Study of moduli of indecomposable modules and degeneration of modules

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As a joint work with Osamu Iyama (Nagoya Univ.), we have defined the mutation as an action of braid group on the set of indecomposable objects in a general triangulated category. We have developed a general theory of mutation and have applied it to the classification problem of rigid Cohen-Macaulay modules. In particular, we succeeded to describe the perfect classification of rigid Cohen-Macaulay modules over a Veronese subring of dimension 3 and of degree 3. Through this consideration of the mutation, we are able to obtain further examples as well where we can classify the rigid Cohen-Macaulay modules. Actually the study was done by considering the maximal orthogonal subcategories in the stable category of Cohen-Macaulay modules. An investigation of the similar problem in a derived category is now in progress.We also made a study on the deformation and the degeneration of modules. And we succeeded to construct a noncommutative parameter space of the universal deformations of a module. By this, we are now able to understand well the classical obstruction theory of modules. This noncommutative deformation theory is also in progress.