Deformation Theory of Algebras and related topics

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Algebraic deformation theory was found by Gerstenhaber in 1963. Now this has become one of the most important theory and the core tool to investigate mathematical physics. In this project I have mainly worked on the deformation of Lie triple systems (Lts).Achievements (1)I have laid the foundation of the deformation theory of Lts'. I defined the basic concepts such as infinitesimal, equivalence, rigid, integrable for a deformation of a Lts. The most important discovery is to find out that Yamaguchi cohomology theory is well suited for the investigation of the deformation theory of Lts'.(2)To understand a difference between perturbation and deformation has been known to be very difficult. My answer is as follow : In the perturbation any calculation is done by original product, but it is done by deformed product in the deformed world.Accomplishment This project has been basically done by the discussions with Professor Gerstenhaber. I visited him at University of Pennsylvania in September 2003 and he visited me at Hiroshima University in March 2005. Main subjects of our discussions are as follows : 1)Yamaguchi cohomology, 2)History of Deformation Theory based on the mathematicians Reimann, Kodaira-Spencer and Gerstenhaber, 3)My idea of the application of Deformation Theory to Inverse problems and 4)Deformation Theory in the quantum mechanics.Problems and forward In this project, I have just shown the starting point to research the deformation theory of Lts'. I know that there are many obstructions to go forward. Even for the definition of the representation of a Lts there are at least three proposals which seem to be different for me. That of Yamaguchi is not the ordinal one and requires more works. As for the inverse problem in engineering, the deformation theory will take the very important role, I believe.