This chapter describes the essential aspects of geometry definition of flow channels, blades and vanes in radial compressors. The impeller blades and flow channel make up a complex three-dimensional shape, and a general method for defining such geometries using Bezier surfaces is described. The meridional channel is defined as a series of Bezier patches whose geometry is parametrised to allow changes to be made quickly and efficiently during the design. Radial compressor impeller blades are typically defined as a distribution of camber line angle and thickness on the hub and casing meridional sections. A three-dimensional shape is generated by joining the hub and casing sections by straight lines to make up a ruled surface. This simplifies simplify manufacture by flank milling. In transonic flows, a more complex definition is used in which the geometry is defined along a series of separate meridional planes. A description of the definition of asymmetric components, such as the volute, is also provided.

The first lecture introduces the study of closed symplectic manifolds via closed pseudoholomorphic curves, focusing in particular on McDuff’s famous theorem from 1990 that characterizes symplectic ruled surfaces via symplectically embedded spheres. The theorem is stated in a slightly more modern formulation using Lefschetz fibrations, and a proof modulo of some technical lemmas is sketched. The topic of intersection theory is then introduced by considering the natural question of what kinds of holomorphic curves can be proven to form leaves of smooth foliations by holomorphic curves.

This lecture concludes our survey of closed holomorphic curves with a discussion, in the first section, of local intersection numbers, positivity of intersections and the adjunction formula for closed holomorphic curves, and then, in the second section, with an explanation of how these figure into the proof of McDuff’s theorem on symplectic ruled surfaces. The last two sections then begin a shift in focus toward punctured holomorphic curves: this discussion starts with a general introduction to contact manifolds and their symplectic fillings and then continues by defining the moduli space of punctured asymptotically cylindrical holomorphic curves in a completed symplectic cobordism between contact manifolds.