Collaborative Research: Vector Bundles on Projective Spaces

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abstractThe investigator and his colleagues study small rank vector bundles on projective spaces. New and simplified constructions are obtained both in finite and in zero characteristics. In the case of rank two bundles, these constructions on projective four space are valid only in positive characteristics. The investigator and his colleagues work on the question of extending these constructions to characteristic zero. This is related to the deformation theory of these bundles. They investigate whether these bundles can be deformed from positive to zero characteristic. The deformation theory of such bundles also has applications to questions regarding degenerating sums of line bundles and the existence of exotic components of the Hilbert scheme.The investigator and his colleagues give explicit constructions of objects called vector bundles. Vector bundles are devices which encode algebraic information about huge numbers of geometric figures like curves and surfaces. With their explicit knowledge of vector bundles, the investigator and his colleagues can then construct geometric figures with desired properties. The work is done using matrices which are easily implemented on computer algebra systems. Much of the work is done over finite fields which allows the use of computers to give exact answers. Problems involving geometric modeling in the real world require approximating the answers, one method being the use of such finite fields. The project studies the interplay between geometric objects existing in the real world (over fields of characteristic zero) and their approximations over finite fields.