There are introductory graduate courses in Algebra, Analysis, and Geometry-Topology. Each is offered in two versions: a masters level version and a Ph.D. level version. The masters level algebra course is Math 502/503, and the Ph.D. level course is Math 602/603. In analysis, the corresponding numbers are 508/509 and 608/(609 or 610). In geometry-topology, the corresponding numbers are 500/501 and 600/601.
All Ph.D. students are required to take (or place out of) these 600-level courses, although it is possible to take the 500-level courses first if this seems appropriate.
In addition, all Ph.D. students are required to take a year of algebraic topology, Math 618/619. Below are descriptions of Math 602/603, 608/(609 or 610), and 600/601. See the full list of course descriptions for information about other courses.
The First Year Curriculum in Algebra
Groups, normal subgroups and conjugacy classes, finite groups of order 12.
Readings: Lang, Algebra, Chapter 1; Jacobson, Basic Algebra, Part I, chapter 1.
Rings, polynomial rings in one variable, unique factorization, non-commutative rings - matrix ring.
Readings: Lang, Chapters 5,6,9,10 (para. 1-4), 17. Jacobson, Part I, Chap. 2, Part II, Chap. 4 (para. 1-6), Chap. 7. Symmetric and Hermitian matrices, spectral theorem, quadratic forms and signature.
Readings: Lang, Chap. 3,13,14,15,16. Jacobson, Part I, Chap. 3,6 (para. 1-3), Part II, Chap. 3,5 (para. 1-3).
Definition of a field, field of fractions of an integral domain.
Course content, Algebra I,II (Math 602, 603)
Groups: Sylow's theorem and its applications, finite abelian groups, nilpotent and solvable groups.
Rings: commutative noetherian rings, Hilbert basis theorem, prime and maximal ideals and localizations, primary decomposition, integral extensions and normal rings, Dedekind domains, Eisenstein irreducibility criteria, group ring, semisimple rings and Wedderburn's theorem.
Modules: tensor product, symmetric and exterior algebras and induced maps, exact functors, projective and injective modules, finitely generated modules over a Principal Ideal Domain with application to canonical forms of a matrix over a field, elementary theory of group representations.
Field extensions and Galois theory: separable and inseparable extensions, norm and trace, algebraic and transcendental extensions, transcendence basis, algebraic closure, fundamental theorem of Galois theory, solvability of equations, cyclotomic extensions and explicit computations of Galois groups.
Readings: Lang, Chap. 7,8,10 (para. 1-4); Jacobson, Part I, Chap. 4, Part II, Chap. 8.
General references: Lang - Algebra; Jacobson - Basic Algebra I, II; Atiyah-Macdonald - Introduction to commutative algebra; Kaplansky - Fields and Rings; Artin - Galois Theory; van der Waerden - Modern Algebra; Kaplansky - Commutative rings; Serre - Linear representations of finite groups; Zariski-Samuel, Commutative algebra (Vol. 1).
The First Year Curriculum in Analysis
Axiomatic development of the real number system, especially the completeness axiom; Abstract metric spaces, open and closed sets, completeness, compactness; Continuous functions from one metric space to another, uniform continuity; Continuous functions on a compact metric space have compact image and are uniformly continuous; Pointwise and uniform convergence of sequences and series of functions; continuity of a uniform limit of continuous functions. Differentiation: mean value theorem, Taylor's theorem and Taylor's series, partial differentiation and total differentiability of functions of several variables.
Riemann integration: definition and elementary properties, fundamental theorem of calculus. Interchange of limit operations, of order of partial differentiation, integration of spaces term-by-term. Implicit function theorem. Fourier analysis up to pointwise convergence for piecewise smooth functions. Use of Fourier analysis to solve heat and vibration equations. Differential equations, solution of common forms. Complex numbers, power series and Fourier series (an undergraduate course in complex analysis would be helpful).
Readings: Except for the material on Fourier analysis, the above is all in Rosenlicht's "Introduction to Analysis", Rudin's "Principles of Mathematical Analysis", Boyce and de Prima's "Elementary Differential Equations" and many other books.
Course Content, Analysis I (Math 608)
The first two-thirds of the semester concerns conplex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. The last third of the semester provides an introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C^0 and the Riesz-Markov theorem, L^p spaces, Fubini theorem.
Course Content, Analysis II (Math 609)
The first third of the semester continues the study of real analysis begun in Math 608. Topics will include: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L^2-theory of the Fourier transform. The last two-thirds of the semester concerns functional analysis: normed linear spaces, convexity, the Hahn-Banach Theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, L^p theory for the Fourier transform.
The First Year Curriculum in Geometry-Topology
Basic familiarity with point-set (general) topology: metric spaces, topological spaces, separation axioms, compactness, completeness.
Course Content, Geometry-Topology I (Math 600)
Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.
Readings: M. Spivak, "A Comprehensive Introduction to Differential Geometry", vol.I, 2nd edition. Publish or Perish, 1979. Supplementary: V. Guillemin & A. Pollack, "Differential topology", Prentice-Hall, 1974.
Course content, Geometry-Topology II (Math 601)
Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.
Readings: M.J. Greenberg & J. Harper, "algebraic Topology, a first course". Math Lecture Note Series, vol.58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. Supplementary: A. Hatcher, "Algebraic topology", Cambridge University Press, 2002.
1. I. Singer - J. Thorpe, Lecture notes on elementary topology and geometry.
2. J. Kelley, General topology (look at the exercises).
3. J. Djugundji, Topology.
4. J. Munkres, Topology: a first course.
5. F. Warner, Foundations of differentiable manifolds and Lie groups (first half).
6. M. Spivak, A comprehensive introduction to differential geometry, Vol. 1.
7. H. Flanders, Differential forms, with applications to the physical sciences.
8. W. Fleming, Functions of several variables.
9. G. Simmons, Introduction to topology and modern analysis.
10. F. Croom, Basic concepts of algebraic topology (good for fundamental groups and homology).
11. W. Massey, Algebraic topology, an introduction (good for fundamental groups).
12. C.T. Wall, A geometric introduction to topology.
13. M.A. Armstrong, Basic topology.
14. J. Milnor, Topology from the differentiable viewpoint.
15. Greenberg, Lectures on algebraic topology (first two chapters are good for fundamental groups and covering spaces).
16. Bourbaki, General topology.
17. Guillemin and Pollack, Differential topology (good exercises).
18. Y. Matsushima, Differentiable manifolds.