This is a course about 3-manifolds and hyperbolic geometry. We will start by understanding some of the key theorems underlying the current study of geometric topology in dimension 3; see [Po2019]. This will follow Thurston's original notes [Th1980] to some degree, but we will suplement with the relevant chapters of Ratcliffe's book [Ra2006]. From there, the course will go in one of (at least) two possible directions: towards the study of character varieties (with some of [CAM2020] as a goal) or in the direction of essential surfaces and mapping class groups (with some of [HS2009] as a goal). Further references will be added as the course proceeds.
As a starting point, this course will assume familiarity with knot theory and three-manifolds. A great reference for this is Dale Rolfsen's classic book Knots and Links [Ro1976].
The image on the right is borrowed from Thurston's famous notes based on courses he gave at Princeton in the early 1980s [Th1980].
[CAM2020] Ted Chinburg, Alan Reid, and Matthew Stover Azumaya Algebras and Canonical Components IMRN (2020)
[NS2009] Hossein Namazi and Juan Souto Heegaard splittings and pseudo-Anosov maps GAFA (2009)
[Po2019] Joan Porti Hyperbolic knot theory Notes from Quy Nhon University (2019)
[Ra2006] John Ratcliffe Foundations of hyperbolic manifolds Graduate Texts in Mathematics (2006)
[Ro1976] Dale Rolfsen Knots and Links Publish or Perish Press (1976)
[Th1980] William Thurston Geometry and topology of three-manifolds Notes from Princeton University (1980)