Part 1: Surfaces and Topology. In this part, we define such
concepts as surface, Euler characteristic, fundamental group, deck
group, and covering space. We prove that the deck group of a surface
and its fundamental group are isomorphic. We also prove, under some
conditions, that a space has a universal cover.
Part 2: Surfaces and Geometry. The first 3 chapters in this
part introduce Euclidean, spherical, and hyperbolic geometry, respec-
tively. (In the Euclidean case, which is so well known, we concentrate
on nontrivial theorems.) Following this, we discuss the notion of a
Riemannian metric on a surface. In the final chapter, we discuss
hyperbolic surfaces, as special examples of Riemannian manifolds.
Part 3: Surfaces and Complex Analysis. In this part, we give a
rapid primer on the main points taught in the first semester of com-
plex analysis. Following this, we introduce the concept of a Riemann
surface and prove some results about complex analytic maps between
Part 4: Flat Cone Surfaces. In this part, we define what is meant
by a flat cone surface. As a special case, we consider the notion of
a translation surface. We show how the “aﬃne symmetry group” of
a translation surface, known as the Veech group, leads right back to
complex analysis and hyperbolic geometry. We end this part with an
application to polygonal billiards.
Part 5: The Totality of Surfaces. In this part, we discuss the
basic objects one considers when studying the totality of all flat or
hyperbolic surfaces, namely moduli space, Teichm¨ uller space, and the
mapping class group. As a warmup for the flat-surface case, we discuss
continued fractions and the modular group in detail.
Part 6: Dessert. In this part, we prove 3 classic results in geome-
try. The Banach–Tarski Theorem says that—assuming the Axiom of
Choice—you can cut up a ball of radius 1 into finitely many pieces
and rearrange those pieces into a (solid) ball of radius 2. Dehn’s
Theorem says that you cannot cut up a cube with planar cuts and re-
arrange it into a regular tetrahedron. The Cauchy Rigidity Theorem
says roughly that you cannot flex a convex polyhedron.