# Prof. Dr. Raimar Wulkenhaar, Mathematisches Institut

Member of CRC 1442 Geometry: Deformations and RigidityInvestigator in Mathematics Münster

Investigator in Mathematics Münster

Private Homepage | http://www.uni-muenster.de/MathPhys/u/raimar/ |

Research Interests | Mathematical Physics Quantum field theory Noncommutative geometry |

Selected Publications | • Branahl, Johannes; Hock, Alexander; Wulkenhaar, Raimar Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures. Communications in Mathematical Physics Vol. 393, 2022 online• Panzer, Erik; Wulkenhaar, Raimar Lambert-W solves the noncommutative \Phi^4-model. Communications in Mathematical Physics Vol. 374, 2020, pp 1935-1961 online• Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar Solution of all quartic matrix models. , 2019 online• Schürmann, Jörg; Wulkenhaar, Raimar An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings Vol. 174 (3), 2023 online• Wulkenhaar, Raimar Quantum field theory on noncommutative spaces. Advances in Noncommutative Geometry, 2019, pp 607-690 online• Grosse, Harald; Wulkenhaar, Raimar Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. to appear in Commun. Math. Phys.Communications in Mathematical Physics Vol. 329 (3), 2014 online• Grosse H, Wulkenhaar R Renormalisation of \phi^4-theory on noncommutative R^4 in the matrix base
. Communications in Mathematical Physics Vol. 256 (2), 2005, pp 305-374 online• Rivasseau, Vincent, Vignes-Tourneret, Fabien; Wulkenhaar, Raimar Renormalisation of noncommutative \phi^4-theory by multi-scale analysis. Communications in Mathematical Physics Vol. 262 (3), 2006, pp 565-594 online• Krajewski T, Wulkenhaar R Perturbative quantum gauge fields on the noncommutative torus. International Journal of Modern Physics A Vol. 15 (7), 2000, pp 1011-1029 online• Grosse H, Wulkenhaar R Power-counting theorem for non-local matrix models and renormalisation. Communications in Mathematical Physics Vol. 254 (1), 2005, pp 91-127 online |

Selected Projects | • EXC 2044 - B3: Operator algebras & mathematical physics The development of operator algebras was largely motivated by physics since they provide the right mathematical framework for quantum mechanics. Since then, operator algebras have turned into a subject of their own. We will pursue the many fascinating connections to (functional) analysis, algebra, topology, group theory and logic, and eventually connect back to mathematical physics via random matrices and non-commutative geometry. online• CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability The project investigates a novel integrable system which arises from a quantum field theory on noncommutative geometry. It is characterised by a recursive system of equations with conjecturally rational solutions. The goal is to deduce their generating function and to relate the rational coefficients in the generating function to intersection numbers of tautological characteristic classes on some moduli space. online• RTG 2149: Strong and Weak Interactions - from Hadrons to Dark Matter The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group. Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC. The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics. online • CRC 878 C04 - Mathematical aspects of QFT and condensed matter We continue our work on solvable quantum field theory models on noncommutative geometries and extend them to several components. In order to give a rigorous foundation to the fractional quantum Hall effect we work on a construction of a KK-element which classifies topological effects in disordered materials. Mathematische Aspekte von Quantenfeldtheorie und kondensierter Materie.online • CRC 478 E04 - Noncommutative geometry and quantisation of physical interactions The aim is to intensify previous research on the intersection of noncommutative geometry with quantum models for physical interactions. This includes the endeavour to renormalise gauge theories on noncommutative geometries on one hand and the analysis of a noncommutative algebra of loops related to quantum gravity on the other hand. Spectral triples play an important rˆole in both directions and a central question is to construct a Dirac operator with the physically desired properties. online |

Project membershipMathematics Münster | B: Spaces and OperatorsC: Models and ApproximationsB3: Operator algebras & mathematical physics C1: Evolution and asymptotics C3: Interacting particle systems and phase transitions |

Current Talks | • Scalar quantum fields on 4-dimensional noncommutative geometry. Exactly solvable models, Wien Slides Link to event• BKP-relations in Kontsevich model with arbitrary potential. The QFT path, Heidelberg Slides Link to event• Quantum fields on noncommutative geometries. North Atlantic Noncommutative Geometry Seminar, Warsaw Slides Link to event• Quantum fields on noncommutative geometries. Workshop on Operator Algebras and Applications: Non-Commutative Geometry, Toronto Slides Link to event• Matrix models and topological recursion. Kolloquium Mathematische Physik, Universität Bielefeld, Bielefeld Slides• Blobbed topological recursion of quartic matrix models. Invitation to Recursion, Resurgence and Combinatorics, Okinawa Slides Link to event• Blobbed topological recursion of the \lambda \Phi^4 matrix model. Quantum Gravity and Random Geometry, Paris Slides Link to event• The Euclidean λφ^4_4-model on noncommutative geometries: a status report. Energy conditions in quantum field theory, Leipzig Slides Link to event• How topological recursion organises quantum fields on noncommutative geometries (3 lectures). Fields Institute Workshop on Noncommutative Geometry, Free Probability Theory and Random Matrix Theory, London (Ontario) Slides Link to event |

Current Publications | • Hock, Alexander; Wulkenhaar, Raimar; Dołega, Maciej (appendix) Blobbed topological recursion of the quartic Kontsevich model II: Genus=0. Annales de l’Institut Henri Poincaré D: Combinatorics, Physics and their Interactions Vol. online first, 2024 online• Borot, Gaëtan; Wulkenhaar, Raimar A note on BKP for the Kontsevich matrix model with arbitrary potential. Symmetry, Integrability and Geometry: Methods and Applications Vol. 20, 2024 online• Grosse, Harald; Kanomata, Naoyuki; Sako, Akifumi; Wulkenhaar, Raimar Real symmetric Φ^4-matrix model as Calogero-Moser model. Letters in Mathematical Physics Vol. 114, 2024 online• Schürmann, Jörg; Wulkenhaar, Raimar An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings Vol. 174 (3), 2023 online• Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar A Laplacian to compute intersection numbers on M_{g,n} and correlation functions in NCQFT. Communications in Mathematical Physics Vol. 399, 2023 online• Hock, Alexander; Wulkenhaar, Raimar Blobbed topological recursion from extended loop equations. , 2023 online• Melong, Fridolin; Wulkenhaar, Raimar Generalized Heisenberg-Virasoro algebra and matrix models from quantum algebra. Journal of Mathematical Physics Vol. 64 (7), 2023 online• de Jong, Jins; Hock, Alexander; Wulkenhaar, Raimar Nested Catalan tables and a recurrence relation in noncommutative quantum field theory. Annales de l’Institut Henri Poincaré D: Combinatorics, Physics and their Interactions Vol. 9 (1), 2022 online• Branahl, Johannes; Hock, Alexander; Wulkenhaar, Raimar Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures. Communications in Mathematical Physics Vol. 393, 2022 online |

Current Projects | • CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability We investigate blobbed topological recursion for the general Kontsevich matrix model, as well as the behaviour of Baker–Akhiezer spinor kernels for deformations of the spectral curve and for the quartic Kontsevich model. We study relations between spin structures and square roots of Strebel differentials, respectively between topological recursion and free probability. We examine factorisation super-line bundles on infinite-dimensional Grassmannians and motivic characteristic classes for intersection cohomology sheaves of Schubert varieties. • EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues. Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online • EXC 2044 - C3: Interacting particle systems and phase transitions The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scalerandomly distributed heterogeneities and small gradient perturbations.We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes. We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder-Schrader axioms. Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We willalso explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model. We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interactingmany-particle models of reinforced random walks. Formally, local cross-diffusion syste ms are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment. online • EXC 2044 - B3: Operator algebras & mathematical physics The development of operator algebras was largely motivated by physics since they provide the right mathematical framework for quantum mechanics. Since then, operator algebras have turned into a subject of their own. We will pursue the many fascinating connections to (functional) analysis, algebra, topology, group theory and logic, and eventually connect back to mathematical physics via random matrices and non-commutative geometry. online |

E-Mail | raimar at uni-muenster dot de |

Phone | +49 251 83-33734 |

Room | 416 |

Secretary | Sekretariat Dierkes Frau Gabi Dierkes Telefon +49 251 83-33730 Zimmer 414 |

Address | Prof. Dr. Raimar Wulkenhaar Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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