I don't have the final version on hand at the moment...

Kevin Purbhoo's Qual Syllabus (First Attempt)
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Minor (C*-algebras)
spectrum
spectral radius
gelfand transform
gelfand theory for comm. C*-algebras
analytic functional calculus
continuous functional calculus for normal operators
extended functional calculus
the spectral theorem (various forms)
unitalization
approximate units
positive elements
polar decomposition
states
representations
GNS construction

Major (Algebraic topology)
Homotopy theory:
homotopy equivalence
fundamental group
covering spaces
higher homotopy groups
relative homotopy groups
long exact sequence of a pair
CW-complexes
cellular approximation
locally trivial bundles
principal bundles
classifying space
weak homotopy equivalence
Whitehead theorem

Homology/Cohomology
singular
cellular
relative homology groups
long exact sequence
mayer-vietoris sequence
universal coefficient formula
Kunneth formula
Hurewicz theorem
Cup and Cap products
Morse theory
Morse-Smale-Witten complex
Poincare duality
Thom's Jet Transversality theorem (vague notions only)
intersection theory
Characteristic classes
obstruction theory
Chern classes, Stieffel-Whitney classes, Pontryagin classes

Major (Symplectic Geometry)
symplectic linear algebra
symplectic manifolds
Darboux theorem
almost complex structures
Kahler manifolds
Hamiltonian vector fields
Poisson bracket
symplectic and Hamiltonian actions
moment maps
obstuctions for moment maps
coadjoint orbits
symplectic reduction
cotangent bundles
cotangent lifts
more (maybe also some differential geometry)