Syllabus Application
MATH 409
Proofs from the Notebook
Instructor(s) Information
Kağan Kurşungöz
- Email: kursungoz@sabanciuniv.edu
Course Information
Catalog Course Description
Learning Outcomes:
| 1. | Upon completion of the course, student should be able to Perceive relations between different branches of mathematics, |
|---|---|
| 2. | Understand advanced creative ideas involved in some of the fundamental results, |
| 3. | Develop proofs using ideas similar to those introduced in the course, |
| 4. | Have a sense of the history of mathematics, |
| 5. | Learn some of the major open problems in mathematics. |
Course Objective
mathematical background as possible, to show how different fields of mathematics are
related to each other, to enhance students' analytical skills.
Although the topics seem to be diverse, the underlying theme will be "ingenious proofs".
Some material will be familiar from your previous courses, such as MATH 102, but the main
difference with MATH 102 will be that we will do proofs of all results we state. Knowledge of
MATH 301 (Introduction to mathematical analysis) is essential.
1 - Pythegorean Theorem and Pythagorean triples,
2 - Prime numbers (we will just skim through this one, it is largely done in number theory),
3 - Fermat's last theorem and Fermat's method of infinite descent,
4 - Elementary Inequalities, infinite products and some applications,
5 - Topology, Metric and Norm,
6 - Power Series,
7 - Stone-Weierstrass approximation theorem,
8 - Banach fixed point theorem,
9 - Selected topics as time allows: fundamental theorem of algebra, the Riemann Zeta function,
Brouwer fixed point theorem, geometry on the sphere.
Course Materials
Resources:
The lecture notes are the primary source of studying.
Auxiliary reading is:
Tosun Terzioğlu, "Bir Analizcinin Defterinden Seçtikleri"
(Nesin Yayınları, ikinci basım, 2014)
It is accessible as online resource through IC.
However, the auxiliary reading is in Turkish.
We will just follow the order of contents there.
Another source for auxiliary reading is:
Martin Aigner, Günter M. Ziegler, “Proofs from THE BOOK”
(Springer, 2nd ed., 2013)