Analysis I-II-III (Math 521-522-621)An introduct ory sequence in analysisAnalysis I - Math 521Prerequisites for Math 521: Previous serious exposure to understanding andwriting proofs is required in Math 521. Any of the courses Math 341, Math 371,Math 375-76, Math 421, or Math 461 provide this experience. Admission to Math521 is also possible with the consent of the instructor.Course Outline1. The real number system.An axiomatic approach to th e real numbers R; the explicit construction is notpart of this course.2. Metric spaces and basic topology.Finite, countable and uncountable sets. Metric spaces, compact sets, connectedsets, perfect sets.3. Sequences and series.Convergence, Cauchy sequences, mono tone sequences, upper and lower limits,general properties of series, series with nonnegative terms, the role of the geometricseries, the number e, su mma tion by parts, absolute and conditional convergence,multiplication of series, rearrangements.Uniform convergence of a sequence (series) of functions.4. Continuity.Limits of functions, continuous functions, continuity and compact ness, co ntinuityand connectedness.5. Topics from differential and integral calculus.Review of the basics, with some proofs (a repetit ion of math 421 is not intende dhere). Uniform continuity and the existence of the integral for continuous func tions.More on the Riemann integral. Fundamental theorem of calculus and Taylor’stheore m.Uniform convergence and integration, uniform convergence and differentiation.Power series.Improper integrals.Note: The order of topics is flexible. It will naturally depend on t he choice ofthe textbook and the preferences of the instructor. For example, one can studysequences and series in the beginning, and introduce metric spaces later. It isimportant to cover uniform conver gence in Math 521.Analysis II - Math 522Prerequisites for Math 522: Mat h 521, a course in linear alge bra (equivalentof Math 340 or 341 or 375) wh ich can also be taken concurrently, or consent ofinstructo r.Course OutlineWe follow here the outline in Rudin’s book. The order of the top ic s in Math 522is quite flexible.Review of some topics that may not have been c overed in Math 521.6. More on convergence.Approximations of the identity. Approximation by polynomials, the Stone-Weierstrass theorem. Infinite products ( optional).7. Special functions.Exponential functions, and more on power series. Algebraic completeness of thecomplex field. Fourier se ries. Stirling’s formula and the Γ-function.8. Compactness in metric spaces.Charact erizations of compactness in metric spaces, The Arzela-Ascoli theore m(with a concrete ap plication such as the Peano’s existe nce th eorem fo r differentialequations).9. The contraction principle.With applications, in particular existence and uniqueness theorems for differen-tial e qu ations.10. Differential calculus in normed spaces.Including the implicit function theore m and applicat ions.11. Other optional topics.Such as:Rectifiability of curves.The co nstruction of real numbers.Baire category with some applications.Analysis III - Math 62 1To be offered in the Spring of 2011.Prerequisites for Math 621. Math 521-522, or consent of instructor.Course OutlineSeveral approaches to the subject are possible. The following is based on Spiva k’stext “Calcu lus on manifolds”, but topics could certainly be covered in differentorder, a nd with different emphasis.1. Differentiation. Short review of differential calculus in sever al variables.2. Integration in Euclidean spaces. Basic definitions, measure and contentzero, and characterization of Riemann integrability (optiona l) . Fubini’s theorem.Partition of unity. Chang es of variables.3. Some multilinear algebra. Review of determinants. Multilinear maps,tensors, alternating tenso rs, wedge pr oduct.4. Fields and forms. Vector fields, differential forms, differential of a form,closed and exact forms.5. Manifolds. Basic concept of a manifold, definition(s) of tangent sp ace, fieldsand forms on manifolds, orientation.6. Integration. Stokes’ theorem on manifolds.Euclidean measure for submanifolds of Rnand the classical theorems in vectoranalysis by Green, Gau ss and Stokes. The form f (z)dz and Cau chy’s integr altheore m.7. Oth er optional topics (if time