## Algebra

### Categories

• natural transformations
• products
• representable functors
• universal properties
• Yoneda lemma

### Group Theory

First 6 chapters (220 pages) of D&F.

• alternating groups;
• Cayley’s Theorem
• Cauchy’s Theorem
• Class Equation;
• Classification of abelian groups
• commutators
• conjugacy
• cosets
• cycle decomposition
• cyclic groups and subgroups;
• derived series
• Dihedral groups
• direct products
• factor groups
• finitely generated groups and their presentations
• free groups
• Fundamental Theorem of Finitely Generated Abelian Groups and applications
• group actions
• group extensions.
• Hölder Program
• homomorphisms
• Isomorphism Theorems (1st - 4th)
• Jordan-Holder theorems;
• Lagrange’s Theorem;
• nilpotent
• normal subgroups
• $$p{\hbox{-}}$$groups
• quotient groups
• semi-direct product groups
• semidirect products.
• simple groups
• simplicity of An
• solvable groups
• subgroup lattice;
• Sylow’s theorems (1-3)
• symmetric groups and permutation representations;
• transpositions

### Commutative Ring Theory

Chapters 7-9 in D&F

• basic theorems about factorization and UFD’s
• bilinear pairings
• chain conditions
• Chinese Remainder Theorem
• completion
• construction of finite fields;
• Dedekind rings
• duality
• Eisenstein’s Criterion
• Euclidean Domains
• factorization in one variable
• free and projective modules
• Gauss’ Lemma
• Hilbert basis theorem
• Hilbert Nullstellensatz
• homomorphisms
• ideals
• integral domains
• integral ring extensions
• irreducibility criteria
• Isomorphism Theorems;
• localization
• maximal ideals
• nilpotence
• Noetherian rings
• polynomial rings
• prime ideals
• Principal Ideal Domains
• quadratic integer rings;
• quotient rings
• rings and subrings;
• rings of fractions;
• structure theory of modules over a PID
• symmetric and alternating forms.
• tensor products
• Unique Factorization Domains
• Zorn’s lemma

### Field Theory

Chapters 13-14 in D&F

• algebraic and transcendental extensions
• algebraic closures
• algebraic extensions
• Computation of Galois groups of polynomials
• cyclic extensions and Kummer theory.
• cyclotomic extensions
• cyclotomic polynomials.
• degrees
• existence and uniqueness of finite fields
• field extensions
• finite fields
• minimal polynomials of algebraic elements
• normal extensions
• separable and inseparable extensions
• separable extensions
• solvability by of polynomials radicals
• splitting fields
• straightedge and compass constructions
• transcendental extensions

### Galois Theory

• abelian extensions of $${\mathbb{Q}}$$
• Compute galois groups for small degree examples
• discriminants
• field automorphisms
• fixed fields
• Fundamental Theorem of Algebra
• Galois theory of finite fields
• insolvability of the quintic.
• Primitive Element Theorem
• roots of polynomials of degree ≤ 4
• solvable and radical extensions
• symmetric functions
• the Fundamental Theorem of Galois Theory
• the Fundamental Theorem of Symmetric Functions

### Module Theory:

Sections 10.1,2,3 and 12.1,2,3.

• direct sums
• free modules
• generation of modules
• modules and submodules
• modules over PIDs
• projective modules
• quotient modules
• rank
• the Fundamental Theorem of Finitely Generated Modules over a P.I.D.
• the Isomorphism Theorems

### Noncommutative Ring Theory

• Artin-Wedderburn theorem;
• group rings
• indecomposable modules
• irreducible modules
• Krull-Schmidt theorem;
• non-semisimple rings
• Semisimple rings

### Representations of Groups

• character tables
• characters of finite groups
• complete reducibility of representations
• esp. finite groups.
• invariant inner products
• matrix coefficients
• parametrization of complex representations by characters
• Peter-Weyl theorem
• Schur orthogonality

## Real Analysis

• $$L^p$$ spaces
• approximate identities;
• Arzela-Ascoli theorem
• closed graph theorem
• compact subsets of C(X)
• convergence theorems
• convolutions on $${\mathbb{R}}^n$$
• covering lemmas
• differentiation of measures
• distribution functions
• dominated convergence theorem
• Egoroff’s theorem
• Fatou’s lemma
• Fourier analysis
• Fubini’s theorem;
• Functional analysis
• functions of bounded variation
• Hahn decompositions
• Hahn-Banach theorem
• Hausdorff measures
• Hilbert space
• Holder’s inequality
• Jensen’s inequality
• Jordan decomposition theorem
• Lebesgue decomposition theorem
• Lebesgue dominated convergence theorem
• Lebesgue integration;
• linear functionals
• linear operators
• Lusin’s theorem
• measure theory;
• Minkowski inequalities
• monotone convergence theorem
• open mapping theorem
• orthonormal systems
• Plancherel’s theorem
• Poisson summation formula
• Riemann-Lebesgue lemma
• Riesz representation theorem
• self-adjoint linear operators and their spectra;
• signed measures
• strong/weak/weak* topologies
• the Baire Category theorem
• The spaces C(X)
• the Stone-Weierstrass theorem.
• the Tychonoff theorem
• trigonometric series
• uniform boundedness principle
• uniform convergence
• Urysohn’s lemma
• Weak $$L^p$$ spaces

## Complex Analysis

• Analytic functions
• Casorati-Weierstrass theorem
• Cauchy’s theorem on multiply connected domains
• Cauchy’s theorem: Goursat’s proof
• Compact families of analytic and harmonic functions:
• Conformal mappings
• conjugate functions
• consequences of Cauchy integral formula
• Dirichlet problem for a disk
• Elementary mappings
• elliptic functions.
• exponential and logarithm functions
• fractional linear transformations
• Harmonic functions:
• Harnack’s principle
• Hurwitz theorem
• infinite products
• isolated singularities
• mapping of polygons
• mappings of finitely connected domains.
• maximum modulus principle
• maximum principle
• mean value property
• Mittag-Leffler theorem
• Mobius transformations and spherical representation.
• Morera’s theorem
• open mapping theorem
• Picard’s theorem.
• Poisson integrals
• Poisson- Jensen formula.
• reflections across analytic boundaries
• residue theorem
• Riemann mapping theorem
• Rouche’s theorem
• Schwarz lemma
• Schwarz reflection principle.
• series and product developments
• Subharmonic functions
• such as Liouville’s theorem
• sums of power series
• the argument principle
• the Dirichlet problem
• The evaluation of definite integrals.
• the hyperbolic metric.
• The monodromy theorem
• Weierstrass product theorem

## Differential Topology

• degree theory
• Ehresmann’s theorem that proper submersions are locally trivial fibrations
• embedding theorem
• Euler characteristic
• integral curves
• manifolds
• Sard’s Theorem on the measure of critical values
• smooth maps
• tangent bundle
• tangent vectors
• the Lefshetz Fixed Point Theorem
• transversality
• vector bundles in general
• vector fields

## Differential Geometry

• de Rham cohomology
• degree theory and Euler characteristic from the viewpoint of de Rham cohomology
• differential forms
• integrable distributions and the Frobenius Theorem
• integration and Stokes’ Theorem
• interpretation of the classical integral theorems as aspects of Stokes’ Theorem for differential forms
• Lie derivatives
• Poincare duality
• Riemannian metrics
• the Mayer-Vietoris sequence
• deRham’s theorem $$H_\text{sing} \cong H_{\text{DR}}$$
• Thom classes
• volume forms

## Algebraic Topology

• axioms of homology theory
• calculation of homology and cohomology of standard spaces
• cell complexes and cellular homology
• cohomology theory
• covering spaces
• fundamental group
• homotopy theory
• Mayer-Vietoris sequence
• singular homology