Non-commutative Algebraic Geometry

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The principal investigator and his colleagues study non-commutativealgebraic goemetry with particular emphasis on projectivesurfaces. General abstract methods are developed and are applied tospecific important classes such as quantum ruled surfaces, and blowups ofnon-commutative analogues of the projective plane. The positive cone ofcurves on these surfaces is studied, with the goal of betterunderstanding the intersection theory and using the Picard group as aninvariant. The possibility of using cyclic homology as a new invariant isalso under examination. Non-commutative algebraic geometry extends the fruitfulinteraction of algebra and geometry that lies at the root of modernalgebraic geometry. The modern origin dates from Descarte's introductionof coordinate axes, but until recently the geometric perspective onalgebra has been restricted to commutative algebra. However, modernphysics shows that nature behaves non-commutatively at the quantum level(Heisenberg's uncertainty priciple says that the measurementof momentum and position do not commute -that is, the order in which onemeasures them affects the values obtained). Other fundamental mathematicaltools that appear in physics,like Lie groups and differential operators,are also non-commutative. Thus, development of non-commutative geometry ispart of understanding the geometry underlying the mathematical structuresof modern physics.