshop on "Functional Analysis, Operator Theory and Dynamical Systems",  Universität Potsdam, 09/2022, Link

Euler-Lecture, Universität Potsdam, 05/2019, 05/2021, 05/2022, 05/2023, 05/2024 Link

PhD seminar, Friedrich-Schiller-Universität Jena, 03/2013 – 09/2016

Colloquium: "Job opportunities for mathematicians", Friedrich-Schiller-Universität Jena, 2013 – 2015

PhD symposium at the TU Chemnitz, within the Fall school Dirichlet forms, operator theory and mathematical physics, 02/2013

    Talks

    Below you can find a selected list of talks I presented in the past including some slides or recordings.

    At international conferences

    • 07/2024 XXI International Congress on Mathematical Physics, Strasbourg (France): Slides
      The Dry Ten Martini Problem for Sturmian Schrödinger Operators
    • 05/2024 Workshop "Point sets: Dynamics, Sampling, and Operator Algebras", University of Oslo (Norway):
      Spectral convergence rates and the underlying dynamics
    • 08/2023 Workshop "Aspects of Aperiodic Order", MFO Oberwolfach (Germany):
      The Dry Ten Martini problem for Sturmian dynamical systems
    • 06/2022 Workshop "Ergodic Operators and Quantum Graphs", Simons Center for Geometry and Physics (USA)
      Spectral approximations beyond dimension one
    • 04/2022 Workshop "Almost-Periodic Spectral Problems", Banff International Research Station (Canada)
      The table and the chair: Spectral approximations beyond dimension one Link to the video
    • 01/2021 Workshop "Geometry, Dynamics and Spectrum of Operators on Discrete Spaces", MFO Oberwolfach (Germany):
      Primitive substitutions beyond abelian groups: The Heisenberg group
    • 07/2019 International Workshop on Operator Theory and its Applications (IWOTA 2019), Instituto Superior Técnico, Lisbon (Portugal):
      Hunting the spectra via the underlying dynamics
    • 05/2019 8th Miniworkshop on Operator Theoretic Aspects of Ergodic Theory, Leipzig (Germany):
      Hunting the spectra via the underlying dynamics
    • 01/2018 Hardy-type inequalities and elliptic PDEs, Midreshet Sde Boker (Israel), Poster:
      Spectral Approximation of Schrödinger Operators
    • 10/2017 Workshop "Spectral Structures and Topological Methods in Mathematical Quasicrystals", MFO Oberwolfach (Germany):
      Spectral stability of Schrödinger operators in the Hausdorff metric
    • 07/2017 Analysis and Geometry on Graphs and Manifolds, Universität-Potsdam (Germany):
      Shnol type Theorem for the Agmon ground state
    • 05/2017 Israel Mathematical Union 2017, Acre (Israel):
      The space of Delone dynamical systems and related objects
    • 01/2017 Workshop on Mathematical Physics, Weizmann Institute of Science, Rehovot (Israel), Poster:
      Spectral Approximation of Schrödinger Operators
    • 06/2016 Thematic School "Transversal Aspects of Tilings", Oleron (France):
      Continuity of the spectra associated with Schrödinger operators
    • 09/2015 CMO-BIRS, Workshop on "Spectral properties of quasicrystals via analysis, dynamics and geometric measure theory", Oaxaca (Mexiko):
      Spectral approximation of Schrödinger operators: continuity of the spectrum, Link to the video
    • 07/2015 Young researcher symposium, Pontificia Universidad Catolica de Chile, Santiago de Chile (Chile):
      Spectral study of Schrödinger operators with aperiodic ordered potential in one-dimensional systems
    • 06/2015 Workshop on "Time-frequency analysis and aperiodic order", Norwegian University of Science and Technology, Trondheim (Norway):
      An approximation theorem for the spectrum of Schrödinger operators related to quasicrystals

    In seminars and colloquia

    • 10/2024 The University of Glasgow (Great Britain):
      Dry Ten Martini Problem for Sturmian Dynamical Systems
    • 04/2023 Universität Münster (Germany):
      Dry Ten Martini Problem for Sturmian Dynamical Systems
    • 03/2023 Institute Foruier - Grenoble (France):
      Dry Ten Martini Problem for Sturmian Dynamical Systems
    • 02/2023 University of Potsdam (Germany) - Maria-Weber lecture
      Dry Ten Martini Problem for Sturmian Dynamical Systems
    • 04/2022 University of California - Irvine (USA):
      Sturmian dynamical systems and the Kohmoto butterfly
    • 01/2020 Universität Leipzig (Germany):
      When do the spectra of self-adjoint operators converge?
    • 05/2019 Justus-Liebig-Universität Gießen (Germany):
      Hunting the spectra via the underlying dynamics
    • 07/2018 Technische Universität München (Germany):
      When do the spectra of self-adjoint operators converge?
    • 10/2017 Pontificia Universidad Catolica de Chile, Santiago (Chile):
      Shnol type Theorem for the Agmon ground state
    • 10/2017 Hebrew University of Jerusalem (Israel):
      When do the spectra of self-adjoint operators converge?
    • 08/2017 University of Oslo (Norway):
      Spectral approximation via an approach from C*-algebras
    • 07/2017 Friedrich-Alexander Universität Erlangen-Nürnberg (Germany):
      The space of Delone dynamical systems and its application
    • 07/2017 RWTH Aachen (Germany):
      Shnol type Theorem for the Agmon ground state
    • 09/2016 Aalborg University (Danemark):
      Continuous variation of the spectra: A characterization and a tool
    • 07/2016 Universität Bielefeld (Germany):
      Hölder-continuous behavior of the spectra associated with self-adjoint operators
    • 05/2015 Technische Universität Chemnitz (Germany):
      Schrödinger operators on quasicrystals
    • 04/2015 Israel Institute of Technology (Technion), Haifa (Israel):
      The role of Gähler-Anderson-Putnam graphs in the view of Schrödinger operators
    • 04/2014 University of Alabama at Birmingham (USA):
      Gähler-Anderson-Putnam graphs of 1-dimensional Delone sets of finite local complexity
    • 01/2014 Technische Universität Hamburg-Harburg (Germany):
      Wannier transformation for Schrödinger operators with aperiodic potential

    Teaching

    Autumn semester 2025

    • Exercises: Mathematik für Wirtschaftswissenschaften
    • Exercise: Functional Analysis 1

    Summer semester 2024

    • Lecture and Exercise: Aperiodische Ordnung (Moodle course)
    • Erweitertes Fachwissen für den schulischen Kontext in Mathematik (Moodle course)
    • Exercise: Functional Analysis 2

    Autumn semester 2023

    • Exercise: Functional Analysis 1
    • Exercise and Tutorial: Mathematik für Wirtschaftsinformatiker

    Summer semester 2023

    Autumn semester 2022

    • Exercise: Mathematik für Physik 3
    • Lecture and Exercise: Functional Analysis 1

    Summer semester 2022

    • Exercise: Mathematik für Physik 2
    • Seminar: Random operators

    Autumn semester 2021

    • Exercise: Mathematik für Physik 1
    • Exercise and Tutorial: Mathematik für Wirtschaftsinformatiker
    • Seminar: Aperiodische Ordnung / Aperiodic Order

    Summer semester 2021

    Autumn semester 2020

    Summer semester 2020

    Autumn semester 2019

    Summer semester 2019

    Autumn semester 2018

    Anschrift

    Universität Potsdam
    Institut für Mathematik
    Campus Golm, Haus 9
    Karl-Liebknecht-Straße 24-25
    14476 Potsdam