Algebra
Categories
- adjoint functors
- 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
- nilradicals
- 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 \({\mathbf{Q}}\)
- 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 \({\mathbf{R}}^n\)
- 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
- Radon-Nikodym theorem
- 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
Topology
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
- gradients
- 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