# Preliminary Examinations and Basic Course Sequences

A three-course sequence in each of the three fields of algebra, analysis and geometry-topology is offered each year. Preliminary examinations (prelims) are given for each sequence at the beginning, middle and end of each academic year. Students may take the preliminary exams as often as they wish. In the case where only one student arrives at the exam room to take a preliminary exam, that student may either go ahead and take the exam, or opt to take the exam with another topic group (if the schedule allows).

**TOPICS AND SYLLABI FOR BASIC COURSES**

**ALGEBRA**

**MATH 200 Algebra 1 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 Algebra I* by N. Jacobson, *Abstract Algebra* by D. Dummit and R. Foote, *Algebra* by M. Artin.

**MATH 201 Algebra II 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: *Basic Algebra I* by N. Jacobson, *Abstract Algebra* by D. Dummit and R. Foote, *Algebra* by M. Artin.

**MATH 202 Algebra III 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: *Basic Algebra I* by N. Jacobson, *Abstract Algebra* by D. Dummit and R. Foote, *Algebra* by M. Artin.

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

**MATH 203 Algebra IV:** 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: *Basic Algebra I* by N. Jacobson, *Abstract Algebra* by D. Dummit and R. Foote, *Algebra* by M. Artin.

**ANALYSIS**

**MATH 204 Analysis 1 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.

Textbooks 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

**MATH 205 Analysis II 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,.

Textbooks 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.

**MATH 206 Analysis III 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.

Textbooks 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.

**MATH 207 Complex Analysis:** 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**

**MATH 208 Manifolds I 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

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, *Calculus on Manifolds* by Michael Spivak

**MATH 209 Manifolds II 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.

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, *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.

**MATH 210 Manifolds III 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.

Textbooks 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.

**MATH 212 Differential Geometry:** 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.

Textbooks and references: *Riemannian Geometry* by Peter Peterson, *Riemannian Geometry* by John Lee, *Foundations of Differential Manifolds and Lie Groups* by Frank W. Warner, *A Comprehensive Introduction to Differential Geometry* by Michael Spiuvak, *Riemannian Geometry* by do Carmo.