| Prerequisites: | Multivariable
Calculus, Linear Algebra, Real Analysis or consent of the instructor. |
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| Exams: | 1 self-scheduled midterm exam, and 1 self-scheuled
final exam. The midterm exam will be held Oct 21-24. |
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| Grading Policy: | Homework 45%; midterm 15%;
project 15%; final exam 30%.
The class will not be
graded "on a
curve": if everyone deserves an A, everyone
will get an A. |
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| Attendance: | Attendance will not be taken at
each class. However, it is much harder to learn the material on your own, so you are strongly
encouraged to attend each class. You must complete the midterm exam and final exam. Make-up
exams will only be given in special circumstances. |
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| Drop date: | The drop date for the course
is Friday September 18. |
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Due Date: Draft: 4pm November 20, Essay: 4pm December 11, Talks Dec 14, 15, 16.
Outline: The aim of the project is to allow you to read and learn about a part of topology which is
related to the course, but not in it. You will write an essay of about 5 pages in length and give a 20 minute talk on
what you have learned.
Grading Policy: The grade for the project is worth 15% of your
course grade and consists of three parts:
preparation 3%, essay 7% and talk 5%. You will need to read and think
about your project during the semester, it
is not something to complete the weekend before it is due! In order to get full credit for preparation you
must meet with me (at least) three times during the semester to discuss
your project. (Twice for the essay and once for the talk.) You must also
submit
a draft (1-2 pages) of your essay by November 20 (that is before Thanksgiving break).
Topics: Here are some topics you might choose.
- Constructing the real numbers. There are two ways to do this. This is a nice topic for those who
have never seen
this before and would like to follow up on the analysis connection. The properties of the reals are used in the proof
of the Heine-Borel Theorem for example. Reference: W. Rudin Principles of Mathematical Analysis 3rd Edition Ch
1 and part of Ch 2.
- Classifying surfaces. Understand 2-dimensional surfaces and classify compact ones completely.
Some key ideas used
include Euler Characteristic, genus and triangulation. Indeed, understanding the triangulation of
surfaces is a project in its own right. Reference: M.A. Armstrong Basic Topology, Sue.E.
Goodman Beginning Topology. Bill Meeks "The shape of space". Lei Lei
- Topology and art. There are some beautiful pieces of art and sculpture that rely heavily on
mathematics for their
beauty. Use genus and other topological characteristics to appreciate these works of art in a new way.
References: Start with http://www.isama.org/. Also try "Two Theorems, Two Sculptures, Two Posters." Helaman Ferguson
American Mathematical Monthly, 97 no. 7, 1990 p.586-610. Rebecca
- Drawing correct models of topological spaces. It is surprisingly difficult to draw some of the
spaces we are
learning about in 2-dimensions (on a piece of paper or the blackboard). Take a journey into how to do this and
discover some new spaces along the way. One nice project is to understand the many different immersions
of RP^2 into R^3. Reference: G. Francis "A topological picture book". Katie
- Knot theory. Understand some of the basic ideas behind this subject. References: Colin Adams "The
Knot
Book", K. Murasugi "Knot Theory and its applications". Abbi
- Knot theory and DNA Learn how topology can help biochemists understand some of the mysteries of
long molecules
like proteins and DNA. References: De Witt Sumners "Lifting the Curtain: using topology to probe the hidden
action of enzymes" Notices of the AMS 42, (5) May 1995 p.529. E. Flapan
"When topology meets chemistry".
Dianna
- More point set toplogy or algebraic topology. Learn about other properties of topological spaces, for example
- Under
what conditions does a topological space have a metric? This is Uryson's Metrization theorem. Reference: J.R. Munkres
"Topology".
- Give one of the two proofs of the Jordan Curve theorem found in Chapter 9 of our textbook.
- Understand the beginnings of simplicial homology. Found in Chapter 10/11 of our text, plus in
"Knots and Links" by Peter Cromwell. This project could tie in with a project on a proof of the Jordan Curve theorem
or with a project on knot theory. Sylvia
- Understand De Rham Cohomology. This theory starts from multivariable calculus and linear algebra and
develops a tool to measure capture the "holes" in a space. Key ideas includes tensor products and dual vector spaces.
This is a piece of mathematics you can develop for yourself following a set of exercises.
- Topology of the p-adics. Understand how completeness (a topological idea from metric space
theory) plays into the theory of the p-adic numbers. Best for a student with a strong algebra background.
Reference: "p-adic Numbers: An Introduction" by Fernando Gouvea. Leah
- Topology and Market Economics: Understand how topology is the language of dyanamical
systems and study applications to economics. References: "Introduction to Topology: pure and applied",
Colin Adams and Robert Fransoza,
"Topology and Markets", American Mathematical Society and the Fields Institute for Mathematical Sciences, Toronto, Canada, 1999. Xiao
- I could probably come up with some more projects. Please ask me for more details.
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