Professor of Mathematics

at

University of Bergen

Birdwatcher and Nature Photographer

Maze-maker

Contact Hans Munthe-Kaas:

  • Email: hans.munthe-kaas@uib.no
  • Homepage: http://hans.munthe-kaas.no
  • Phone office: +47 55584179
  • Cell phone: +47 45238987
  • Office: Realfagbygget, Allégaten 41, Room 4A7d
  • Office mail address:

    University of Bergen
    Department of Mathematics
    Postbox 7803
    NO-5020 BERGEN
    NORWAY

  • Private address:

    Mølleveien 4
    NO-5067 BERGEN
    NORWAY

Research overview

Overview

My research is in the borderland between pure- and applied mathematics and computer science. A recurring theme is the role of mathematical abstractions in computational science, both as a tool for constructing efficient algorithms and as an organizing principle for computational software. Thus, as an overall label, most of my work is within Foundations of Computational Mathematics.

I am interested in applications of differential geometric techniques, geometric integration and structure preserving algorithms for solution of differential equations. Lie group integrators are numerical integration methods for differential equations built from coordinate independent operations such as Lie group actions on a manifold. Lie group integrators have been developed in tight cooperation between the research groups in Bergen, DAMTP Cambridge and mathematics NTNU.

Analysis of numerical Lie group integrators lead us to the study of new types of formal power series for flows on manifolds. Lie-Butcher theory combines classical B-series for integration schemes with Lie series. This research activity has connections to many areas of mathematical research such as control theory, stochastic differential equations, renormalization, combinatorial Hopf algebra theory and non-commutative symmetric functions.

In my PhD (1989) I, among other things, investigated FFTs for functions possessing crystallographic symmetries. After 2004, I have again found interest in this topic, due to connections between this theory and multivariate versions of Chebyshev polynomials. This is applied in the construction of spectral element methods based on triangular and simplicial subdivisions of the domain. A related topic is applications of Fourier transforms on groups, with applications in numerical linear algebra and image processing.

My PhD work, for which I was awarded the Esso prize in 1999, was on numerical linear algebra and parallel algorithms. I was interested in applications of group theory in the construction of fast elliptic solvers, as well as an organizing principle for massively parallel computations. In the early 1990s I was writing software and developing algorithms for massively parallel computers, in particular FFTs and various algebraic techniques for routing permutations on massively parallel computers. I was also interested in construction of permutation networks, in particular I studied some generalized shuffle-exchange networks. These are based on interesting connections between permutation networks and the theory of linear shift registers.

Object oriented program design is founded on the distinction between specification and implementation, or ‘what’ and ‘how’. This is a well established principle in core computer science. In computational mathematics, however, software has traditionally been organized around coordinate based formulations and discrete representations. In the SOPHUS project, initiated around 1991, we investigated coordinate free formulations for numerical algorithm design and for numerical software. In particular we developed coordinate free formulations of tensor computations and this was also an initial motivation for the study of Lie group integrators. The DiffMan Matlab toolbox for solving differential equations on manifolds is based on the ideas of the SOPHUS C++ library.

Teaching

Teaching ‘Mattesirkelen’ for high school students

Munthe-Kaas has over more than three decades been lecturing courses in applied mathematcs, pure mathematics, and computer science at all university levels, from advanced seminars to introductory courses. Internationally he has given advanced courses at AIMS South Africa, at a CIMPA school in Brazil, at Morningside Centre in Beijng, Uppsala University Sweden, and he has given lecture series various research semesters.

Teaching autumn 2021: Mat101 (precalculus).

Students who want master thesis projects: Please take contact.

Outreach

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Publications

Reverse chronologic order:
[1] Hans Z Munthe-Kaas, Ari Stern, and Olivier Verdier. Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds. SIAM Journal on Applied Algebra and Geometry, 4(1):49-68, 2020. [ bib | .pdf ]
[2] Charles Curry, Kurusch Ebrahimi-Fard, Dominique Manchon, and Hans Z Munthe-Kaas. Planarly branched rough paths and rough differential equations on homogeneous spaces. Journal of Differential Equations, 269(11):9740-9782, 2020. [ bib | .pdf ]
[3] Hans Z Munthe-Kaas and Alexander Lundervold. Correction to: On post-Lie algebras, Lie-Butcher series and moving frames. Foundations of Computational Mathematics, 19(1):241-241, 2019. [ bib | .pdf ]
[4] Gunnar Fløystad, Dominique Manchon, and Hans Z Munthe-Kaas. The universal pre-lie-rinehart algebras of aromatic trees. In International conference on Geometric and Harmonic Analysis on homogeneous spaces and Applications in Honor of Professor Takaaki Nomura, pages 137-159. Springer, 2019. [ bib | .pdf ]
[5] Hans Z Munthe-Kaas and Kristoffer K Føllesdal. Lie-butcher series, geometry, algebra and computation. In Discrete Mechanics, Geometric Integration and Lie-Butcher Series, pages 71-113. Springer, 2018. [ bib | .pdf ]
[6] E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard, and H.Z. Munthe-Kaas, editors. Computation and Combinatorics in Dynamics, Stochastics and Control. Abel Symposia. Springer, 2018. [ bib | .pdf ]
[7] Kurusch Ebrahimi-Fard, Igor Mencattini, and Hans Munthe-Kaas. Post-Lie algebras and factorization theorems. Journal of Geometry and Physics, 119:19-33, 2017. [ bib | .pdf ]
[8] Robert I McLachlan, Klas Modin, Hans Munthe-Kaas, and Oliver Verdier. Butcher series - A story of rooted trees and numerical methods for evolution equations. Asia Pacific Mathematics Newsletter, 7(1):1-11, 2017. [ bib | .pdf ]
[9] Charles Curry, Kurusch Ebrahimi-Fard, and Hans Munthe-Kaas. What is a post-Lie algebra and why is it useful in geometric integration. In European Conference on Numerical Mathematics and Advanced Applications, pages 429-437. Springer, 2017. [ bib | .pdf ]
[10] Hans Z Munthe-Kaas. Groups and symmetries in numerical linear algebra. In Exploiting Hidden Structure in Matrix Computations: Algorithms and Applications, pages 319-406. Springer, 2016. [ bib | .pdf ]
[11] Hans Munthe-Kaas and Olivier Verdier. Aromatic Butcher series. Foundations of Computational Mathematics, 16(1):183-215, 2016. [ bib | .pdf ]
[12] Robert I McLachlan, Klas Modin, Hans Munthe-Kaas, and Olivier Verdier. B-series methods are exactly the affine equivariant methods. Numerische Mathematik, 133(3):599-622, 2016. [ bib | .pdf ]
[13] Hans Munthe-Kaas and Olivier Verdier. Integrators on homogeneous spaces: isotropy choice and connections. Foundations of Computational Mathematics, 16(4):899-939, 2016. [ bib | .pdf ]
[14] Gunnar Fløystad and Hans Munthe-Kaas. Pre-and post-lie algebras: The algebro-geometric view. In The Abel Symposium, pages 321-367. Springer, 2016. [ bib | .pdf ]
[15] Kurusch Ebrahimi-Fard, Alexander Lundervold, and Hans Z Munthe-Kaas. On the lie enveloping algebra of a post-lie algebra. Journal of Lie Theory, 25(4):1139-1165, 2015. [ bib | http ]
[16] Alexander Lundervold and Hans Z Munthe-Kaas. On algebraic structures of numerical integration on vector spaces and manifolds. Faà di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series, pages 219-263, 2015. [ bib | http ]
[17] Kurusch Ebrahimi-Fard, Alexander Lundervold, Igor Mencattini, Hans Z Munthe-Kaas, et al. Post-Lie algebras and isospectral flows. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 11:093, 2015. [ bib | .pdf ]
[18] Hans Z Munthe-Kaas, Gilles Reinout W Quispel, and Antonella Zanna. Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT Numerical Mathematics, 54(1):257-282, 2014. [ bib | .pdf ]

Earlier publications


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