实分析
- $\sigma$ -algebras
- abstract measure theory
- integration
- bounded variation
- absolute continuity
- complex measures
- Radon-Nikodym theorem
- Baire category theorem
- dual spaces
- Hahn-Banach theorem
- Hilbert space
- orthogonality, projections
- $L^{p}$ spaces
- distribution functions
- interpolation of $L^{p}$ spaces
- the dual of $C_{0}(X)$
- Fourier series
- the Fourier transform
- Plancherel’s theorem
- Fourier analysis on lca groups
- Sobolev spaces
- Lebesgue measure theory on $\mathbb{R}^{N}$
- measurable functions
- product measures, Fubini’s theorem
- monotonic functions
- signed measures
- differentiation of measures
- Lebesgue’s theorems
- Fatou’s lemma
- Banach spaces
- uniform boundedness principle
- open mapping principle
- Riesz representation theorem
- spectral theory
- duality of $L^{p}$ spaces
- weak $L^{p}$
- positive functionals on $C_{c}(X)$
- the Haar integral
- summability methods
- $L^{2}$ theory in Fourier analysis
- Fourier inversion
- distributions
复分析
- complex numbers
- harmonic functions
- $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$
- elementary functions $\left(e^{z}, \log z, \arg z\right)$
- power series
- Liouville’s theorem
- Schwarz-Pick lemma
- Laurent series
- Casorati-Weierstrass theorem
- Cauchy integral theorem
- Runge’s theorem
- existence of primitives and logs
- real integrals by residues
- argument principle
- local mapping theorem
- Ascoli-Arzéla theorem
- Riemann mapping theorem
- Schwarz-Christoffel formula
- theta functions
- Weierstrass factorization
- complex differentiation
- harmonic conjugates
- linear fractional transformations
- elementary Riemann surfaces
- complex integration
- maximum modulus principle
- mean value property (and converse)
- classification of singularities and zeros
- residues
- Cauchy integral formula
- simply connected domains
- residue theorem
- Poisson integral formula
- Rouché’s theorem
- normal families
- Montel’s theorem
- Schwarz reflection principle
- mappings of rectangles,elliptic integrals
- Mittag-Leffier theorem
- Blaschke products
- factoring by Blaschke products
- Hadamard theorem on lacunarity
- product representation
- Stirling’s formula
- Riemann $\zeta$-function
- functional equation for $\zeta$
- periodic functions
- covering spaces
- fundamental domain of modular group
- $\sigma$ -function
- congruence subgroup mod 2
- analytic continuation
- Picard’s little theorem
- subharmonic functions
- growth and zeros of entire functions
- univalent functions
- infinite products
- Weierstrass theorem
- Jensen’s formula
- boundary theory of $H^{p}$ functions
- inner and outer functions
- gamma function
- Bohr-Mollerup theorem
- beta function
- Euler product formula
- prime number theorem
- elliptic functions
- modular group
- Weierstrass $\mathcal{P}$ -function
- modular function
- fundamental domain for congruence subgroup
- monodromy theorem
- Picard’s great theorem
- solution of Dirichlet problem
- Nevanlinna characteristic
几何/拓扑
- open, closed sets
- homeomorphisms
- homology
- differential calculus in $\mathbb{R}^{N}$
- smooth maps and their differentials
- rank theorem
- exterior derivative
- exact and closed forms
- differentiable manifolds
- forms and vector fields on a manifold
- orientation
- Stokes’s theorem
- surfaces in $\mathbb{R}^{3}$
- Gaussian curvature
- the Levi-Civita connection
- geodesics and curvature
- geodesic convexity
- symmetric spaces
- the Lie derivative
- foliations
- universal covering space
- covering transformations
- van Kampen theorem
- fundamental group of a surface
- Riemann surfaces
- Lie groups and algebras
- 1-parameter subgroups
- closed subgroup theorem
- matrix groups $(S O(3), S L(2, \mathbb{R}),$ etc. $)$
- symmetric spaces
- homotopy
- vectors and $k$ -forms
- inverse and implicit function theorems
- wedge product
- pull-backs of vectors and forms
- line integrals of 1-forms
- the tangent bundle
- manifolds with boundary
- partitions of unity
- Poincaré’s lemma
- the Gauss map
- intrinsic geometry of surfaces
- Gauss-Bonnet theorem
- completeness and the Hopf-Rinow theorem
- spaces of constant curvature
- the flow of a vector field
- Frobenius theorem
- covering spaces
- path and homotopy lifting
- fundamental group
- classification of compact,oriented surfaces
- cohomology on surfaces
- de Rham cohomology and theorem
- correspondence of Lie algebra and group
- exponential map
- adjoint representation
- homogeneous spaces
- elementary representation theory
- separation axioms
- cohomology
代数
- group theory
- solvability
- free abelian groups
- polynomial rings
- localization
- modules
- direct products of modules
- free modules
- dual spaces
- modules over principal rings
- polynomials over a factorial ring
- Hilbert’s theorem
- symmetric polynomials
- field extensions
- splitting fields
- separable extensions
- inseparable extensions
- roots of unity
- norm and trace
- solvable and radical extensions
- integral ring extensions
- Galois cohomology
- alg. indep. of homomorphisms
- Sylow theorems
- direct sums
- free groups
- group rings
- principal and factorial rings
- homomorphisms of modules
- sums of modules
- vector spaces
- dual modules
- polynomials in one variable
- criteria for irreducibility
- partial fractions
- formal power series
- algebraic closure
- normal extensions
- finite fields
- Galois extensions
- linear independence of characters
- cyclic extensions
- abelian Kummer theory
- integral Galois extensions
- non-abelian Kummer extensions
- normal basis theorem
- infinite Galois extensions
- extensions of homomorphisms
- Hilbert’s Nullstellensatz
- associated primes
- Nakayama’s lemma
- ordered fields
- real zeros and homomorphisms
- dependence and independence
- finite extensions
- complex fields
- characteristic polynomials
- Jordan normal form
- flat modules
- tensor algebra of a module
- exterior algebra
- universal derivations
- homological algebra
- Grothendieck group
- homotopies of morphisms
- delta functors
- homology sequence
- the modular connection
- Noether normalization theorem
- Noetherian rings and modules
- primary decomposition
- discrete valuations
- real fields
- absolute values
- completions
- valuations
- polynomials in complete fields
- eigenvalues
- tensor products
- functorial isomorphisms
- symmetric products
- alternating products
- the de Rham complex
- Euler characteristic
- injective modules
- derived functors
- spectral sequences
参考资料:
-
Steven G. Krantz的A Mathematician’s Survival Guide: Graduate School and Early Career Development
-
你可以通过这里生成Qual了
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普林斯顿过来人的经验分享大全