Title for thesis:

	'Differential geometric and symplectic interpretations
of stability in the sense of Gieseker'.



Abstract for Thesis:

	For  holomorphic  vector  bundles  over  compact  Kahler  manifold, 
stability in geometric  invariant theory can be  understood in differential 
geometric way.  This completes the previous  works along  this direction by 
Donaldson, Uhlenbeck and Yau.  It gives naturally a  new  system  of Monge-
Ampere equations closely related to the Atiyah Singer Index theorem. 

	From the symplectic point of view,  a one parameter family of gauge 
invariant symplectic form on the space of  connection is introduced and the 
limiting  symplectic  reduction can  be interpreted  as the moduli space in 
the sense geometric invariant theory. A natural lagrangian using Bott-Chern 
form is also constructed which generalized Donaldson lagrangian.