Cantor Kümesi

Part of the 2018 Aratatil Yüksel Onaran Lise-Lisans-Lisansüstü Programı at the Nesin Mathematics Village, January 15–21, 2018.

Lecture notes, edited January 29, 2018: 58 pages, size A5, 12 point type.

Başlık:

Cantor Kümesi/The Cantor Set

Eğitmen:

Prof. David Pierce

Kurum:

MSGSÜ

Tarih:

15-21 Ocak 2018

Önkoşul:

Kalkülüs

Seviye:

İleri seviye lisans, lisansüstü

İçerik:

Doğal sayılar kümesi, sayılabilir bir kümedir, ama Cantor Teoremi`ne göre bu kümenin kuvvet kümesi sayılamazdır. Tihonov Topolojisi altında bu kuvvet kümesi, elemanları gerçel sayı olan Cantor Kümesi`ne homeomorftur. Özel olarak Cantor Kümesi tıkızdır. Bunlar ve ilgili teoremleri kanıtlayacağız, örneğin Heine-Borel Teoremi, mantık için bir tıkızlık teoremi, ve Stone Temsilcilik Teoremi.

Finite Fields

Part of the 2018 Aratatil Lise-Lisans-Lisansüstü Cebir Programı at the Nesin Mathematics Village, January 22–28, 2018.

Lecture notes, last edited March 30, 2018: 52 pages, size A5, 12 point type.

(I needed to turn page 35 in the pdf file with the command pdftk finite-fields-2018-revised-2.pdf cat 1-34 35north 36-end output finite-fields-2018-revised-2-turned.pdf)

Başlık:: Sonlu Cisimler/Finite Fields
Eğitmen:: Prof. David Pierce
Kurum:: MSGSÜ
Tarih:: 22-28 Ocak 2018
İçerik:: Toplama ve çarpma işlemleri altında tamsayılar, verilen pozitif bir modüle göre, değişmeli bir halka oluşturur. Bazı modüller için, halkada çarpmaya göre tersler vardır, dolayısıyla halka cisimdir; ayrıca bu cisim üzerinde sonlu-boyutlu vektör uzayları cisim olur. Bunu göstermek için indirgenemezlik, asallık, ve cebirsel kapanış kavramlarını ve Öklid Algoritması`nı geliştireceğiz.

Continued fractions

2018 TMD Lisans ve Lisansüstü Yazokulu

Lecture notes, edited August 22, 2018: 57 pages, size A5, 12 point type:

Title of the course:: Continued fractions
Instructor:: Prof. David Pierce
Institution:: MSGSÜ
Dates:: 13–19 August 2018
Prerequisites:: None. We shall study some topics that may be studied in a second-semester number-theory course; however, no specific results are required from a first-semester course.
Level:: Graduate, advanced undergraduate
Abstract:: We accept from childhood that multiplication of whole numbers is commutative; but Euclid gives a rigorous proof based on what we now call the Euclidean algorithm. This algorithm can be used to write any real number as a continued fraction. The continued fraction repeats when the real number is the solution of a quadratic equation. This case yields the solutions of a so-called Pell equation, x² − dy² = 1, an example of a Diophantine equation.
Textbook or/and course webpage:: [this page]
Language:: TR; EN

I expect to use my own notes from a Number Theory II course. I revised the notes extensively last winter, and they contain such diagrams as the one above.

Prime numbers

2018 TMD Lisans ve Lisansüstü Yazokulu

Lecture notes, edited August 29, 2018: 58 pages, size A5, 12 point type:

Title of the course:: Prime numbers
Instructor:: Prof. David Pierce
Institution:: MSGSÜ
Dates:: 20–26 August 2018
Prerequisites:: Some elementary number theory
Level:: Graduate, advanced undergraduate
Abstract:: We shall work through the Prime Number Theorem, that the probability that a given number N is prime is about 1⁄log(N).
Language:: TR, EN
Textbook or/and course webpage:: [this page]

Some basic ideas are illustrated by proving Bertrand’s Postulate, that for every counting number n, some prime p satisfies n < p ≤ 2n. A little more work yields Chebyshev’s Theorem, that the ratio of π(x) (the number of primes no greater than x) and x⁄log(x) stays within fixed bounds. The ratio tends to unity, by the Prime Number Theorem, which takes quite a bit more work. The work seems to be most conveniently done with some complex analysis; D. Zagier streamlines the proof of D. J. Newman in five pages (“Newman’s short proof of the prime number theorem,” Amer. Math. Monthly 104 [1997], no. 8, 705–708).