Basic Courses for Graduate Students

Analysis Courses

MATH 4130 Honors Introduction to Analysis I (Fall)

Textbook (Fall 94): The Way of Analysis by Robert S. Strichartz

Topics:

  1. The real number system: Cauchy sequences, limits and completeness
  2. Topology of the real line: the theory of limits, open and closed sets, compact sets
  3. Continuous and differentiable functions: continuous functions, uniform continuity, monotone functions, derivatives of functions, intermediate and mean value theorems, product and chain rules, inverse function theorem, higher derivatives, Taylor series
  4. Riemann integral: existence, fundamental theorem of calculus, integration by parts, change of variable formula
  5. Sequences and series of functions: complex numbers, convergence and absolute convergence, uniform convergence, integration and differentiation of limits, power series, radius of convergence, approximation by polynomials, equicontinuity, Arzela-Ascoli theorem

MATH 4140 Honors Introduction to Analysis II (Spring)

Textbook (Spring 93): The Way of Analysis by R. Strichartz

Topics: Chapters 9-14 were covered.

  1. Euclidean space and metric spaces, completeness, compactness, continuous functions.
  2. Differentiation, the chain rule, higher derivatives.
  3. Ordinary differential equations: existence and uniqueness of solutions.
  4. Fourier series, conditions for convergence.
  5. Implicit function theorem.
  6. Lebesgue integral

MATH 6110 Real Analysis (Fall)

Textbook (Fall 94): Real and Complex Analysis by W. Rudin

Topics: The first nine chapters were covered.

  1. Measures and abstract integration,
  2. Lebesgue measure, spaces,
  3. Hilbert spaces, Banach spaces,
  4. Fourier series,
  5. Hahn-Banach theorem, Radon-Nikodym theorem, Riesz representation theorem,
  6. Differentiation,
  7. Integration on product spaces, Fubini's theorem,
  8. Fourier transforms, Fourier inversion and Plancherel theorems

MATH 6120 Complex Analysis (Spring)

Textbook (Spring 97): Real and Complex Analysis by W. Rudin and notes by Cliff Earle

Topics:

  1. chapter 10 in Rudin
  2. Normal families of holomorphic functions
  3. Riemann mapping theorem
  4. Runge's approximation theorem
  5. The equation
  6. Mittag-Leffler theorem
  7. Weierstrass theorem
  8. Analytic continuation
  9. Harmonic functions
  10. Riemann surfaces

Algebra Courses

MATH 4330 Honors Introduction to (Linear) Algebra (Fall)

Textbook (Fall 1994): Linear Algebra, 2nd edition by Hoffman and Kunze.

Topics:

  1. Matrices and linear equations, vector spaces, bases and dimension.
  2. Linear transformations, representation by matrices.
  3. The algebra of polynomials (over a field), its ideals, factorization of polynomials.
  4. Inner products and bilinear forms. Determinants, multi-linear products.
  5. Characteristic values and polynomials, Cayley-Hamilton theorem.
  6. Canonical forms of matrices.

MATH 4340 Honors Introduction to (Abstract) Algebra (Spring)

Textbook (Spring 1995): Abstract Algebra by Doomed and Foote. The course covered chapters 1, 2, 3.1-3.3, 4, 5.1, 5.4, 5.5, 6.3, 7, 8, 9.1-9.4, 10, 12.1, parts of 13 and 14 (sketch)

Topics:

  1. Group theory: quotients, group actions, Sylow theorems, direct and semi-direct products, simplicity of alternating groups, presentations by generators and relations
  2. Ring theory: ideals, quotient rings, root adjunction, localization, Chinese remainder theorem, PIDs and UFDs, structure of finitely generated modules over PIDs
  3. Introduction to Galois theory (sketch with few proofs): algebraic elements, splitting field, Galois extension, Galois group, Galois correspondence

MATH 6310 Algebra

Textbooks: Algebra by Hungerford, Algebra by S. Lang or Basic Algebra by N. Jacobson. The course is more abstract than 431 and does not necessarily follow a single textbook. In Lang's book see chapters 1-7, 13, 14, 16, 19. See also chapters 1-4 in volume 1, and chapter 1 in volume 2 in Jacobson.

Topics:

  1. Group theory: definitions, cyclic groups, normal subgroups, Sylow theorems, composition series, etc.
  2. Ring theory: definitions, modules, exact sequences, tensor products, symmetric and alternating products, PIDs and UFDs
  3. Galois theory: fields, field extensions, Galois groups, solvability of equations by radicals, compass and ruler constructions.

Topology Courses

MATH 6510 Introductory Algebraic Topology (Spring)

Textbook (Spring 1997): Algebraic Topology I by Allen Hatcher. The course covers chapters 0, 1 and 2. Available online at http://www.math.cornell.edu/~hatcher

Topics:

  1. Cell complexes, homotopy equivalence.
  2. Fundamental group, van Kampen's theorem.
  3. Covering spaces.
  4. Singular homology: verifying the axioms, applications of degree.
  5. Cellular homology, Euler characteristic, Lefschetz fixed point theorem.

Last modified:September 22, 2008