*Introduction to 4-Manifolds* Winter 2018

### Instructors: Danny Calegari; Benson Farb

### MWF 1:30-2:20 Eckhart 203

### Description of course:

This course is an introduction to the geometry and topology of smooth
4-manifolds, especially those with `extra' structure (eg symplectic, complex).

### Cancellations:

None yet.

### Notices:

None yet.

### Syllabus:

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.

### Notes from class:

Notes from class will be posted online

here and updated as we go along.

### References:

The following references are just suggestions:

- R. Bryant, An Introduction to Lie Groups and Symplectic Geometry, in: Geometry and Quantum Field Theory, IAS/Park City Mathematics Series Vol. 1
- R. Gompf and A. Stipcicz, 4-Manifolds and Kirby Calculus, AMS GSM 20
- R. Kirby, The Topology of 4-Manifolds; Springer LNM 1374
- J. Moore, Lectures on Seiberg-Witten Invariants; Springer LNM 1629