Preliminary Examination Topics

Algebra Topics | Analysis Topics | Geometry-Topology Topics

 | Topics and Syllabi for Core Sequences

Algebra Topics

1. 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 representrations 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
2. 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
3. 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  
4. 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

Analysis Topics

1. 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
2. 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
3. Metric Spaces
   • Distance function, metric spaces
   • Convergence, Cauchy sequences, completeness
   • The contraction mapping theorem
   • Continuous functions on metric spaces
   • Arzela-Ascoli theorem and applications
4. 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
5. 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
6. 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.

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Geometry-Topology Topics

1. 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
2. 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
3. Fundamental Group and Covering Space
   • Fundamental groups, calculations, Van Kampen theorem
   • Covering spaces, properties, classification of covering spaces
4. (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 (available online), 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.

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