The Erlangen program of F. Klein (influenced
by S. Lie) defines geometry as a study of invariants under a certain
group action. This approach proved to be fruitful much beyond the
traditional geometry. For example, special relativity is the study of
invariants of Minkowski space-time under the Lorentz group action.
Another example is complex analysis as study of objects invariant
under the conformal maps.
In this course we consider in detail the group SL2(ℝ) and the
corresponding geometrical and analytical invariants with their
interrelations. Consequently the course has a multi-subject nature
touching algebra, geometry and analysis. No special knowledge beyond a
standard undergraduate curriculum is required.
The considered topics are:
the group SL2(ℝ) and Möbius transformations of the real
line.
Vladimir V. Kisil.
Meeting Descartes and Klein somewhere in a noncommutative space.
In A. Fokas, J. Halliwell, T. Kibble, and B. Zegarlinski, editors,
Highlights of mathematical physics (London, 2000), pages 165–189.
Amer. Math. Soc., Providence, RI, 2002.
arXiv:math-ph/0112059.
Vladimir V. Kisil.
Erlangen program at large–0: Starting with the group SL2(R).
Notices Amer. Math. Soc., 54(11):1458–1465, 2007.
arXiv:math/0607387,
On-line.
Zbl# 1137.22006.
Vladimir V. Kisil.
Fillmore-Springer-Cnops construction implemented in
GiNaC.
Adv. Appl. Clifford Algebr., 17(1):59–70, 2007.
On-line. A more
recent version: arXiv:cs.MS/0512073. The latest documentation, source
files, and live ISO image are at the project page:
http://moebinv.sourceforge.net/. Zbl# 05134765.
Vladimir V. Kisil.
Erlangen programme at large: An overview.
In S.V. Rogosin and A.A. Koroleva, editors, Advances in Applied
Analysis, chapter 1, pages 1–94. Birkhäuser Verlag, Basel, 2012.
arXiv:1106.1686.
Vladimir V. Kisil.
Geometry of Möbius Transformations: Elliptic, Parabolic
and Hyperbolic Actions of SL2(R).
Imperial College Press, London, 2012.
Includes a live DVD. Zbl# 1254.30001.
Figure 1: Unification of elliptic, parabolic and hyperbolic cycles