Linear Algebra

• Matrices, determinants, vector spaces, subspaces, bases, dimensions

• Linear maps, isomorphisms, kernel, image, rank

• Characteristic polynomial, eigenvalues, eigenvectors

• Vector spaces with symmetric and alternating inner products

• Matrix representations of linear maps and inner products

• Normal forms for symmetric, hermetian, and general linear maps, diagonalization

• Orthogonal, unitary, hermitian matrices

• Multilinear algebra: tensor products, exteriors, symmetric algebras

Group Theory

• Groups, subgroups, cosets, Lagrange’s theorem, the homomorphism theorems, quotient groups

• Permutation groups, alternating groups, matrix groups, dihedral groups, quaternion groups

• Free groups, groups described by generators and relations, free abelian groups

• Automorphisms, direct and semidirect products

• P-groups, the class equation, applications

• Group actions on a set, Sylow theorems

• Nilpotent and solvable groups, simple groups

Ring and Module Theory

• Ideals, integral domains, quotients rings, polynomial rings, matrix rings

• Euclidian domains, principal ideal domains, unique factorization

• Chinese Remainder Theorem, prime ideals, localization

• Modules over a PID, applications to a normal form

• Free modules, short exact sequences  

Field Theory

• Algebraic and transcendental extensions, normal extensions, separability

• Finite algebraic extensions, splitting fields, Galois theory, perfect fields

• Finite field

• Cyclotomic polynomials, cyclotomic extensions of the rationals and of finite fields

Textbooks and references

• Algebra by Artin

• Abstract Algebra second edition by Dummit and Foote

• Algebra by Lang

• Topics in Algebra by Herstein

• Algebra by Hungerford

• Algebra by Jacobson

Basic Analysis

• Sequences and series functions, uniform convergence, Fourier series

• Differentiation and integration of real and complex valued functions

• Functions of bounded variation, the Riemann-Stieltjes integral

• The implicit function theorem, the inverse function theorem

General Topology

• Open and closed sets, topological spaces, bases, Hausdorff spaces

• Continuous functions, the product topology, Tychonoff theorem

• Locally compact spaces, Urysohn’s lemma, partition of unity

• Nowhere dense set, set of the first category, Baire category theorem

Metric Spaces

• Distance function, metric spaces

• Convergence, Cauchy sequences, completeness

• The contraction mapping theorem

• Continuous functions on metric spaces

• Arzela-Ascoli theorem and applications

Measure and Integration

• Lebesgue measure, Borel sets, measurable sets, additivity

• Abstract measure, o-algebra, construction of measure, Caratheodory criterion

• Measurable function, Egorov theorem

• Pointwise convergence, uniform convergence, Vitali-Lusin theorem

• Lebesgue integration, monotone convergence theorem, Fatou lemma

• Lebesgue dominated convergence theorem, convergence in measure

• Relations between different notions of convergence

• Product measure, complete measure, Fubini theorem

• Lp space, Holder and Minkowski inequalities, Cheveshev’s inequality

• Radon-Nikodym theorem, Lebesgue theorem

Complex Analysis

• analytic functions, Cauchy-Riemann equations

• Cauchy integral theorem, Cauchy integral formula

• Singularities, poles, the theory of residues, evaluation of integrals

• Maximum modulus theorem

• Argument principle and Rouche’s theorem

• Linear fractional transformation

Functional Analysis

• normal linear space, Banach space

• Linear functional, linear operator, continuity and boundedness

• Hahn-Banach theorem

• Uniform boundedness theorem

• Open mapping and closed graph theorems

• Weak and weak* topology, reflexive space, Banach-Alaoglu theorem

• Inner product, Hilbert space, orthonormal bases, Riesz representation theorem

• Self-adjoint operator, compact operator, and their spectrum

• Fredholm alternative property, Fredholm operator

• Fourier transform, rapidly decreasing function, Fourier transform on L2

Textbooks and references: 

• The Way of Analysis by Robert Strichartz

• Principles of Mathematical Analysis by Walter Rudin

• Elementary Real Analysis by Brian Thomson

• Judith Bruckner and Andrew Bruckner

• Real and Complex Analysis by Walter Rudin

• Real Variable and Integration by John Benedetto

• Real Analysis by Royden

• Measure and Integration Theory by H. Widom

• Complex Analysis by Ahlfors

• Complex Variables and Applications by Churchill

• Functional Analysis by Rudin

• Functional Analysis by Ronald Larson

• Functional Analysis by Yosida

• Partial Differential Equations by Evans

Manifold and Tangent Bundle

• Examples of manifolds, orientation

• Inverse function theorem and implicit function theorem, immersion, submersion

• Partition of unity, embedding, Whitney embedding theorem

• Sard’s theorem

• Tangent vector, tangent bundle, push-forward

• ODE on manifolds, existence and uniqueness theory

• Flows, Lie bracket, Forbenius’ theorem

• Riemannian metrics, examples

• Basic Lie groups

Differential Forms and Integration on Manifolds

• Cotangent bundle, exterior differentiation, contraction, Lie derivative, de Rham differential, Cartan formula

• Integration on manifolds, Stokes’ theorem

• De Rham cohomology, de Rham theorem, examples

• More applications of Stokes; theorem, degree and winding number

• Frobenius’ theorem, foliations, non-integrable distributions

Fundamental Group and Covering Space

• Fundamental groups, calculations, Van Kampen theorem

• Covering spaces, properties, classification of covering spaces

(Co)homology

• Simplicial and CW complexes, examples

• Singular (co)homology, properties, calculations, exact sequences for singular (co)homology

• Betti number, Euler number

• Eilenberg-Steenrod axioms for homology

• Mayers-Vietoris sequences

• Cup and cap products, and Poincare duality for manifolds

• Degree, Euler characteristics, applications

• Lefschetz fixed point theorem and applications

Textbooks and references: 

• Introduction to Smooth Manifolds by John M. Lee

• Foundations of Differential Manifolds and Lie Groups by Frank W. Warner

• An Introduction to Differentiable Manifolds and Riemannian Geometry by W. M. Boothby

• Algebraic Topology by Allen Hatcher

• Introduction to Topology by V. A. Vassiliev

• A Basic Course in Algebraic Topology by W. S. Massey

• Algebraic Topology by Marvin J. Greenburg

• Riemannian Geometry by Manfredo do Carmo