The principal investigator and his colleagues study non-commutative algebraic goemetry with particular emphasis on projective surfaces. General abstract methods are developed and are applied to specific important classes such as quantum ruled surfaces, and blowups of non-commutative analogues of the projective plane. The positive cone of curves on these surfaces is studied, with the goal of better understanding the intersection theory and using the Picard group as an invariant. The possibility of using cyclic homology as a new invariant is also under examination.
Non-commutative algebraic geometry extends the fruitful interaction of algebra and geometry that lies at the root of modern algebraic geometry. The modern origin dates from Descarte's introduction of coordinate axes, but until recently the geometric perspective on algebra has been restricted to commutative algebra. However, modern physics shows that nature behaves non-commutatively at the quantum level (Heisenberg's uncertainty priciple says that the measurement of momentum and position do not commute -that is, the order in which one measures them affects the values obtained). Other fundamental mathematical tools that appear in physics,like Lie groups and differential operators, are also non-commutative. Thus, development of non-commutative geometry is part of understanding the geometry underlying the mathematical structures of modern physics.
