This course is an introduction to the geometry and topology of smooth 4-manifolds, especially those with `extra' structure (eg symplectic, complex).
In the first half of the course we will introduce smooth 4-manifolds. We will briefly survey the topological classification of (simply-connected) 4-manifolds, and then start to explain how the smooth theory diverges from the topological one, via old tools (eg Rochlin's theorem) and more recent ones (eg Seiberg-Witten) leading to a range of interesting geometric phenomena absolutely unique to 4-dimensions. We hope to get to the point of answering questions such as: what is the minimal genus of a smooth embedded curve in the (complex) projective plane representing a given homology class?
The second half will be a crash course on algebraic surfaces. We will try to assume as little background as possible, and will introduce important concepts (eg the Picard group) via examples.