Math 347     Advanced Real Analysis     Summer 2011

Instructor: Sunil Chebolu

• Office: Stevenson 303B
• Email: szchebol@ilstu.eduz (with the two "z"'s removed)
• Office Hours: TTh 3:15 - 4:15 in STV 303B or by appointment.
• Class meeting time: MTWTh 1:00 - 3:15 in STV 126.
• Text: Analysis W/Intro To Proof (ISBN 9780131481015) by S. Lay, 4th edition.

Announcements:

• (June 18) Pick a project from the suggested list of project below and start working on it soon.

Projects and presentations: Each group will work on an assigned project and will present it in the last week of classes. Here are some interesting and exciting topics from which you can pick. You can borrow the relevant references from me.

• The sum of k-th powers of the first n-natural numbers. The sum of the first n natural numbers is well-known; baby Gauss formula. The sum of the squares and cubes of the first n natural numbers is also well-known. But what is not so well-known is the formula for the sum of the k-th powers of the first n natural numbers. This is a beautiful formula which is worth investigating. It is a miracle that one can use analytic methods to arrive at this formula which seems to live in the discrete world. The best reference for this is a text book on number theory by Shafarevich.

• The lion and the man. This is an interesting problem proposed by Richard Rado in 1932. Suppose a man finds himself in the redbird arena with a lion. Assume that both the man and the lion can move the entire arena at the same maximum speed. Without leaving the arena, can the man pursue a course of motion ensuring that the lion will never catch him? This problem can be solved using some facts about the harmonic series. A good reference for this problem is "Famous Puzzles of Great Mathematicians" by Miodrag Petkovic.

• The fundamental theorem of algebra. This theorem states that every polynomial of degree n over the field of complex numbers has exactly n roots counting multiplicity. This is a very fundamental theorem and has numerous application. A nice analytic proof of this theorem is given by Anindhya Sen in American Math Monthly "Yet another proof of the fundamental theorem of Algebra." The project is to study this proof and some applications of the theorem.

• Brower's fixed point theorem and the Hex game. This is a superb theorem in topology. A simple version of this states that any continuous self map of the closed unit disc has a fixed point, i.e., a point on the disc that is sent to itself. It turns out that this topological theorem is equivalent to the determinacy of the game of Lex; that is, the game of Lex can never end in a draw. A good reference is a paper by David Gale.

• Continuous everywhere but nowhere differentiable. We are all familiar with functions that are continuous but not differentiable. The famous example is the continuous function f(x) = |x| which fails to admit a derivative at the origin. However, this is just one point where it is not differentiable. Does there exists a continuous function that is nowhere differentiable? Your first guess is probably "no." But as it turns out, there are plenty of functions which have this bizarre property. A famous example is the path of Brownian motion and there are other examples in the world of fractals. See theorem 36.9 in our textbook; page 333.

• The Cantor set and the space filling curve. The Cantor set is perhaps the most famous subset of the real line with endless surprises. It can be used to construct space filling curves-- curves which are defined on the unit interval (I) and whose range is the I x I. Wikipedia has some useful information for this project.

4th Edition

Section 5 1, 2, 4, 15, 18 1,2, 4, 5, 26
Section 6 1, 4, 6, 11 1, 10, 12, 19
Section 7 1, 8, 13, 16, 18 1, 10, 19, 26, 30
Section 8 1, 3, 9 13 1, 3, 9, 13
Section 9 skip skip
Section 10 3, 4, 6, 10, 12 3, 4, 6, 14, 16
Section 11 3, 7, 6, 8 3, 7, 6, 8
Section 12 1, 3, 6, 10, 11 1, 3, 6, 10, 11
Section 13 1, 3, 5, 7, 11, 13, 15, 17 1, 3, 5, 11, 15, 17, 19, 21
Section 14 1, 3, 4, 6, 8, 11 1, 3, 4, 6, 8, 11
Section 15 1, 3, 8, 9, 10, 15, 17 1, 3, 10, 11, 12, 17, 19
Section 16 1,5, 7, 8, 13 1, 7, 9, 10, 15
Section 17 1, 6, 7, 9, 15 1, 6, 9, 9, 15
Section 18 1, 2, 4, 6 1, 2, 4, 10
Section 19 1, 3, 5, 6 1, 3, 5, 8
Section 20 1, 3, 7, 11, 13, 15 1, 3, 9, 13, 15, 17
Section 21 1, 3, 5, 9, 12, 17 1, 3, 5, 9, 12, 17
Section 22 1, 3, 4, 6, 7, 9 1, 3, 4, 6, 7, 9
Section 23 1, 3, 5, 6, 11, 14 1, 3, 5, 6, 11, 15
Section 24 skip skip
Section 25 3, 5, 7, 10, 17 4, 5, 7, 11, 18
Section 26 1, 5, 6, 10, 12, 14, 15 1, 5, 8, 12, 14, 16, 17
Section 27 3, 4, 6, 7, 8 3, 4, 6, 7, 8
Section 28 1, 3, 4, 6, 13 1, 3, 4, 6, 13

Presentations:

Here is the schedule of the presentations for the final week. It is important that you attend ALL talks and not just your own talk. There is a lot that you (and I) can learn from these talks. So please don't skip any talks. In fact, you may not get any credit if you don't attend all presentations.

Your goal: A student who walks out of the classroom after your presentation should think: "wow! That's pretty cool stuff. I really understood it." That is your goal -- make every student in the class feel that way. To this end, you may want to keep the following things in mind.

1. Explain very clearly at the beginning what you are going to present.
2. Try to convey the big picture of the topic/proof before getting into technicalities which might obscure the beautiful underlying ideas. If time is short, you don't have to present all the proofs. Mention the key steps and explain how you can link them to get the result.
3. Your board work should be tidy and your handwriting should be legible.
4. Go slow and pause for questions.
5. Use pictures whenever appropriate to illustrate ideas.
6. Use of technology in the classroom is encouraged but certainly not necessary. A traditional chalk and board discussion is good enough.
7. Finally make sure you follow the c^5 rule: Be Clear, Concise, Coherent, Creative, and Cool. That is the key to learning the art of writing or presenting beautiful mathematics. As the famous English Mathematician G. H. Hardy said, "There is no permanent place for ugly mathematics in the world."

Abstract

Monday, July 18, 1:00 - 2:00 On the sum of the kth powers of the first n natural numbers Amber Anderson
Lindsey Gaworski
Jill Horne
TBA
Monday, July 18, 2:15 - 3:15 The fundamental Theorem of Algebra Chris Pedersen
Sam Stalter
and Beth Waller
TBA
Tuesday, July 19, 1:00 - 2:00 The Cantor set and the space filling curve Christy Engel
Kate Lawrence
Xiheng Tong
Lisa Schuckman
TBA
Tuesday, July 19, 2:15 - 3:15 Continuous but nowhere differentiable curves Will Cragoe
Sara Poland
Michael Sanchez
TBA.
Wednesday July 20, 1:00 - 2:00 How can a man escape from a lion in a circular arena? Eric Larson
Mike Mayers
Jason Robart
David Rogers
TBA