Seminar Spring 2015:

Group theory in Computational Mathematics

 

Initial plan:

Part 1: (6-8 hours)

Title: “Groups and Symmetries in Numerical Linear Algebra”

These are lectures I will also be giving at a EMS Summer School in Calabria, Italia, June 2015 http://web.math.unifi.it/users/cime/


Part 2: (6-8 hours)

Title: “Numerical integration, structure preservation and combinatorics”

These are lectures I will also be giving at CIMPA Summer School in Sao Paulo, Brazil, August 2015. http://www.cimpa-icpam.org/ecoles-de-recherche/ecoles-de-recherche-2015/liste-chronologique-des-ecoles-de/article/new-interactions-of-combinatorics-665?lang=en


MORE DETAILED PLANS (and what we have done):

PART I:

  1. •Lecture 1: (Feb. 3) Introduction to groups, actions, subgroups, quotients. Locally compact Abelian groups (LCA). Pontryagin duality. The Fourier transform on LCA.

  2. •Lecture 2: (Fri. Feb. 6, 12.30-14.00 Room UA13B Realfagbygget).
    More on LCA. Lattices, sampling and interpolation. Lattice rules. The Heissenberg group in Fourier analysis. The fast fourier transform and the Weil - Brezin map.

  3. •Lecture 3: (Tue. March 3, 12.30-14.00, old lunchroom, 4F18A). More on the Fast Fourier Transform, Zak transform, Lattice sampling theory.

  4. •Lecture 4: (Fri. March 6, 12.30-14.00, old lunchroom, 4F18A). (Generalized) Dirichlet kernels and interpolation theory. Lebesgue numbers. We start discussing Weyl groups (kaleidoscopes of mirrors) and their relationship to multivariate Chebyshev polynomials.

  5. •Week march 9-13 no lectures.


PART II:

  1. •Lie-Butcher theory. (Starting from March 17 every Tuesday and Friday 12:30-14:00). I will discuss topics related to integration of differential equations on Lie groups and other manifolds. There is a rich algebraic theory of flows which in numerics goes back to John Butcher (seminal papers 1963-1972). More recently this turned out to be connected to many branches of mathematics (renormalisation in physics, stochastics, control, ... (list is longer!). In our lectures we will emphasize our modern view, where classical B-series appear as a special case of Lie-Butcher series, a combination of classical B-series ideas with Lie series, post-Lie algebras ....

  2. •March 17 (new lunchroom). Introduction to Lie group integration. Chapter 1-2 of Lie group methods (see link under Literature, below).

  3. •March 20 (old lunchroom). Start discussion of connections on manifolds and algebras associated with these.

  4. •Easter week March 28- April 6: No lecture.




Literature:

  1. •Lecture notes will appear here.

  2. •Part I follows the article:
    H. Z. Munthe-Kaas, M. Nome, B. N. Ryland: Through the Kaleidoscope; Symmetries, Groups and Chebyshev Approximations from a Computational Point of View, Foundations of Computational Mathematics, Budapest 2011 403, 188. London Mathematical Society Lecture Note Series.

  3. •Part II initiates with a basic discussion of Lie groups and actions, following Ch 1-2 of:
    A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, A. Zanna: Lie-group methods, Acta Numerica (2000), pp. 215–365, CUP.

  4. •HMK, A Lundervold: On post-Lie algebras, Lie-Butcher series and moving frames. J. FoCM, vol. 13, no. 4, 2013.

  5. •K.Ebrahimi-Fard, A. Lundervold, H. Munthe-Kaas: On the Lie enveloping algebra of a post-Lie algebra.