Mathematics 4801/5609

Mathematics 4801/5609 Topics in Combinatorics

Winter 2019, Math 4801/5609
School of Mathematics and Statistics, Carleton University

Instructor: Daniel Panario
Office: #4372 HP, Tel: (613) 520 2600 (Ext. 2159)
Email: daniel@math.carleton.ca
Lectures: Tuesdays 17:35-20:25. Room: HP 4369
Office hours: Tuesdays 16:35-17:25, or by appointment (send me mail or talk with me).

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, 6th edition, Springer, 2018.
Classes begin: Tuesday, January 8, 2019.
Classes end: Tuesday, April 9, 2019.
Winter break: February 18-22, 2019.
The lecures of February 12 and of March 19 will be covered the week before, respectively.
Evaluation:
There will be three assignments (15% each), an oral presentation of a topic covered in the textbook (40%), and a project (15%).
The tentative schedule of assignments is below.

Assignment Hand-out Date Due Date Worth
1 January 15 February 5 15%
2 February 5 March 5 15%
3 March 5 March 26 15%

There is also an oral presentation (worth 40%) on any topic in the textbook not covered in class. This mark is formed by 25% for the understanding of the proof, and 15% for the oral presentation. Student presentations will be on the last week of classes (April 8-12, 2019).

Finally, in addition to the three assignments and oral presentation, there is an individual project worth 15% 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 proof. More information about the project will be given in class in late February. The lecturer and the student must agree on the topic of the project by the end of February, 2019. The deadline to hand-in the project is end of March, 2019.

There is no final exam.
Withdrawal: The last day for withdrawal from the course without penalties is the last day of classes.
Academic Accommodation:

Assignment	Hand-out Date	Due Date	Worth
1	January 15	February 5	15%
2	February 5	March 5	15%
3	March 5	March 26	15%

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.

Tentative lecture schedule - subject to change

WEEK

DATES

CHAPTER

TOPICS

1

Jan. 7-11

1
Introduction to the course. Six proofs of the infinitude of primes and proofs that the sum of the reciprocal of the primes diverges.

2

Jan. 14-18

2
Bertrand's postulate. [A1 out]

3

Jan. 21-25

6
Every finite division ring is a field.

4

Jan. 28 - Feb.1

8
Some irrational numbers.

5

Feb. 4-8

9
Riemann Zeta function and three times pi^2/6. [A1 in/A2 out]

6

Feb. 11-15

9
Three times pi^2/6 (cont).

~

Feb. 18-22

READING WEEK

7

Feb. 25 - Mar. 1

28
Pigeonhole principle and double counting.

8

Mar. 4-8

28
Double counting and extremal graph theory. [A2 in/A3 out]

9

Mar. 11-15

30
Three famous theorems in finite sets.

10

Mar. 18-22

31
Shuffling cards.

11

Mar. 25-29

31; 35
Shuffling cards (cont.); Kakeya problem. [A3 in] [Deadline to submit project]

12

Apr. 1-5

31; 35
Kakeya problem (cont.).

13

Apr. 8-12

Student presentations (lectures from textbook).

The above weekly outline is subject to change depending on the progress of the course. The chapters are from the textbook.

WEEK	DATES	CHAPTER	TOPICS
1	Jan. 7-11	1	Introduction to the course. Six proofs of the infinitude of primes and proofs that the sum of the reciprocal of the primes diverges.
2	Jan. 14-18	2	Bertrand's postulate. [A1 out]
3	Jan. 21-25	6	Every finite division ring is a field.
4	Jan. 28 - Feb.1	8	Some irrational numbers.
5	Feb. 4-8	9	Riemann Zeta function and three times pi^2/6. [A1 in/A2 out]
6	Feb. 11-15	9	Three times pi^2/6 (cont).
~	Feb. 18-22		READING WEEK
7	Feb. 25 - Mar. 1	28	Pigeonhole principle and double counting.
8	Mar. 4-8	28	Double counting and extremal graph theory. [A2 in/A3 out]
9	Mar. 11-15	30	Three famous theorems in finite sets.
10	Mar. 18-22	31	Shuffling cards.
11	Mar. 25-29	31; 35	Shuffling cards (cont.); Kakeya problem. [A3 in] [Deadline to submit project]
12	Apr. 1-5	31; 35	Kakeya problem (cont.).
13	Apr. 8-12		Student presentations (lectures from textbook).