Some topics to learn for graduate school in Mathematics
While preparing to apply to graduate schools, I searched around a few math department websites to get some idea of topics they expected students to know upon entry or within their first year.
Some of these lists were sourced from the Math department pages at Columbia and Harvard, but unfortunately I’ve forgotten where the rest came from. I’ve just decided to toss all of it online for anyone that might find such a thing useful!
Undergraduate
Linear Algebra
 Finite dimensional vector spaces (over $\RR$)
 And linear maps between them
 Subspaces
 Quotient spaces
 Dimension
 Bases
 Matrix representations
 Positive definite inner products
 Orthonormal bases
 Extensions of orthonormal subsets
 Eigenvalues and eigenvectors for automorphisms
 Characteristic polynomials
References:
 M. Artin, ‘Algebra‘ (Prentice Hall, 1991), Chapters 1,3,4
 K. Hoffman and R. Kunze, ‘Linear Algebra‘, Chapters 16, (PrenticeHall, 1971)
Abstract Algebra
 Definitions of groups, rings, fields, and modules over a ring
 Homomorphisms of these objects
 Subgroups, normal subgroups, quotient groups
 Cyclic groups
 The structure theorem for finitely generated abelian groups
 Ideals, prime and maximal and their quotients
 Basic examples such as $\ZZ$, $k[x]$, rings of algebraic integers
 Field extensions
 Splitting fields of polynomials
 Normal extensions
References:
 M. Artin, ‘Algebra‘, Chapters 2, 10, 11, 12, 13, 14
 I. Herstein, ‘Topics in Algebra‘ (Blaisdell Publishers, 1964)
PointSet Topology
 Open and closed sets
 Continuous functions
 Connectedness
 Compactness
 Hausdorff
 Normality
 Metric spaces, $\RR^n$
 HeineBorel theorem
Reference:
 J. Munkres, ‘Topology, A First Course‘, Part I (PrenticeHall)
Calculus
 Differential of a smooth mapping between open subsets in Euclidean spaces
 Matrix of partial derivatives
 Inverse and implicit functions
 Multivariable Riemann integration
References:
 W. Rudin, ‘Principles of Mathematical Analysis‘ (McGrawHill, 1964)
 A. Browder, ‘Mathematical Analysis: An Introduction‘ (Springer, 1996)
Complex Analysis:
 Definition of holomorphic functions
 Cauchy integral formula
 Power series representations of holomorphic functions
 Radius of convergence
 Meromorphic functions
 Residues
Reference:
 L. Ahlfors, ‘Complex Analysis‘, (McGrawHill, 1973), Chapters 1 5
Real Analysis:

A thorough working knowledge of advanced calculus, at the level of the books of W. Rudin or A. Browder as listed under Calculus
 Pointwise uniform convergence of functions
 Equicontinuity
 $\ell^2$, $L^2(S^1)$
 Hilbert spaces
 Orthonormal bases
First Year Graduate
Algebra
 Group theory:
 Sylow theorems
 $p$groups
 Solvable groups
 Free groups
 Rings and modules:
 Tensor products
 Determinants
 Jordan canonical form
 PID’s
 UFD’s
 Polynomials rings
 Field theory:
 Splitting fields
 Separable and inseparable extensions
 Galois theory:
 Fundamental theorems of Galois theory
 Finite fields
 Cyclotomic fields
 Representations of Finite Groups:
 Character theory
 Induced representations
 Structure of the group ring
 Basics of Lie groups and Lie algebras:
 Exponential map
 Nilpotent and semisimple Lie algebras and Lie groups
References:
 Dummit and Foote: Abstract Algebra, 2nd edition, except chapters 15, 16 and 17
 Serre: Representations of Finite Groups (Sections 16)
 FultonHarris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics)
 Lie groups and algebras, Chapters 710
Algebraic Geometry
 Affine and projective varieties;
 Regular functions and maps;
 Cones and projections
 Projective space and Grassmannians
 Ideals of varieties;
 The Nullstellensatz
 Rational functions
 Rational maps and blowing up
 Dimension and degree of a variety;
 The Hilbert function and Hilbert polynomial
 Smooth and singular points of varieties;
 The Zariski tangent space;
 Tangent cones;
 Dual varieties
 Families of varieties (Chow varieties and Hilbert schemes)
 Algebraic curves:
 Genus;
 The genus formula for plane curves
 The RiemannHurwitz formula
 RiemannRoch theorem
References:
 Shafarevich: Basic Algebraic Geometry 1, 2nd edition
 Harris: Algebraic Geometry: A First Course
Complex Analysis
 Holomorphic and meromorphic functions
 Conformal maps
 Linear fractional transformations
 Schwarz’s lemma
 Complex integrals:
 Cauchy’s theorem
 Cauchy integral formula
 Residues
 Harmonic functions:
 The mean value property;
 The reflection principle;
 Dirichlet’s problem
 Series and product developments:
 Laurent series
 Partial fractions expansions
 Canonical products
 Special functions:
 The Gamma function
 The zeta functions
 Elliptic functions
 Basics of Riemann surfaces
 Riemann mapping theorem
 Picard theorems
References:
 Ahlfors: Complex Analysis (3rd edition)
Algebraic Topology
 Fundamental groups
 Covering spaces
 Higher homotopy groups
 Fibrations and the long exact sequence of a fibration
 Singular homology and cohomology
 Relative homology
 CW complexes and the homology of CW complexes
 MayerVietoris
 Universal coefficient theorem
 Kunneth formula
 Poincare duality
 Lefschetz fixed point formula
 Hopf index theorem
 Čech cohomology and de Rham cohomology
 Equivalence between singular, Čech and de Rham cohomology
References:
 A. Hatcher: Algebraic Topology, W. Fulton: Algebraic Topology
 E. Spanier: Algebraic Topology, Greenberg and Harper: Algebraic Topology: A First Course
Differential Geometry
 Basics of smooth manifolds:
 Inverse function theorem
 Implicit function theorem
 Submanifolds
 Integration on manifolds
 Basics of matrix Lie groups over $\RR$ and $\CC$:
 The definitions of $\operatorname{Gl}(n)$, $\operatorname{SU}(n)$, $\operatorname{SO}(n)$, $\operatorname{U}(n)$
 Their manifold structures
 Lie algebras
 Right and left invariant vector fields
 Differential forms
 The exponential map
 Bundles:
 Definition of real and complex vector bundles
 Tangent and cotangent bundles
 Basic operations on bundles such as
 Dual bundle
 Tensor products
 Exterior products
 Direct sums
 Pullback bundles
 Differential forms:
 Definition of a differential form
 Exterior product
 Exterior derivative
 De Rham cohomology
 Behavior under pullback
 Metric Geometry:
 Metrics on vector bundles
 Riemannian metrics
 Definition of a geodesic
 Existence and uniqueness of geodesics
 Definition of curvature, flat connections, parallel transport
 Definition of LeviCevita connection
 Properties of the Riemann curvature tensor
 Metrics on vector bundles
 Principal Bundles:
 Definition of a principal Lie group bundle for matrix groups
 Associated vector bundles: Relation between principal bundles and vector bundles
 Definition of covariant derivative for a vector bundle
 Connection on a principal bundle
References:
 Taubes: Differential geometry: Bundles, Connections, Metrics and Curvature
 Lee: Manifolds and Differential Geometry (Graduate Studies in Math 107, AMS)
 S. Kobayashi and K. Nomizu: Foundations of Differential Geometry
Real Analysis
 Measure Theory
 Borel measure
 Complex measures of bounded variation
 RadonNikodym theorem
 Lebesgue differentiation theorem
 Lebesgue Integration
 Jensen’s inequality
 Convergence theorems for integrals
 $L^p$ spaces
 Duality of $L^p$ space
 Fubini theorem
 ODE/PDE Theory
 Fourier series
 Fourier transform
 Convolution
 Heat equation
 Dirichlet problem
 Fundamental solutions
 Probability Theory
 Central limit theorem
 Law of large numbers
 Conditional probability
 Conditional expectation
 Functional Analysis
 Distributions
 Sobolev embedding theorem
 Hilbert spaces
 Riesz representation theorem
 Maximum principle
References:
 Rudin: Real and complex analysis
 Stein and Shakarchi: Real analysis
 LiebLoss: Analysis, Chapter 2. Fourier series:
 Stein and Shakarchi: Fourier Analysis
 Evans: Partial Differential Equations. Chapter 5
 Shiryayev: Probability
 Feller: An Introduction To Probability Theory And Its Applications
 Durrett: Probability: Theory And Examples