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 1-6, (Prentice-Hall, 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)

Point-Set Topology

• Open and closed sets
• Continuous functions
• Connectedness
• Compactness
• Hausdorff
• Normality
• Metric spaces, $\RR^n$
• Heine-Borel theorem

Reference:

• J. Munkres, ‘Topology, A First Course‘, Part I (Prentice-Hall)

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‘ (McGraw-Hill, 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‘, (McGraw-Hill, 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 semi-simple 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 1-6)
• Fulton-Harris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics)
• Lie groups and algebras, Chapters 7-10

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 Riemann-Hurwitz formula
• Riemann-Roch 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
• Mayer-Vietoris
• 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
    • Pull-back bundles
• Differential forms:
  • Definition of a differential form
  • Exterior product
  • Exterior derivative
  • De Rham cohomology
• Behavior under pull-back
• 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 Levi-Cevita connection
  • Properties of the Riemann curvature tensor
• 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
  • Radon-Nikodym 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
• Lieb-Loss: 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