Log Canonical and Rational Singularities

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AbstractSandor Kovacs98 18357 Professor Kovacs will study log canonical singularities with special regard to deformations, vanishing theorems, connections with Du Bois singularities, Kollar's conjecture, and cohomologically insignificant singularities. He will try to establish the validity of the conjecture that non-isomorphic minimal models of an algebraic variety cannot be deformed into each other. He will also consider generalizations of the relative de Rham complex similar to Du Bois' absolute complex in an attempt to build a relative de Rham cohomology theory for non smooth morphisms. This is a project in algebraic geometry. In this important branch of modern mathematics, geometric objects like curves and surfaces are modeled using algebraic constructions. The abstract nature of the now algebraic objects makes them easier for mathematicians to study. Nevertheless, newly discovered properties of the algebraic models translate back to new properties of the geometric curves and surfaces. Prof. Kovacs is particularly interested in singularities, places where the geometric objects sort of collapse on themselves. He also hopes to establish the complete validity of an educated guess that says small geometric changes to certain important objects, algebraic varieties, cannot change the algebraic properties of the varieties. As these powerful algebraic techniques are developed, other scientists and mathematicians will have additional tools to use on their geometric problems. Already algebraic geometry has become an important tool in theoretical physics.