Graduate Topics

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 \({\mathbb{Q}}\)
  • 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
  • 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