Algebraic Topology
Andrew Blumberg, Moon Duchin
Brisk review of point-set topology, including second-countability and paracompactness. Many examples, including projective spaces and the compact-open topology. One-point compactification. Covering spaces, fundamental group, and homotopy.
handout: Topology--pdf/dvi

Tools of Analysis
Justin Holmer, Sharon McCathern
Basic definitions: measure and measurable functions. Lebesgue measure, the Stone-Weierstrass Theorem, Hilbert spaces, Banach spaces.
handout: Analysis Defs and Exercises--pdf/dvi

Groups and Galois theory
Brian Johnson, Haris Skiadas
Focus on Galois theory: Integral domains, fields and extensions, algebraic and transcendental elements. Separability, splitting fields, algebraic closure. Examples: C and Q, quadratic extensions of Q. Normal extensions, Galois extensions, the Fundamental Theorem of Galois Theory.
handout: Basic algebra review--pdf/dvi
handout: Galois theory --pdf/dvi

Linear Algebra
Dan Grossman
Vector spaces, endomorphisms, normal forms (Jordan, rational canonical), bilinear forms and adjoints, dual vector spaces. Piles of examples. Tensor products, exterior powers, symmetric powers.
handout: Linear algebra--pdf/dvi

Manifolds
Mark Behrens, Ben Lee
Definition: Whitney Embedding to motivate the definition (eg, need second-countability to rule out Long Line, which does not embed). Partitions of unity. Atlases: smooth and other structures. The tangent space defined via embeddings, through velocity of curves, and through derivations.
handout: Manifold Basics--pdf/dvi

Vector Bundles
Moon Duchin, Steve Wang
Definition, triviality vs. local triviality, classifying line bundles over the circle as a first example. Sections; vector fields as an example (section of the tangent bundle). Associated bundles: product, quotient, subbundle, etc. The tangent bundle and parallelizability of manifolds. Notion of a principal bundle.
handout: Exercises--pdf/dvi

Fourier Series
Mark Behrens, Justin Holmer
Motivation, definitions, tools: Riemann-Lebesgue Lemma and Dirichlet kernel. Criteria for pointwise and absolute convergence. Example: a continuous function whose Fourier series diverges at a point. L^2 theory, Plancherel.
handout: basic Fourier analysis--pdf/dvi

Riemannian metrics
Pallavi Dani, Dan Margalit
General definition; hyperbolic plane H^2 as the main example.
handout: Isom(H^2)--pdf/dvi

Lie groups
David Ben-Zvi, Karin Melnick
Lie groups as groups of symmetries arising in geometry; homogeneous spaces focusing on SL_2 / H^2 and SO_3 / S^2. Definition, left-invariant vector fields and Lie algebras. Computation and examples. Exponential map, done explicitly for linear groups. Lie group-Lie algebra correspondence.
handout: Lie groups--pdf/dvi

Forms and homology
Mark Behrens, David Ben-Zvi
Differential forms on subsets of R^n with a strictly calculus perspective. Homology defined for a space with a specific finite triangulation, in terms of chains and boundaries. Everything calculuated explicitly on the annulus. Existence of a closed but not exact form related to existence of a chain which is a cycle but not a boundary.