Über mich

Kristina Schubert

Mathematikerin, Stochastikerin

 

Forschungsthemen

Theorie zufälliger Matrizen: invariante Ensembles und Matrizen mit abhängigen Einträgen, repulsive Teilchensysteme

Phasenübergänge in Spinsystemen: Ising Modelle mit Blockstruktur und/oder zufälligen Interaktionen

Anschrift

Dr. Kristina Schubert
Technische Universität Dortmund
Fakultät für Mathematik
Lehrstuhl LSIV
Vogelpothsweg 87
44227 Dortmund
Raum M 625

Veröffentlichungen

Preprints

    Veröffentlichte Artikel

    • Löwe, Matthias; Schubert, Kristina; Vermet, Franck (2020): „Multi-group Binary Choice with Social Interaction and a Random Communication Structure - a Random Graph Approach“. In: Physica A: Statistical Mechanics and its Applications. 556 , p. 124735, doi: 10.1016/j.physa.2020.124735.

      We construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size. We concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups. Given that the resulting graph is sufficiently dense we show that, with probability one, the average decision in each of the two groups is the same as in the fully connected model. In particular, we show that there is a phase transition: If the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated. We also compute the free energy per particle in our model.

      @article{Ising_binary_choice,
        author = {Löwe, Matthias and Schubert, Kristina and Vermet, Franck},
        title = {Multi-group Binary Choice with Social Interaction and a Random Communication Structure - a Random Graph Approach},
        volume = {556},
        pages = {124735},
        year = {2020},
        month = oct,
        day = {15},
        issn = {0378-4371},
        doi = {10.1016/j.physa.2020.124735},
        journal = {Physica A: Statistical Mechanics and its Applications},
        archiveprefix = {arXiv},
        eprint = {1904.11890}
      }
      
    • Kabluchko, Zakhar; Löwe, Matthias; Schubert, Kristina (2020): „Fluctuations of the Magnetization for Ising Models on Erdős-Rényi Random Graphs – the Regimes of Small p and the Critical Temperature“. In: Journal of Physics A: Mathematical and Theoretical. IOP Publishing 53 (35), p. 355004, doi: 10.1088/1751-8121/aba05f.

      We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph. This graph is constructed on N vertices and every edge has probability p to be present. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993] and analyzed by the authors in a previous note, in which we consider the case of p=p(N) satisfying p^3N^2 \to ∞ and β<1 . In the current note we prove a quenched Central Limit Theorem for the magnetization for p satisfying pN \to ∞ in the high-temperature regime β<1 . We also show a non-standard Central Limit Theorem for p^4N^3 \to ∞ at the critical temperature β=1 . For p^4N^3 \to 0 we obtain a Gaussian limiting distribution for the magnetization. Finally, on the critical line p^4N^3 \to c the limiting distribution for the magnetization contains a quadratic component as well as a x^4 -term. Hence, at β=1 we observe a phase transition in p for the fluctuations of the magnetization.

      @article{kabluchko2019fluctuations,
        title = {Fluctuations of the Magnetization for Ising Models on Erdős-Rényi Random Graphs -- the Regimes of Small p and the Critical Temperature},
        author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
        doi = {10.1088/1751-8121/aba05f},
        year = {2020},
        month = aug,
        publisher = {{IOP} Publishing},
        volume = {53},
        number = {35},
        pages = {355004},
        journal = {Journal of Physics A: Mathematical and Theoretical},
        archiveprefix = {arXiv},
        eprint = {1911.10624}
      }
      
    • Kabluchko, Zakhar; Löwe, Matthias; Schubert, Kristina (2020): „Fluctuations for the partition function of Ising models on Erdös-Rényi random graphs“. In: Ann. Inst. H. Poincaré Probab. Statist. (submitted; in revision). Institut Henri Poincaré.

      We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with N vertices and edge probability p=p(N) that were introduced by Bovier and Gayrard [J. Statist. Phys., 72(3-4):643–664, 1993] and investigated in two previous articles by the authors. We prove Central Limit Theorems for the partition function of the model and – at other decay regimes of p(N) – for the logarithmic partition function. We find critical regimes for p(N) at which the behavior of the fluctuations of the partition function changes

      @article{kabluchko2020fluctuations,
        title = {Fluctuations for the partition function of Ising models on Erdös-Rényi random graphs},
        author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
        journal = {Ann. Inst. H. Poincaré Probab. Statist. (submitted; in revision)},
        publisher = {Institut Henri Poincaré},
        year = {2020},
        eprint = {2002.11372},
        archiveprefix = {arXiv}
      }
      
    • Löwe, Matthias; Schubert, Kristina (2020): „Exact recovery in block spin Ising models at the critical line“. In: Electron. J. Statist. The Institute of Mathematical Statistics and the Bernoulli Society 14 (1), pp. 1796–1815, doi: 10.1214/20-EJS1703.

      We show how to exactly reconstruct the block structure at the critical line in the so-called Ising block model. This model was re-introduced by Berthet, Rigollet and Srivastava in a recent paper. There the authors show how to exactly reconstruct blocks away from the critical line and they give an upper and a lower bound on the number of observations one needs; thereby they establish a minimax optimal rate (up to constants). Our technique relies on a combination of their methods with fluctuation results for block spin Ising models. The latter are extended to the full critical regime. We find that the number of necessary observations depends on whether the interaction parameter between two blocks is positive or negative: In the first case, there are about N \log N observations required to exactly recover the block structure, while in the latter \sqrt N \log N observations suffice.

      @article{Ising_exact_recovery,
        author = {Löwe, Matthias and Schubert, Kristina},
        doi = {10.1214/20-EJS1703},
        fjournal = {Electronic Journal of Statistics},
        journal = {Electron. J. Statist.},
        number = {1},
        pages = {1796--1815},
        publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
        title = {Exact recovery in block spin Ising models at the critical line},
        volume = {14},
        year = {2020},
        eprinttype = {arxiv},
        eprint = {1906.00021}
      }
      
    • Knöpfel, Holger; Löwe, Matthias; Schubert, Kristina; et al. (2020): „Fluctuation Results for General Block Spin Ising Models“. In: J. Stat. Phys. 178 (5), pp. 1175–1200, doi: 10.1007/s10955-020-02489-0.

      We study a block spin mean-field Ising model, i.e. a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices, and where pair interactions are given according to their blocks. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein’s method we establish a rate of convergence in the Central Limit Theorem for the block magnetization vector in the high temperature regime.

      @article{loeweSinulis,
        author = {Knöpfel, Holger and Löwe, Matthias and Schubert, Kristina and Sinulis, Arthur},
        title = {Fluctuation Results for General Block Spin Ising Models},
        journal = {J. Stat. Phys.},
        fjournal = {Journal of Statistical Physics},
        volume = {178},
        year = {2020},
        number = {5},
        pages = {1175--1200},
        issn = {0022-4715},
        mrclass = {60F05 (60F10 82B20)},
        mrnumber = {4081224},
        doi = {10.1007/s10955-020-02489-0},
        eprinttype = {arxiv},
        eprint = {1902.02080}
      }
      
    • Kabluchko, Zakhar; Löwe, Matthias; Schubert, Kristina (2019): „Fluctuations of the Magnetization for Ising Models on Dense Erdős–Rényi Random Graphs“. In: Journal of Statistical Physics. doi: 10.1007/s10955-019-02358-5.

      We analyze Ising/Curie-Weiss models on the (directed) Erdős-Rényi random graph on N vertices in which every edge is present with probability p . These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime β<1 when p=p(N) satisfies p^3N^2 \to ∞ .

      @article{Ising_random_interaction1,
        author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
        year = {2019},
        title = {Fluctuations of the Magnetization for Ising Models on Dense Erdős–Rényi Random Graphs},
        journal = {Journal of Statistical Physics},
        issn = {1572-9613},
        doi = {10.1007/s10955-019-02358-5},
        eprinttype = {arxiv},
        eprint = {1905.12326}
      }
      
    • Löwe, Matthias; Schubert, Kristina (2019): „On the Limiting Spectral Density of Random Matrices filled with Stochastic Processes“. In: Random Operators and Stochastic Equations. 27 (2), doi: 10.1515/rose-2019-2008.

      We discuss the limiting spectral density of real symmetric random matrices. Other than in standard random matrix theory the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process, which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

      @article{LS,
        author = {Löwe, Matthias and Schubert, Kristina},
        title = {On the Limiting Spectral Density of Random Matrices filled with Stochastic Processes},
        journal = {Random Operators and Stochastic Equations},
        eprinttype = {arxiv},
        eprint = {1512.02498},
        volume = {27},
        year = {2019},
        number = {2},
        doi = {10.1515/rose-2019-2008}
      }
      
    • Löwe, Matthias; Schubert, Kristina (2018): „Fluctuations for block spin Ising models“. In: Electron. Commun. Probab. The Institute of Mathematical Statistics and the Bernoulli Society 23 , pp. 12 pp., doi: 10.1214/18-ECP161.

      We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet, Rigollet and Srivastava. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we show the validity of a non-standard Central Limit Theorems for the magnetization.

      @article{loewe2018,
        author = {Löwe, Matthias and Schubert, Kristina},
        doi = {10.1214/18-ECP161},
        fjournal = {Electronic Communications in Probability},
        journal = {Electron. Commun. Probab.},
        pages = {12 pp.},
        pno = {53},
        publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
        title = {Fluctuations for block spin Ising models},
        volume = {23},
        year = {2018},
        eprinttype = {arxiv},
        eprint = {1806.06000}
      }
      
    • Schubert, Kristina (2016): „Spectral Density for Random Matrices with Independent Skew-Diagonals“. In: Electron. Commun. Probab. The Institute of Mathematical Statistics and the Bernoulli Society 21 , pp. 12 pp., doi: 10.1214/16-ECP3.

      We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same skew-diagonal and we distinguish between two types of such correlations, a rather weak and a rather strong one. For weak correlations the limiting distribution is Wigner’s semi-circle distribution; for strong correlations it is the free convolution of the semi-circle distribution and the limiting distribution for random Hankel matrices.

      @article{Schubert_Hankel,
        author = {Schubert, Kristina},
        title = {Spectral Density for Random Matrices with Independent Skew-Diagonals},
        doi = {10.1214/16-ECP3},
        fjournal = {Electronic Communications in Probability},
        journal = {Electron. Commun. Probab.},
        pages = {12 pp.},
        pno = {40},
        publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
        volume = {21},
        year = {2016},
        eprinttype = {arxiv},
        eprint = {1510.06448}
      }
      
    • Schubert, Kristina (2015): „Spacings in orthogonal and symplectic random matrix ensembles“. In: Constr. Approx. 42 (3), pp. 481–518, doi: 10.1007/s00365-015-9274-6.

      We consider the universality of the nearest neighbour eigenvalue spacing distribution in invariant random matrix ensembles. Focussing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution converges in a uniform way. More precisely, the main result states that the expected Kolmogorov distance of the empirical spacing distribution from its universal limit converges to zero as the matrix size tends to infinity.

      @article{MR3416165,
        author = {Schubert, Kristina},
        title = {Spacings in orthogonal and symplectic random matrix ensembles},
        journal = {Constr. Approx.},
        fjournal = {Constructive Approximation. An International Journal for
          Approximations and Expansions},
        volume = {42},
        year = {2015},
        number = {3},
        pages = {481--518},
        issn = {0176-4276},
        mrclass = {15B52},
        mrnumber = {3416165},
        doi = {10.1007/s00365-015-9274-6},
        eprinttype = {arxiv},
        eprint = {1501.05637}
      }
      
    • Schubert, Kristina; Venker, Martin (2015): „Empirical spacings of unfolded eigenvalues“. In: Electron. J. Probab. 20 , pp. Paper No. 120, 37, doi: 10.1214/EJP.v20-4436.

      We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that the density of particles is asymptotically constant. This allows to consider all empirical spacings, where previous results were restricted to a tiny fraction of the particles. Moreover, we prove bounds on the rates of convergence. The main ingredient for the proof, a strong bulk universality result for correlation functions in the unfolded setting including optimal rates, should be of independent interest.

      @article{MR3425540,
        author = {Schubert, Kristina and Venker, Martin},
        title = {Empirical spacings of unfolded eigenvalues},
        journal = {Electron. J. Probab.},
        fjournal = {Electronic Journal of Probability},
        volume = {20},
        year = {2015},
        pages = {Paper No. 120, 37},
        issn = {1083-6489},
        mrclass = {60B20 (82C22)},
        mrnumber = {3425540},
        eprinttype = {arxiv},
        eprint = {1505.07664},
        doi = {10.1214/EJP.v20-4436}
      }
      
    • Kriecherbauer, Thomas; Schubert, Kristina; Schüler, Katharina; et al. (2015): „Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory“. In: Markov Process. Related Fields. 21 , pp. 639–694.

      The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the Christoffel-Darboux kernel [59] become the key for studying universality. In this paper the existing results on the CD-kernel will be extended in two directions. Firstly, in order to analyze the transition from the universal to the non-universal regime, we provide leading order asymptotics that are global rather than local. This allows e.g. to describe the moderate deviations for the largest eigenvalues of unitary ensembles ( β= 2 ), where such a transition occurs. Secondly, our asymptotics will be uniform under perturbations of the probability measure that defines the matrix ensemble. Such information is useful for the analysis of a different type of ensembles [25], which is not known to be determinantal and for which the method of orthogonal polynomials cannot be used directly. The just described applications of our results are formulated in this paper but will be proved elsewhere. As a byproduct of our analysis we derive first order corrections for the 1-point correlation functions of unitary ensembles in the bulk. Our proofs are based on the nonlinear steepest descent method [20]. They follow closely [17] and incorporate improvements introduced in [36, 64]. The presentation is self-contained except for a number of general facts from Random Matrix theory and from the theory of singular integral operators.

      @article{KSSV,
        author = {Kriecherbauer, Thomas and Schubert, Kristina and Schüler, Katharina and Venker, Martin},
        title = {Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory},
        journal = {Markov Process. Related Fields},
        eprinttype = {arxiv},
        eprint = {1401.6772},
        volume = {21},
        year = {2015},
        pages = {639--694},
        url = {http://math-mprf.org/journal/articles/id1379/}
      }
      
    • Kriecherbauer, Thomas; Schubert, Kristina (2013): „Spacings: an example for universality in random matrix theory“. In: Random matrices and iterated random functions. Springer, Heidelberg (Springer Proc. Math. Stat.), pp. 45–71, doi: 10.1007/978-3-642-38806-4_3.

      Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the empirical spacing distribution and its Kolmogorov distance from the universal limit. We describe new results, some analytical, some numerical, that are contained in [27]. A large part of the paper is devoted to explain basic definitions and facts of Random Matrix Theory, culminating in a sketch of the proof of a weak version of convergence for the empirical spacing distribution.

      @incollection{MR3095193,
        author = {Kriecherbauer, Thomas and Schubert, Kristina},
        title = {Spacings: an example for universality in random matrix theory},
        booktitle = {Random matrices and iterated random functions},
        series = {Springer Proc. Math. Stat.},
        volume = {53},
        pages = {45--71},
        publisher = {Springer, Heidelberg},
        year = {2013},
        mrclass = {60B20},
        mrnumber = {3095193},
        doi = {10.1007/978-3-642-38806-4_3},
        eprinttype = {arxiv},
        eprint = {1510.06597}
      }
      

    Abschlussarbeiten

    • Schubert, Kristina (2012): „On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles“. (Dissertation) Germany: Ruhr-Universität Bochum.

      One of the most striking phenomena of Random Matrix Theory is that several of their local eigenvalue statistics define universal distributions in the limit of large matrix dimensions. In the first part of this thesis we focus on invariant matrix ensembles including the classical Gaussian ensembles GOE, GUE and GSE. In the main theorem we consider the empirical spacing distribution and show that the expected Kolmogorov distance from the universal limit converges to zero as the matrix size tends to infinity. In the second part of this thesis we present numerical experiments to determine rates of convergence for general beta Hermite and Wigner ensembles.

      @thesis{Diss,
        author = {Schubert, Kristina},
        title = {On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles},
        school = {Ruhr-Universität Bochum},
        address = {Germany},
        type = {Dissertation},
        year = {2012},
        url = {https://hss-opus.ub.ruhr-uni-bochum.de/opus4/frontdoor/index/index/year/2019/docId/3280}
      }
      
    • Schubert, Kristina (2007): „Über die Abweichung der Abstandsverteilung benachbarter Eigenwerte zufälliger Matrizen vom universellen Gesetz“. (Diplomarbeit) Germany: Ruhr-Universität Bochum.
      @thesis{Dipl,
        author = {Schubert, Kristina},
        title = {Über die Abweichung der Abstandsverteilung benachbarter Eigenwerte zufälliger Matrizen vom universellen Gesetz},
        school = {Ruhr-Universität Bochum},
        address = {Germany},
        type = {Diplomarbeit},
        year = {2007},
        url = {https://www.uni-muenster.de/Stochastik/mitarbeiter/schubert/dipl.pdf}
      }
      

    Curriculum Vitæ

    Vertretung Professur (W2) „Mathematische Methoden der Datenanalyse“

    Fachbereich Mathematik/Informatik

    since 10/2020

    Universität Osnabrück

    Akademische Rätin

    Fakultät für Mathematik, Lehrstuhl für Stochastik und Analysis

    since 10/2018

    Technische Universität Dortmund

    Wissenschaftliche Mitarbeiterin

    Institut für Mathematische Stochastik

    04/2013–09/2018

    Westfälische Wilhelms-Universität Münster

    Wissenschaftliche Mitarbeiterin

    Mathematisches Institut, Lehrstuhl Nichtlineare Analysis und Mathematische Physik

    04/2011–03/2013

    Universität Bayreuth

    Promotion

    Fakultät für Mathematik. Dissertation: On the Convergence of the Nearest Neighbour Eigenvalue Spacing Distribution for Orthogonal and Symplectic Ensembles

    06/2012

    Ruhr-Universität Bochum

    Wissenschaftliche Mitarbeiterin

    Fakultät für Mathematik, Lehrstuhl für Analysis

    07/2007–03/2011

    Ruhr-Universität Bochum

    Diplom Mathematik

    Fakultät für Mathematik. Diplomarbeit: Über die Abweichung der Abstandsverteilung benachbarter Eigenwerte zufälliger Matrizen vom universellen Gesetz

    04/2007

    Ruhr-Universität Bochum

    Ausgewählte Vorträge

    2020

    Random Matrix and Probability Theory Seminar, Harvard University, Cambridge, Massachusetts

    2019

    Minikurs Konzentrationsungleichungen (4× 90 Min.), Sommerschule des GRK 2131, Lingen

    Seminar Stochastik und Geometrie, Ruhr-Universität Bochum

    Seminar des GRK 2131, Technische Universität Dortmund

    2018

    Oberseminar Stochastik, Universität Osnabrück

    Oberseminar Stochastik und Analysis, Technische Universität Dortmund

    Workshop Women in Probability 2018, Technische Universität München

    2017

    Séminaire Géométrie, Physique et Probabilités, Université catholique de Louvain, Louvain-la-Neuve, Belgien

    2016

    Seminar Random Matrices, Universität Bielefeld

    Seminar Classical Analysis, Katholieke Universiteit Leuven, Belgien

    2015

    Summer-School on probability and mathematical physics, SFB/TR 12, Oggebbio/Ghiffa, Italien

    2013

    Oberseminar Mathematische Stochastik, Westfälische Wilhelms-Universität Münster

    2012

    Summer-School on probability and mathematical physics, SFB/TR 12, Oggebbio/Ghiffa, Italien

    DMV Tagung, Minisymposium Free probability theory and random matrices, Universität des Saarlandes, Saarbrücken

    2011

    Conference on Random Matrix Theory and Applications in Theoretical Sciences (Poster), Universität Bielefeld

    2010

    Summer-School on probability and mathematical physics, SFB/TR 12, Oggebbio/Ghiffa, Italien

    2007

    Summer-School on probability and mathematical physics, SFB/TR 12, Oggebbio/Ghiffa, Italien

    Lehre

    WS 2020/21

    Vorlesung Multivariate Statistik, Universität Osnabrück

    Vorlesung Konzentrationsungleichungen, Universität Osnabrück

    SS 2020

    Vorlesung Konzentrationsungleichungen (4-std. Vorl. mit 2-std. Üb.), Technische Universität Dortmund

    WS 2019/20

    Seminar Statistik, Technische Universität Dortmund

    Tutorübung Analysis III, Technische Universität Dortmund

    SS 2019

    Globalübung, Tutorübung Analysis II, Technische Universität Dortmund

    WS 2018/19

    Globalübung Analysis I, Technische Universität Dortmund

    Masterseminar Markov-Ketten, Technische Universität Dortmund

    SS 2018

    Koordination des Übungsbetriebs Wahrscheinlichkeitstheorie, Westfälische Wilhelms-Universität Münster

    WS 2017/18

    Masterseminar Random Walks in Random Environment, Westfälische Wilhelms-Universität Münster

    SS 2017

    Vorlesung Konzentrationsungleichungen (4-std. Vorl. mit 2-std. Üb.), Westfälische Wilhelms-Universität Münster

    Übung Konzentrationsungleichungen, Westfälische Wilhelms-Universität Münster

    WS 2016/17

    Übung Wahrscheinlichkeitstheorie II, Westfälische Wilhelms-Universität Münster

    SS 2016

    Übung Höhere Finanzmathematik, Westfälische Wilhelms-Universität Münster

    WS 2015/16

    Übung Stochastische Analysis, Westfälische Wilhelms-Universität Münster

    SS 2015

    Übung Höhere Finanzmathematik, Westfälische Wilhelms-Universität Münster

    Bachelorseminar Zufällige Graphen, Westfälische Wilhelms-Universität Münster

    WS 2014/15

    Übung Stochastische Analysis, Westfälische Wilhelms-Universität Münster

    SS 2014

    Übung Wahrscheinlichkeitstheorie I, Westfälische Wilhelms-Universität Münster

    WS 2013/14

    Bachelorseminar Wahrscheinlichkeitstheorie, Westfälische Wilhelms-Universität Münster

    SS 2013

    Masterseminar Extremwerttheorie, Westfälische Wilhelms-Universität Münster

    WS 2012/13

    Übung Analysis I, Universität Bayreuth

    SS 2012

    Übung Einführung in die partiellen Differentialgleichungen, Universität Bayreuth

    WS 2011/12

    Übung Analysis II, Universität Bayreuth

    SS 2011

    Übung Analysis I, Universität Bayreuth

    WS 2010/11

    Übung Analysis I, Ruhr-Universität Bochum

    SS 2010

    Übung Funktionalanalysis, Ruhr-Universität Bochum

    WS 2009/10

    Übung Mathematik für Maschinenbau-, Bauing. und Umwelttechniker I, Ruhr-Universität Bochum

    SS 2009

    Übung Mathematik für Physiker II, Ruhr-Universität Bochum

    WS 2008/09

    Übung Mathematik für Physiker I, Ruhr-Universität Bochum

    SS 2008

    Übung Analysis II, Ruhr-Universität Bochum

    WS 2007/08

    Übung Analysis I, Ruhr-Universität Bochum

    SS 2007

    Übung Mathematik für Elektrotechniker, Ruhr-Universität Bochum

    WS 2006/07

    Übung Mathematik für Maschinenbau-, Bauing. und Umwelttechniker III, Ruhr-Universität Bochum

    SS 2006

    Übung Mathematik für Maschinenbau-, Bauing. und Umwelttechniker II, Ruhr-Universität Bochum

    WS 2005/06

    Übung Mathematik für Maschinenbau-, Bauing. und Umwelttechniker I, Ruhr-Universität Bochum