Mathematics 4801/5609 Topics in Combinatorics

General Information

• Course goals: This course is about beautiful proofs in Mathematics, with special emphasis on Combinatorics and Number Theory. We follow very closely the textbook "Proofs From The Book" to learn techniques used in many beautiful and elegant proofs. Each chapter of the textbook is self-contained. However, some of the topics require previous notions; we cover the required material during the course.
• Prerequisites: mathematical maturity is recommended. Although not required, at least one undergraduate abstract algebra course and one analysis course may be useful; undergraduate courses in discrete mathematics and number theory could be helpful.
• The material we plan to cover in each lecture is below.
• Textbook: Proofs from THE BOOK by Martin Aigner and Gunter Ziegler, 5th edition, Springer, 2014.
• Classes begin: Thursday, September 7, 2017.
  Classes end: Thursday, December 7, 2017.
  Fall break: October 23-27, 2017.
• Evaluation:
  There will be three assignments (15% each), an oral presentation of a topic covered in the textbook (35%), and a project (20%).
  The tentative schedule of assignments is below.

  Assignment	Hand-out Date	Due Date	Worth
  1	September 21	October 12	15%
  2	October 12	November 9	15%
  3	November 9	November 30	15%

  There is also an oral presentation (worth 35%) on any topic in the textbook not covered in class. This mark is formed by 20% for the understanding of the proof, and 15% for the oral presentation. Student presentations will be on the last week of classes (December 5-7, 2017).
  

  Finally, in addition to the three assignments and oral presentation, there is an individual project worth 20% of the final mark. Projects should have about 5 pages, and can not exceed 10 pages. They can be about any beautiful proof in Mathematics not covered in lectures or assignments. Students should explain the proof and then tell about what is beautiful or surprising about that prof. More information about the project will be given in class in October. The lecturer and the student must agree on the topic of the project by October 19, 2017. The deadline to hand-in the project is Thursday November 30, 2017.
  

  There is no final exam.

• Withdrawal: The last day for withdrawal from the course without academic penalties is the last day of classes.
  
• Academic Accommodation:
You may need special arrangements to meet your academic obligations during the term. For an accommodation request the processes are as follows:
• Pregnancy or religious obligation: Write me with any requests for academic accommodation during the first two weeks of class, or as soon as possible after the need for accommodation is known to exist. For more details, see the Student Guide.
• Students with disabilities requiring academic accommodations: Register with the Paul Menton Centre for Students with Disabilities (PMC) for a formal evaluation of disability-related needs. Documented disabilities include but are not limited to mobility/physical impairments, specific Learning Disabilities (LD), psychiatric/psychological disabilities, sensory disabilities, Attention Deficit Hyperactivity Disorder (ADHD), and chronic medical conditions. Registered PMC students are required to contact the PMC every term to have a Letter of Accommodation sent to the Instructor by their Coordinator. In addition, students are expected to confirm their need for accommodation with the Instructor no later than two weeks before the first assignment is due or the first in-class test/midterm. If you require accommodations only for formally scheduled exam(s) in this course, you must request accommodations by the official accommodation deadline published on the PMC website.

Tentative lecture schedule - subject to change

WEEK	DATES	CHAPTER	TOPICS
1	Sep. 6-8	1	Introduction to the course. Six proofs of the infinitude of primes.
2	Sep. 11-15	1	More on six proofs of the infinitude of primes and proofs that the sum of the reciprocal of the primes diverges.
3	Sep. 18-22	2	Bertrand's postulate. [A1 out]
4	Sep. 25-29	6	Every finite division ring is a field.
5	Oct. 2-6	8	Some irrational numbers.
6	Oct. 10-13	9	Riemann Zeta function and three times pi^2/6. [A1 in/A2 out]
7	Oct. 16-20	9	Three times pi^2/6 (cont).
~	Oct. 23-27		READING WEEK
8	Oct. 30 - November 3	27	Pigeonhole principle and double counting.
9	Nov. 6-10	27	Double counting and extremal graph theory. [A2 in/A3 out]
10	Nov. 13-17	29	Three famous theorems in finite sets.
11	Nov. 20-24	30	Shuffling cards.
12	Nov. 27 - December 1	30; 34	Shuffling cards (cont.); Kakeya problem. [A3 in] [Deadline to submit project]
13	Dec. 4-7		Student presentations (lectures from textbook).