part ¢¡ geometries and general group actions  geometry of singular space shing-tung yau    1  the development of modern geometry that influenced our concept of space    2  geometry of singular spaces    3  geometry for einstein equation and special holonomy group    4  the laplacian and the construction of generalized riemannian geometry in terms of operators    5  differential topology of the operator geometry    6  inner product on tangent spaces and hodge theory    7  gauge groups, convergence of operator manifolds and yang-mills theory    8  generalized manifolds with special holonomy groups    9  maps, subspaces and sigma models    10  noncompact manifolds    11  discrete spaces    12  conclusion    13  appendix    references  a summary of topics related to group actions  lizhen ji    1  introduction    2  different types of groups    3  different types of group actions    4  how do group actions arise    5  spaces which support group actions    6  compact transformation groups    7  noncompact transformation groups    8  quotient spaces of discrete group actions    9  quotient spaces of lie groups and algebraic group actions    i0  understanding groups via actions    11  how to make use of symmetry    12  understanding and classifying nonlinear actions of groups    13  applications of finite group actions in combinatorics    14  applications in logic    15  groups and group actions in algebra    16  applications in analysis    17  applications in probability    18  applications in number theory    19  applications in algebraic geometry    20  applications in differential geometry    21  applications in topology    22  group actions and symmetry in physics    23  group actions and symmetry in chemistry    24  symmetry in biology and the medical sciences    25  group actions and symmetry in material science and engineering    26  symmetry in arts and architecture    27  group actions and symmetry in music    28  symmetries in chaos and fractals    29  acknowledgements and references    referencespart ¢¢ mapping class groups and teichmiiller spaces  actions of mapping class groups  athanase papadopoulos    1  introduction    2  rigidity and actions of mapping class groups    3  actions on foliations and laminations    4  some perspectives    references  the mapping class group action on the horofunction compactification of teichmiiller space  weixu su    1  introduction    2  background    3  thurston's compactification of teichmiiller space    4  compactification of teichmfiller space by extremal length    5  analogies between the thurston metric and the teichmiiller metric    6  detour cost and busemann points    7  the extended mapping class group as an isometry group    8  on the classification of mapping class actions on thurston's metric    9  some questions    references  schottky space and teichmiiller disks  frank herrlich    1  introduction    2  schottky coverings    3  schottky space    4  schottky and teichmfiller space    5  schottky space as a moduli space    6  teichmiiller disks    7  veech groups    8  horizontal cut systems    9  teichmiiller disks in schottky space    references  topological characterization of the asymptotically trivial mapping class group  ege fujikawa    1  introduction    2  preliminaries    3  discontinuity of the teichmfiller modular group action    4  the intermediate teichmiiller space    5  dynamics of the teichmiiller modular group    6  a fixed point theorem for the asymptotic teichmiiller modular group    7  periodicity of asymptotically teichmiiller modular transformation    references  the universal teichmiiller space and diffeomorphisms of the circle with hslder continuous derivatives  katsuhiko matsuzaki    1  introduction    2  quasisymmetric automorphisms of the circle    3  the universal teichmiiller space    4  quasisymmetric functions on the real line    5  symmetric automorphisms and functions    6  the small subspace    7  diffeomorphisms of the circle with hsld