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Information
| Code | MT0022 |
| Name | Selected topics in number theory |
| Term | 2022-2023 Academic Year |
| Semester | . Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Doktora Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator |
Course Goal / Objective
To see how different areas of mathematics play a role in the solution of the problems that comes from number theory.
Course Content
Selected topics in analytic and algebraic number theory: an overview to problems in multiplicative and additive number theory, additive patterns in integers and additive combinatorics, solving number theory problems with probabilistic methods, applications of algebraic methods in cryptography
Course Precondition
None.
Resources
Additive Combinatorics, T. Tao and V. Vu, 2006, Cambridge University Press. An Introduction to Sieve Methods and Their Applications, A. C. Cojocaru and M. R. Murty, 2005, Cambridge University Press.
Notes
The Probabilistic Method, N. Alon and J. Spencer, 3rd ed., 2008, John Wiley & Sons Inc. The Theory of Probability: Explorations and Applications, S. S. Venkatesh, 2013, Cambridge University Press.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | (S)he understands the main problem of the Ramsey theory. |
| LO02 | (S)he learns the applications of the probability theory in number theory. |
| LO03 | (S)he learns the basic techniques in the analytic number theory. |
| LO04 | (S)he learns the facts that can be deduced by the basic sieve theory. |
| LO05 | (S)he learns the applications of the algebraic number theory in computer science. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows the results of previous research in a special field of mathematics | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 4 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | 5 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics | 4 |
| PLO05 | Bilgi - Kuramsal, Olgusal | It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. | |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics | 5 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Sets up original problems in her field and offers different solution techniques | |
| PLO08 | Bilgi - Kuramsal, Olgusal | It carries out original and qualified scientific studies on the subject related to its field. | 4 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | 3 |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. | 2 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have foreign language knowledge at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | 4 |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | It presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Ramsey Theory | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Basic probability theory | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Introduction to the probabilistic method and Ramsey numbers | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | van der Waerden's theorem and van der Waerden numbers | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Expectation and the Markov inequality | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | Sum-free sets and Erdös' Theorem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | The concept of variance and Chebyshev's inequality | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Distinct sums and the Erdös-Moser Theorem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | The partial summation technique and Mertens' theorems | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | Turan's proof of the Hardy-Ramanujan theorem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | Eratosthenes' Sieve and Rankin's Trick | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Smooth numbers and the factoring problem | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | An introduction to the algebraic number theory | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | The Number Field Sieve | Study the relevant sections in the textbook and solve problems | Öğretim Yöntemleri: Anlatım, Tartışma |
| 16 | Term Exams | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Reviewed of the topics discussed in the lecture notes and source again | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 5 | 70 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 15 | 15 |
| Final Exam | 1 | 30 | 30 |
| Total Workload (Hour) | 157 | ||
| Total Workload / 25 (h) | 6,28 | ||
| ECTS | 6 ECTS | ||