MATH S4061.002: Intro to Modern Analysis
  Summer 2014

Lecture: TR 6:15pm - 7:50pm, TBA

• Course Details
• Schedule of Lectures and Homeworks

Announcements

All announcements will be posted in CourseWorks.

Course Syllabus

Instructor:  Michael Woodbury (x4-4988, 427 Mathematics, woodbury@math.columbia.edu)

Office Hours: TBD; by appointment

Teaching Assistants: Graduate TA: Mike Wong

Text: Rudin Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill, 1976.

You are not required to purchase this book, in fact, I recommend that if you do buy the book, you avoid looking at it until after you've completed your homework. However, if you are planning to take Intro to Modern Analysis II, this book will be required for that course.

Course description: Material (roughly) equivalent to chapters 1-7 of Rudin

The algebra of sets; ordered sets, the real number system, Euclidean space. Finite, countable, and uncountable sets. Elements of general topology: metric spaces, open and closed sets, completeness and compactness, perfect sets. Sequences and series of real numbers, especially power series; the number e. Continuous maps.

Functions of real variable: continuity and differentiability, the chain and L'Hopital rules. The Riemann integral: characterizations, mean-value theorems, the fundamental theorem of calculus. Uniform convergence; its relevance in continuity, integration and differentiation. Sequences and series of functions; double series.

Approximations: the Stone-Weierstrass theorem, Bernstein polynomials. Euler/Mac Laurin, DeMoivre, Wallis and Stirling. Taylor approximations, Newton's method. The DeMoivre/Laplace and Poisson approximations to the bimomial distribution; examples.

Monotone functions, functions of finite variation. Infinitely-differentiable functions. Continuous functions which are nowhere differentiable. Convex sets, their separation properties. Convex functions, their differentiability and their relevance.

Prerequisites: Calculus IV (Math V1202) and Lineral Algebra (Math V2010).

Important Dates:

1. May 27: First day of classes
2. August 14: Last day of classes
3. Final Exam: TBD

1. Don't procrastinate. Research has shown to we learn better if we are consistent in our efforts to learn something new. (To help you in your efforts to not procrastinate, homework will be collected daily.)
2. Be an active learner. Watching me or a TA/tutor do math without doing work on your own will not be enough to be able to reproduce the work yourself on exams. (Participation in class is crucial. Even when you aren't sure whether your solution is correct, or even if you only have an inkling of what to do, you'll learn from presenting your unpolished thoughts.)
3. Ask questions in class, and utilize office hours if needed.
4. As unnatural as it may seem, be willing to struggle with the material without searching for others' solutions to the problems. Also be willing to present your ideas in class even if you know you haven't reached a conclusive solution. Forcing yourself to work this way will help you develop strategies and confidence to solve problems on your own. (A good internet search will reveal solutions/help to most of the problems you encounter, but I want you to learn to think for yourself.)
5. Do your best on the daily homework. As discussed below, from a grade standpoint, you need only try all of the problems to "do well" on the daily homework. However, if you think deeply about the problems before class, the classroom discussion will be so much more valuable.
6. Realize that it's okay to be stuck--this is what math is about: getting stuck and trying to invent ways to work around the obstacles.

Resources: My office hours and the helproom will give you the chance to talk to someone who has a "big picture view." I also encourage you to use your classmates as a sounding board. Collaboration is highly recommended.

Homework: There are two types of written homework.

• Daily homework will be collected each day of class. It will be graded on neatness and completeness. Neatness means that in addition to being legible, you should not turn in pages with frayed edges (i.e. from a spiral bound notebook), and you should staple neatly. Completeness means that you make a reasonable effort to solve each problem.
• Weekly homework will be chosen (by me) problems from the daily homework of the previous week. This must be typed. It will be completed in groups.

Some additional thoughts on homework:

• You are welcome--in fact, encouraged--to collaborate. On daily homework, each student must write his/her own solutions, and names of collaborators should be included.
• Late homework will not be accepted for a grade. (Late weekly homework will be evaluated for feedback, but a score of zero will be given.)

Exams: We will have one midterm, and one final

• See schedule below for dates. Note that these dates are tentative and may change.
• The final exam will be TBA. Exams will be either in-class or take-home, to be determined.
• There will be no make-up exams without a note from a doctor or a dean.
• Let me know immediately if there's going to be a conflict

Grading: The grading scheme is as follows:

• Participation: 25%
• Homework: 30% (split between weekly and daily)
• Midterm: 20%
• Final: 25%

• Grading of Presentations. Your participation grade will be determined mostly by how many times you present. This means that getting up in front of the class is more important that getting up in front of the class and perfectly presenting a perfect solution. However, I will keep track of your presentations by assigning a score according to the following table. (This score may have a mild impact on your overall presentation grade.)

• Grading of Weekly Homework. Typically, the weekly homework will be comprised of eight problems to be completed by groups of four students. Each student will take charge of two problems. Each problem will be scored out of four possible points according to the table below. In calculating the student's overall score, both the scores on the problems he/she is directly in charge of, as well as the other problems will be used.

• Grading of Daily Homework. Presumably there will be 18-20 total daily homework assignments. The lowest 3-4 scores will be dropped before computing the final grade. Each daily homework will be scored as either ✓+, ✓, or ✓- according to the following guidelines. (Your score will go down by one grade if you fail to meet "neatnes" requirements: stapled, no frayed edges, legible, ...)