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--ps/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.

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

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 calculated 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.

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