# Douglas Lundholm

## Researcher

**Now at Uppsala University as a member of the group Analysis, PDEs and Applications.**

During the summer term 2019 I was a professor (W3, Vertretung for Christian Hainzl) at LMU Munich.

Recipient of a Göran Gustafsson Prize for Young Researchers, 2018.

Funded by a Swedish Research Council Junior Researcher Project Grant for the period 2014-2017/18 with my project "Spectral theory of quantum systems with exotic symmetries".

During the spring 2013 I was a CARMIN postdoc visiting:

IHÉS, January - April, 2013.

IHP, April - July, 2013, for the trimester program Variational and Spectral

Methods in Quantum Mechanics.

During the fall 2012 I was a postdoc at Institut Mittag-Leffler in the program Hamiltonians in Magnetic Fields.

During 2011-2012 I was an EPDI laureate visiting:

IHÉS, September - October, 2011.

FIM, ETH Zurich, November - December, 2011, and April - August, 2012.

INIMS Cambridge, January - March, 2012.

During 2010-2011 I was a postdoc at the University of Copenhagen.

Address: |
Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden |

Visiting address: |
Room 3646, Floor 6, Lindstedtsv. 25, KTH |

Phone: | +46 8 790 7504 |

Fax: | +46 8 723 1788 |

Department phone: | +46 8 790 6000 |

E-mail: | dogge (ª) math.kth.se |

**Research interests**

Mathematical physics, including: spectral theory of quantum mechanical systems (typically incolving supersymmetry - such as supermembrane matrix models, or exotic particle statistics - anyons), Clifford (geometric) algebras and their applications, quantum gravity and quantum geometry.

Click here to read more about my research and find some ideas for student projects. I encourage any interested students to contact me if you are looking for a suitable project.

**Curriculum vitae:**PDF

**Preprints and publications**

A generalized Liouville equation and magnetic stability, (with A. Ataei and D.-T. Nguyen), April 2024. |

"The six blind men and the elephant": An interdisciplinary selection of measurement features, (with A. Ellingsen and J.-P. Magnot), October 2023. Proceedings of the XL Workshop on Geometric Methods in Physics, Bialowieza, 2023. |

Properties of 2D anyon gas, Invited contribution to the Encyclopedia of Condensed Matter Physics, 2nd edition, 2023 (edited by T. Chakraborty). [DOI]. |

Quantum statistics transmutation via magnetic flux attachment, (with G. Lambert and N. Rougerie), Prob. Math. Phys. 4 (2023) 803-848 [DOI]. |

Emergence of Anyons on the Two-Sphere in Molecular Impurities, (with M. Brooks, M. Lemeshko and E. Yakaboylu), Atoms 9 (2021) 106 [DOI]. Also in Physics of Impurities in Quantum Gases. |

Exchange and exclusion in the non-abelian anyon gas, (with V. Qvarfordt), September 2020. |

Molecular Impurities as a Realization of Anyons on the Two-Sphere, (with M. Brooks, M. Lemeshko and E. Yakaboylu), Phys. Rev. Lett. 126 (2021), 015301 [DOI]. |

A Quantum Impurity Model for Anyons, (with E. Yakaboylu, A. Ghazaryan, N. Rougerie, M. Lemeshko and R. Seiringer), Phys. Rev. B. 102 (2020) 144109 (Editors' Suggestion) [DOI]. |

Vortex patterns in the almost-bosonic anyon gas, (with M. Correggi, R. Duboscq and N. Rougerie), EPL 126 (2019) 20005 [DOI]. |

Lieb-Thirring inequalities for wave functions vanishing on the diagonal set, (with S. Larson and P. T. Nam), Ann. Henri Lebesgue 4 (2021), 251-282 [DOI]. |

Methods of modern mathematical physics: Uncertainty and exclusion principles in quantum mechanics, lecture notes for a master-level course given at KTH in 2017, revised May 2018, last revision 2019 LMU. |

Fermionic behavior of ideal anyons, (with R. Seiringer), Lett. Math. Phys. 108 (2018), 2523-2541 [DOI]. |

Local density approximation for almost-bosonic anyons, (with M. Correggi and N. Rougerie), Contribution to the proceedings of QMath13: Mathematical Results in Quantum Physics, 8-11 October 2016, Atlanta, US [DOI]. |

Local density approximation for the almost-bosonic anyon gas, (with M. Correggi and N. Rougerie), Analysis & PDE 10 (2017), 1169-1200 [DOI]. |

Many-anyon trial states, Phys. Rev. A 96 (2017), 012116 [DOI] [kaleidoscope]. |

Exclusion bounds for extended anyons, (with S. Larson), Arch. Rational Mech. Anal. 227 (2018) 309-365. [DOI]. |

Emergence of fractional statistics for tracer particles in a Laughlin liquid, (with N. Rougerie), Phys. Rev. Lett. 116 (2016), 170401 [DOI]. |

The average field approximation for almost bosonic extended anyons, (with N. Rougerie), J. Stat. Phys. 161 (2015), 1236-1267 [DOI]. |

Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems, (with P. T. Nam and F. Portmann), Arch. Rational Mech. Anal. 219 (2016), 1343-1382 [DOI]. |

Lieb-Thirring Bounds for Interacting Bose Gases, (with F. Portmann and J. P. Solovej), Commun. Math. Phys. 335 (2015), 1019-1056 [DOI]. |

Anyon wave functions and probability distributions, IHES/P/13/25, August 2013. |

Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics, (with J. P. Solovej), Ann. Henri Poincaré 15 (2014), 1061-1107 [DOI]. |

Local exclusion principle for identical particles obeying intermediate and fractional statistics, (with J. P. Solovej), Phys. Rev. A 88 (2013), 062106 [DOI]. |

Hardy and Lieb-Thirring inequalities for anyons, (with J. P. Solovej), Commun. Math. Phys. 322 (2013), 883-908 [DOI]. |

Geometric extensions of many-particle Hardy inequalities, J. Phys. A: Math. Theor. 48 (2015) 175203 [DOI]. |

Partial Hamiltonian reduction of relativistic extended objects in light-cone gauge, (with J. de Woul and J. Hoppe), JHEP 01 (2011) 031 [DOI]. |

Zero-energy states in supersymmetric matrix models, PhD thesis, May 2010. Corrections |

A dynamical symmetry for supermembranes, (with J. de Woul, J. Hoppe and M. Sundin), JHEP 03 (2011) 134 [DOI]. |

Some spectral bounds for Schrödinger operators with Hardy-type potentials, November 2009 (published as part of Ph.D. thesis). |

Clifford algebra, geometric algebra, and applications, (with L. Svensson), July 2009 (lecture notes), last revision 2016. |

Weighted Supermembrane Toy Model, Lett. Math. Phys. 92 (2010), 125-141 [DOI]. |

Spin(9) Average of SU(N) Matrix Models I. Hamiltonian (with J. Hoppe and M. Trzetrzelewski), J. Math. Phys. 50 (2009), 043510 [DOI]. |

Construction of the Zero-Energy State of SU(2)-Matrix Theory: Near the Origin (with J. Hoppe and M. Trzetrzelewski), Nucl. Phys. B 817 (2009), 155-166 [DOI]. |

Octonionic twists for supermembrane matrix models (with J. Hoppe and M. Trzetrzelewski), Ann. Henri Poincaré 10 (2009), 339-356 [DOI]. |

On the Geometry of Supersymmetric Quantum Mechanical Systems, J. Math. Phys. 49 (2008), 062101 [DOI]. |

Dynamical Symmetries in Supersymmetric Matrix Models (with V. Bach and J. Hoppe), Documenta Math. 13 (2008), 103-116 [DOI]. |

On the Construction of Zero Energy States in Supersymmetric Matrix Models IV (with J. Hoppe), June 2007. |

Geometric (Clifford) algebra and its applications, MSc thesis, January 2006. |

**Talks and lectures**

**Supervision**

Simon Larson, Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization, PhD thesis, 2019. |

Gustav Brage and Erik Sönnerlind, Braid group statistics and exchange matrices of non-abelian anyons, with representations in Clifford algebra, BSc thesis, 2018 [PDF]. |

Joel Wiklund, Braiding non-abelian anyons with representations in Clifford algebra, with representations in Clifford algebra, BSc thesis, 2018 . |

Elias Riedel Gårding, Geometric algebra, conformal geometry and the common curves problem, BSc thesis, 2017 [PDF]. |

Viktor Qvarfordt, Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation, MSc thesis, 2017 [PDF]. |

Emil Jacobsen and Erika Lind, Triangles in particle interactions and applications of Clifford algebra, BSc thesis, 2016 [PDF]. |

Oskar Weinberger, The braid group, representations and non-abelian anyons, BSc thesis, 2015 [PDF]. |

**Courses**

Real Analysis, UU, Winter 2020/21 (BSc course).

Mathematical Quantum Mechanics II, LMU, Summer 2019 (Master course).

Clifford algebra, geometric algebra, and applications, LMU, Summer 2019 (seminar course).

Topics in mathematics IV: Methods of modern mathematical physics, KTH, Spring 2017 (Master course).

Clifford algebra, geometric algebra, and applications, KTH, Spring 2009 and Spring 2016 (PhD course).

**Conferences**

Correlations in Mathematical Quantum Mechanics, Conference to celebrate the work of Jan Philip Solovej, Copenhagen, June 2023 (co-organizer). |

Archipelagic perspectives on mathematics and physics, Djurö, August/September 2021. |

Mathematical physics of anyons and topological states of matter, Nordita, Stockholm, March 2019. |

The Mathematical Physics session at the 27th Nordic Congress of Mathematicians, Stockholm, March 2016 (co-organizer). |

A one-day conference in honor of the 65th birthday of Lars Svensson, KTH, January 2014 (in Swedish). |

**Seminar series**

The PDEs and Applications seminar at UU (co-organizer).

Mathematical Physics Oberseminar at LMU, Summer 2019 (organizer).

Talks in Mathematical Physics in the Stockholm area, learning seminar (co-organizer).