Geometry of complex 3-folds, and noncommutative deformations

Rational curves on complex 3-folds are a rich source of geometric interest. I will discuss a new approach to their 
birational and enumerative geometry, joint with M. Wemyss. We associate an associative (not necessarily commutative) 
algebra of deformations to suitable such curves, and use this algebra to understand birational modifications of the 
3-fold, and investigate the structure of its derived category. The algebra may be viewed as an enhanced invariant of the 
curve, unifying and generalising other classical invariants.