Hans Z. MuntheKaas
Photo: Melanie Burford
Contact Hans MuntheKaas:
Research
Multivariate Chebyshev polynomials. Domain of A2family.
Multivariate Chebyshev polynomials. Domain of A3family
Coproduct of the MKW Hopf algebra
Lie group integration of Størmer's problem
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. LieButcher theory combines classical Bseries 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 noncommutative 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 shuffleexchange 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
MuntheKaas 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 MuntheKaas, 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):4968,
2020.
[ bib 
.pdf ]

[2]

Charles Curry, Kurusch EbrahimiFard, Dominique Manchon, and Hans Z
MuntheKaas.
Planarly branched rough paths and rough differential equations on
homogeneous spaces.
Journal of Differential Equations, 269(11):97409782, 2020.
[ bib 
.pdf ]

[3]

Hans Z MuntheKaas and Alexander Lundervold.
Correction to: On postLie algebras, LieButcher series and
moving frames.
Foundations of Computational Mathematics, 19(1):241241, 2019.
[ bib 
.pdf ]

[4]

Gunnar Fløystad, Dominique Manchon, and Hans Z MuntheKaas.
The universal prelierinehart algebras of aromatic trees.
In International conference on Geometric and Harmonic Analysis
on homogeneous spaces and Applications in Honor of Professor Takaaki Nomura,
pages 137159. Springer, 2019.
[ bib 
.pdf ]

[5]

Hans Z MuntheKaas and Kristoffer K Føllesdal.
Liebutcher series, geometry, algebra and computation.
In Discrete Mechanics, Geometric Integration and LieButcher
Series, pages 71113. Springer, 2018.
[ bib 
.pdf ]

[6]

E. Celledoni, G. Di Nunno, K. EbrahimiFard, and H.Z. MuntheKaas, editors.
Computation and Combinatorics in Dynamics, Stochastics and
Control.
Abel Symposia. Springer, 2018.
[ bib 
.pdf ]

[7]

Kurusch EbrahimiFard, Igor Mencattini, and Hans MuntheKaas.
PostLie algebras and factorization theorems.
Journal of Geometry and Physics, 119:1933, 2017.
[ bib 
.pdf ]

[8]

Robert I McLachlan, Klas Modin, Hans MuntheKaas, and Oliver Verdier.
Butcher series  A story of rooted trees and numerical methods
for evolution equations.
Asia Pacific Mathematics Newsletter, 7(1):111, 2017.
[ bib 
.pdf ]

[9]

Charles Curry, Kurusch EbrahimiFard, and Hans MuntheKaas.
What is a postLie algebra and why is it useful in geometric
integration.
In European Conference on Numerical Mathematics and Advanced
Applications, pages 429437. Springer, 2017.
[ bib 
.pdf ]

[10]

Hans Z MuntheKaas.
Groups and symmetries in numerical linear algebra.
In Exploiting Hidden Structure in Matrix Computations:
Algorithms and Applications, pages 319406. Springer, 2016.
[ bib 
.pdf ]

[11]

Hans MuntheKaas and Olivier Verdier.
Aromatic Butcher series.
Foundations of Computational Mathematics, 16(1):183215, 2016.
[ bib 
.pdf ]

[12]

Robert I McLachlan, Klas Modin, Hans MuntheKaas, and Olivier Verdier.
Bseries methods are exactly the affine equivariant methods.
Numerische Mathematik, 133(3):599622, 2016.
[ bib 
.pdf ]

[13]

Hans MuntheKaas and Olivier Verdier.
Integrators on homogeneous spaces: isotropy choice and connections.
Foundations of Computational Mathematics, 16(4):899939, 2016.
[ bib 
.pdf ]

[14]

Gunnar Fløystad and Hans MuntheKaas.
Preand postlie algebras: The algebrogeometric view.
In The Abel Symposium, pages 321367. Springer, 2016.
[ bib 
.pdf ]

[15]

Kurusch EbrahimiFard, Alexander Lundervold, and Hans Z MuntheKaas.
On the lie enveloping algebra of a postlie algebra.
Journal of Lie Theory, 25(4):11391165, 2015.
[ bib 
http ]

[16]

Alexander Lundervold and Hans Z MuntheKaas.
On algebraic structures of numerical integration on vector spaces and
manifolds.
Faà di Bruno Hopf Algebras, DysonSchwinger Equations, and
LieButcher Series, pages 219263, 2015.
[ bib 
http ]

[17]

Kurusch EbrahimiFard, Alexander Lundervold, Igor Mencattini, Hans Z
MuntheKaas, et al.
PostLie algebras and isospectral flows.
SIGMA. Symmetry, Integrability and Geometry: Methods and
Applications, 11:093, 2015.
[ bib 
.pdf ]

[18]

Hans Z MuntheKaas, Gilles Reinout W Quispel, and Antonella Zanna.
Symmetric spaces and Lie triple systems in numerical analysis of
differential equations.
BIT Numerical Mathematics, 54(1):257282, 2014.
[ bib 
.pdf ]

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