Torben KrügerTorben Krüger

Associate Professor

Department of Mathematical Sciences
Universitetsparken 5, 2100 København Ø, 04/E, Building: 4.2.12 
tlk@math.ku.dk 
https://www.math.ku.dk/ 
https://qmath.ku.dk/ 

  • Presentation

  • Research

  • Publications

  • CV

About me

Starting July 01, 2020, I will be an associate professor at the Department of Mathematical Sciences of Copenhagen University. Currently, I am working at the Institute for Applied Mathematics at the University of Bonn.

The main focus of my research lies on the areas of random matrix theory (RMT) and mathematical physics. I apply and develop robust mathematical methods in order to rigorously study statistical properties of large complex systems that typically originate from applications in physics, statistics or engineering.

In Copenhagen I will establish a new research group dedicated to the study of disordered quantum systems through the use of RMT. The project title is "Random Matrix Approach to Universality Phenomena in Disordered Quantum Systems" and the project is funded by The Villum Young Investigator Programme (see announcement here).

Research

Currently, I am working on the following topics:

  • Local laws and spectral universality for a wide class of random matrices. 
  • Spectral properties of non-Hermitian random matrix models.
  • Applications of random matrices in physics, neural networks, communication theory and statistics.
  • Operator-valued free probability
  • Fluctuations in specific zero sum games. 
  • Connection between random matrices and random Schrödinger operators in the weak coupling regime.
  • Dyson-Brownian motion
  • Supersymmetric methods in random matrix theory and other disordered systems
  • Quantum diffusion
  • Quantum transport and conductance in disordered media
  • Randomly coupled differential equations

Publications and preprints

21. Scattering in quantum dots via noncommutative rational functions
with L. Erdős and Yu. Nemish 
preprint, arXiv:1911.05112, 2019
20.

Randomly coupled differential equations with correlations
with L. Erdős and D. Renfrew
preprint, arXiv:1908.05178, 2019
19. Spectral radius of random matrices with independent entries
with J. Alt and L. Erdős
preprint, arXiv:1907.13631, 2019 
18. Cusp Universality for Random Matrices II: The Real Symmetric Case
with G. Cipolloni, L. Erdős and D. Schröder 
to appear in Pure and Applied Analysis, 2019 
17.

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
with L. Erdős and D. Schröder 
to appear in Comm. Math. Phys., 2018
16. Local laws for polynomials of Wigner matrices
with L. Erdős and Yu. Nemish 
to appear in J. Funct. Anal., 2018
15. The Dyson equation with linear self-energy: spectral bands, edges and cusps
with J. Alt and 
L. Erdős
preprint, arXiv:1804.07752, 2018
14. Correlated Random Matrices: Band Rigidity and Edge Universality
with J. Alt, 
L. Erdős and D. Schröder
to appear in Ann. Probab., 2018
13. Power law decay for systems of randomly coupled differential equations
with L. Erdős and D. Renfrew
SIAM Journal on Mathematical Analysis, 50(3):3271–3290, 2018
 12. Location of the spectrum of Kronecker random matrices
with J. Alt, L. Erdős and Y. Nemish
Ann. Inst. Henri Poincaré Probab. Stat., 55(2):661–696, 2019
 11. Random Matrices with Slow Correlation Decay
with 
L. Erdős and D. Schröder
Forum of Mathematics, Sigma, 7:e8, 2019
 10.
Local inhomogeneous circular law
with J. Alt and L. Erdős
Ann. Appl. Probab., Vol. 28, No. 1, 148-203, 2016
 9. Local law for random Gram matrices
with J. Alt and L. Erdős
Electron. J. Probab. 22, no. 25, 41 pp., 2017
 8. Stability of the Matrix Dyson Equation and Random Matrices with Correlations
O. Ajanki and L. Erdős
Probab. Theory Related Fields, 173(1):293–373, 2019
 7. Local Spectral Statistics of Gaussian Matrices with Correlated Entries 
with O. Ajanki and L. Erdős
J. Stat. Phys., 163(2):280–302, 2016
 6. Singularities of solutions to quadratic vector equations on complex upper half-plane
with O. Ajanki and L. Erdős
Comm. Pure Appl. Math., 
70(9):1672–1705, 2017
 5. Universality for general Wigner-type matrices
with O. Ajanki and L. Erdős 
Probab. Theory Related Fields, 169(3):667-727, 2017
 4. Quadratic vector equations on complex upper half-plane
with O. Ajanki and L. Erdős
to appear in Mem. Amer. Math. Soc., 2015
 3. Evolutionary games of condensates in coupled birth–death processes
with J. Knebel, M. F. Weber and E. Frey 
Nat Comms 6:6977, 2015
 2. Local semicircle law with imprimitive variance matrix
with O. Ajanki and L. Erdős
Electron. Commun. Probab. 19, no. 33, 1–9, 2014
 1. Coexistence and Survival in Conservative Lotka-Volterra Networks
with J. Knebel, M. F. Weber and E. Frey
Phys Rev Lett, vol. 110 (16) pp. 168106-5, 2013
My papers on arXiv.

Brief Curriculum Vitae:

Employment

From July 1, 2020 Associate Professor at University of Copenhagen
2017 - 2020 Postdoctoral researcher at University of Bonn
2015 - 2017 Postdoctoral researcher at IST Austria
2014 - 2015 Research assistant at IST Austria
2012 - 2014 Research assistant at Ludwig-Maximilians-University Munich (LMU)

Education

2015    Ph.D. in mathematics at LMU Munich
2013  Diploma in physics at LMU Munich
2012  Diploma in mathematics at LMU Munich

Teaching

 2019

Graduate seminar on Supersymmetry at HCM Bonn
 2019

Lecture series at Heidelberg Graduate Days
 2018 - 2019 Graduate seminar on Random Matrices, Spin Glasses and Deep Learning at HCM Bonn

 2018

Graduate seminar on Dyson Brownian motion at HCM Bonn
 2017

Teaching assistant at PCMI 2017
 2014 - 2016 Teaching assistant for lectures at IST Austria, including 'Random Matrices' and 'Random Schrödinger Operators'
 2009 - 2011 Crash courses in 'Stochastics' and 'Probability Theory' at LMU Munich

 2008 - 2013 Teaching assistant for various lectures at LMU Munich, including 'Stochastic Processes', 'Probability Theory', 'Mathematical Quantum Mechanics' and 'Functional Analysis'

Selected Talks

 2019 Plenary Speaker at QMath14 (Aarhus, Denmark)
 2018 Vilnius IMS-2018 (Vilnius, Lithuania)
 2018 Sums & Products of Random Matrices (Bielefeld, Germany)
 2017 PCMI (Park-City, USA)
 2017 QMath Kick-off (Copenhagen, Denmark)
 2016 Methods of Modern Mathematical Physics (Toronto, Canada)
 2015 ICMP (Santiago, Chile)
 2015 Think & Drink talk at IST Austria (Klosterneuburg, Austria)
 2014 SFB TR12 Workshop (Bad Honnef, Germany)

Honors & Awards

2020 Villum Young Investigator Award (9.3 Mio. DKK)
2017 Carathéodory prize for best dissertation
2006 Student scholarship German National Academic Foundation

Thesis

 2015 Local spectral universality for random matrices with independent entries (PhD Thesis)

 2012 Formulierung der Theorie des zweitquantisierten elektromagnetischen Feldes im Schrödingerbild (The external current problem for the second quantized electromagnetic field in the Schrödinger picture) (Physics Diploma Thesis)
 2011 Konstruktion der unitären Dynamik eines Photonenfeldes in L2-Räumen über Distributionen (Construction of unitary dynamics for the photon field in L2-spaces on distributions) (Mathematics Diploma Thesis)
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--- Last updated on January 23, 2020 ---