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 SL_{2}(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 SL_{2}(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 SL_{2}(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 SL_{2}(R).
Imperial College Press, London, 2012.
Includes a live DVD. Zbl# 1254.30001.
Figure 1: Unification of elliptic, parabolic and hyperbolic cycles