Preliminary Examinations And Basic Graduate Sequences

A three-course sequence in each of the three fields of algebra, analysis and geometry-topology will be offered each year. Graduates must complete the required sequences by the end of their third year or be subject to dismissal. A Ph.D. pass on the preliminary examination can be substituted for the course work in that subject.

Preliminary examinations (prelims) will be given for each sequence at the beginning and end of each academic year. The exam will be designed and graded by a committee of three members, at least two of whom have taught a class in the sequence during the academic year closest to that exam (thus the previous academic year for the Fall exam, and the current academic year for the Spring exam). Students must obtain a Ph.D. level pass on two of the three prelims, or a Ph.D. level pass on one and a Master’s level pass on the remaining two.

Preliminary Exam Timelines

Students may take the written examinations as often as they wish; however all graduate students in the Ph.D. program are expected to pass the required preliminary examinations by the beginning of Fall Quarter, two years after matriculation. Students in the Ph.D. program who do not pass the examinations by the end of their third year should expect to be transferred to the M.A. program, or be subject to dismissal.

TOPICS AND SYLLABI FOR BASIC COURSES

ALGEBRA

Algebra 1 (Math 200)

Group and ring theory: Subgroups, cosets, normal subgroups, homomorphisms, isomorphisms, quotient groups, free groups, generators and relations, group actions on a set. Sylow theorems, semi direct products, simple groups, nilpotent groups and solvable groups. Ring theory, including Chinese remainder theorem, prime ideals, localization, Euclidean domains, PIDs, UFDs, polynomial rings.

Textbook and references:  Basic Algebras I by N. Jacobsen, Abstract Algebra by D. Dummit and R. Foote, Algebras by M. Artin.

Algebra II (Math 201)

Linear algebra:  Vector spaces, linear transformations, eigenvalues and eigenvectors, Jordon canonical forms, bilinear forms, quadratic forms, bilinear forms, quadratic forms, real symmetric forms and real symmetric matrices, orthogonal transformations and orthogonal matrices, Euclidean space, Hermitian forms and Hermitian matrices, Hermitian space, unitary transformations and unitary matrices, skew-symmetric forms, tensor products of vector spaces, tensor algebras, symmetric algebras, exterior algebras, Clifford algebras and spin groups.

Textbook and references:  Algebra by M. Martin, Abstract Algebra by D. Dummit and R. Foote, Basic Algebra by N. Jacobson.

Algebra III (Math 202)

Module theory:  Submodules, quotient modules, module homomorphisms, generators of modules, direct sums, free modules, torsion modules, modules over PIDs and applications to rational and Jordan canonical forms. Field theory, including field extensions, algebraic and transcendental extensions, splitting fields, algebraic closures, separable and normal extensions, the Galois theory, finite fields, Galois theory of polynomials

Textbook and references:  Algebra by M. Artin, Abstract Algebra by D. Dummit and R. Foote, Basic Algebra I by N. Jacobson.

Note: The following course is recommended as a continuation course to the algebra sequence, and as preparation for the preliminary examination.

Algebra IV (Math 203)

Topics include Tensor produce of modules over rings, Projective modules and injective modules, Jacobson radical, Weederburns' theorem, category theory, Noetherian rings, Artinian rings, afine varieties, projective varieties, Hilberts Nullstellensatz, prime spectrum, Zariski topology, discrete valuation rings; Dedekind domains.

Textbooks and references: Algebra by M. Artin, Abstract Algebra by D. Dummit and R. Foote, Basic Algebra I by N. Jacobson.

ANALYSIS

Analysis 1 (Math 204)                

Fundamentals of analysis:  Completeness and compactness for real line, sequences and infinite series of functions, Fourier series, calculus on Euclidean space and implicit function theorem, metric spaces and contracting mapping theorem, Arzela-Ascoli theorem, basics of general topological spaces, Baire category theorem, Urysohn’s lemma, Tychonoff theorem.      

Textbook and references:  The Way of Analysis by Robert Stricharz, Principles of Mathematics by Rudin, Elementary Real Analysis by Thomas, Bruckner and Bruckner, Real and Complex Analysis by Rudin

Analysis II (Math 205)

Measure theory and integration:  Lebesgue measure theory, abstract measure theory, measurable functions, integration, space of absolutely integrable functions, dominated convergence theorem, convergence in measure, Riesz representation theorem, product measure the Fubini theorem, Lpspaces, derivative of a measure and Radon-Nikodym theorem, fundamental theorem of calculus,.

Textbook and references: Real and Complex Analysis by Rudin, Real Variable and Integration by John Benedetto, Real Analysis by Royden, Measure and Integrationn Theory by H. Widom.

Analysis III  (Math 206)

Functional analysis:  Banach space, Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem and closed graph theorem, weak and weak* topology and Banach-Alaoglu theorem, Hilbert space, self-adjoint operators, compact operators, spectral theory, Fredholm operators, space of distributions and Fourier transform, Sobolev spaces.

Textbook and references:  Functional Analysis by Rudin, Functional Analysis by Ronald Larson, Functional Analysis by Yosida, Partial Differential Equations by Evans.

Note: The following course is recommended as a continuation course to the analysis sequence, and as preparation for the preliminary examination.

Complex Analysis (Math 207)

Review of the basic theory of one complex variable, the Cauchy-Riemann equations, Cauchy's theorem, power series expansions, the maximum modulus principle, Classification of singularities, Residue theorem, argument principle, harmonic functions, linear fractional transformations, Conformal mappings, Riemann mapping theorem, Picard's theorem, introduction to Riemann surfaces.

Textbooks and references: Complex Analysis by Ahlfors, Functions of One complex variable by Conway, Complex Variables and Applications Churchill, Elementary Theory of Analytic Functions of One or Several Complex Variables by H. Cartan.

GEOMETRY AND TOPOLOGY

Manifolds I (Math 208)

Theory of manifolds:  Definitions of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard’s theorem and the Whitney embedding theorem, differential forms, exterior derivative, Stokes’ theorem, integration, vector fields, flows, Lie brackets, Frobenius’ theorem

Textbook 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, Calculus on Manifolds by Michael Spivak

Manifolds II (Math 209)

Differential forms and analysis on manifolds: Tensor algebra, differential forms and the associated formalism of pullback, wedge product, exterior derivative, Stokes’ theorem, integration, Cartan’s formula for the Lie derivative, cohomology via differential forms, Poincare lemma and the Mayer-Vietoris sequence, theorems of de Rham and Hodge.

Textbook 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, A Comprehensive Introduction to Differential Geometry by Michael Spivak, Analysis on Manifolds by James R. Munkres, Topology from the Differentiable Viewpoint by John W. Milnor, Foundations of Mechanics by Ralph Abraham and Jerrold E. Marsden, Calculus on Manifolds by Michael Spivak, Lie Groups by J. F. Adams, Differential Forms in Algebraic Tropology by Raoul Bott and Loring W. Tu.

Manifolds III (Math 210)

Algebraic topology:  The fundamental group, covering space theory and the Van Kampen’s theorem (with a discussion of free and amalgamated products of groups), CW complexes, higher homotopy groups, cellular and singular cohomology, the Eilenberg-Steenrod axioms, computational tools (including, e.g., Mayer-Vietoris exact sequences), cup products, Poincare duality, Lefschetz fixed point theorem, homotopy exact sequence of a fibration and the Hurewicz isomorphism theorem, remarks on characteristic classes.

Textbook and references: 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. Greenberg.

Note: The following course is recommended as a continuation course to the geometry-topology sequence, and as preparation for the preliminary examination.

Differential Geometry (Math 212)

Principle bundles, associated bundles and vector bundles, connections on principle and vector bundles. More advanced topics: curvature, introduction to cohomology, the Chern-Weil construction and characteristic classes, the Gauss-Bonnet Theorem or Hodge Theory, eigenvalue estimates for Beltrami Laplacian, comparison theorems in Riemannian geometry. (Formerly course 234C.)

Textbooks and references: Riemannian Gemetry by Peter Peterson, Riemannian geometry by John Lee, Foundations of differential maniufolds and Lie Groups by Frank W. Warner, A comprehensive Introduction to Differential Geometry by Michael Spiuvak, Riemannian geometry by do Carmo.

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