**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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