Combinatorics

Spring 2005
Mathematics BC2006
Tues Thurs 2:40-3:55pm
903 Altschul Hall (Barnard Campus)
http://www.math.columbia.edu/~glass/combs05/

Darren Glass
Office: 516 Mathematics Building
Office Phone: (212) 854-5135
glass@math.columbia.edu

The best way to get in touch with me is by email. I am an email addict.

Announcements

Check this part of the website regularly for announcements.

Course Information

Content

Combinatorics is, put simply, the study of counting. We will learn how to count the number of ways to form various committees and we will learn how to count how many ways we can form a baseball schedule and we will count so many balls coming out of urns that we will all be sick of them. Combinatorics is a fun area of mathematics which has deep applications in biology, computer science, other areas of mathematics, and (gasp) real life.

The skills you will strengthen include: the ability to read and interpret what is being asked, the ability to verbalize questions, the willingness to consider many different ideas and theories, the courage to experiment with those ideas, the ability to endure frustration and failure, the wisdom and experience to understand when an approach is not succeeding, and the ability to recognize a solution.

As far as material goes, the prerequisites for this course are all things that come well before calculus. I will assume the type of mathematical sophistication that comes from courses such as Linear Algebra or Honors Calculus or the like. If you have any questions about your background and whether this course will be appropriate for you, please come talk to me.

Structure

This course is going to (probably) be unlike any other math course you have ever taken, as we will be using the method of "guided discovery". Classes will be devoted to a discussion of the problems and the broad themes in combinatorial mathematics the problems illustrate. Each class period will begin and end with short presentations by me and/or by you, but the majority of classtime will be spent working in small groups. These small groups will be made up of three to five students. It is not necessary that these groups remain the same throughout the semester, and you are encouraged to change groups throughtout the semester.

Textbook

We will be working through a soon-to-be-published text developed under a National Science Foundation grant by Kenneth Bogart, a professor at Dartmouth College. The text consists primarily of problems with a minimal amount of prose to guide us to the solutions. Our journey will not only be about discovering the main theorems of combinatorial mathematics and how they are used, but will also be about developing a deeper understanding of how to think mathematically. You can purchase the text as a course packet at Village Copier (on Broadway between 111th and 112th) or you can download it off of the web from Bogart's website. In any event, you MUST have a hard copy and you must bring it to class every day.

Office Hours and Other Help

While guided discovery is going to be fun, and will be a better way of learning the material than conventional lectures, it is also going to be hard work. I expect you to spend at least 8-10 hours per week outside of class, and I expect that you will want to spend even more because it is so much fun. Of course, all of this means that you may want help outside of class. Your first resource should be the other students in the class, but you should also take advantage of the Barnard and Columbia Helprooms, and my office hours.

My Office Hours: Tuesday: 1:30-2:30, Wednesday: 1-2.

Grading

40% - Homework
20% - Midterm
30% - Final Exam
10% - Attendance and Participation

Exams

There will be a midterm exam the day before spring break (March 10th), and there will be a final exam during finals period. Stay tuned for further details.

Homework

Homework will be assigned roughly once a week. You are encouraged to discuss homework with each other, but you must turn in your own work. If you aren't sure exactly where the boundary lies, please ask me rather than make assumptions. Late homework will not be accepted for any reason, but I will drop your two lowest homework grades. All homework will be graded not only on correctness, but also on how well you communicate and explain your answers. Mathematics is a process, not a final answer, and your work should reflect that.

For many of you, this will be the first course where you are writing proofs. Learning to do this well is a slow and difficult process. Some hints (written by Allison Pacelli) which you may find helpful are here. A more detailed introduction to writing for math courses can be found Annalisa Crannell's webpage

At various points during the semester, I may give you the option of resubmitting problems that you have already done to get more credit for that problem. In these instances, you will be responsible for handing in not only the correct new version but also the original submission. For this reason, I recommend that you keep all homeworks in a binder once they have been returned to you.

Homework assigments can be found at this site.
I would like to thank Ken Bogart and Erika L.C. King for useful discussions on designing this course, as well as some shameless plaigirizing of their syllabi.